FINANCIAL STOCHASTICS Notes for the Course by A.W. van der Vaart UPDATE, November 2005 Reading and Believing at Own Risk ii CONTENTS 1. Black-Scholes . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Change of Measure . . . . . . . . . . . . . . . . . . . . . 9 2.1. Exponential Processes . . . . . . . . . . . . . . . . . . 9 2.2. Lévy's Theorem . . . . . . . . . . . . . . . . . . . 11 2.3. Cameron-Martin-Girsanov Theorem . . . . . . . . . . . 12 3. Martingale Representation . . . . . . . . . . . . . . . . . 23 3.1. Representations . . . . . . . . . . . . . . . . . . . . 23 3.2. Stability . . . . . . . . . . . . . . . . . . . . . . . 27 3.3. Stochastic Differential Equations . . . . . . . . . . . . 29 3.4. Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . 33 3.5. Multivariate Stochastic Integrals . . . . . . . . . . . . 37 4. Finite Economies . . . . . . . . . . . . . . . . . . . . . 45 4.1. Strategies and Numeraires . . . . . . . . . . . . . . . 46 4.2. Arbitrage and Pricing . . . . . . . . . . . . . . . . . 51 4.3. Completeness . . . . . . . . . . . . . . . . . . . . . 54 4.4. Incompleteness . . . . . . . . . . . . . . . . . . . . 63 4.5. Utility-based Pricing . . . . . . . . . . . . . . . . . 69 4.6. Early Payments . . . . . . . . . . . . . . . . . . . . 71 4.7. Pricing Kernels . . . . . . . . . . . . . . . . . . . . 72 5. Extended Black-Scholes Models . . . . . . . . . . . . . . . 75 5.1. Arbitrage . . . . . . . . . . . . . . . . . . . . . . 76 5.2. Completeness . . . . . . . . . . . . . . . . . . . . . 79 5.3. Partial Differential Equations . . . . . . . . . . . . . . 81 6. American Options . . . . . . . . . . . . . . . . . . . . . 84 6.1. Replicating Strategies . . . . . . . . . . . . . . . . . 84 6.2. Optimal Stopping . . . . . . . . . . . . . . . . . . . 86 6.3. Pricing and Completeness . . . . . . . . . . . . . . . 87 6.4. Optimal Stopping in Discrete Time . . . . . . . . . . . 89 7. Payment Processes . . . . . . . . . . . . . . . . . . . . 92 8. Infinite Economies . . . . . . . . . . . . . . . . . . . . 97 9. Term Structures . . . . . . . . . . . . . . . . . . . . . 102 9.1. Short and Forward Rates . . . . . . . . . . . . . . . 103 9.2. Short Rate Models . . . . . . . . . . . . . . . . . . 108 9.3. Forward Rate Models . . . . . . . . . . . . . . . . . 113 10. Vanilla Interest Rate Contracts . . . . . . . . . . . . . . . 119 10.1. Deposits . . . . . . . . . . . . . . . . . . . . . . . 120 10.2. Forward Rate Agreements . . . . . . . . . . . . . . . 121 10.3. Swaps . . . . . . . . . . . . . . . . . . . . . . . . 123 10.4. Caps and Floors . . . . . . . . . . . . . . . . . . . 125 10.5. Vanilla Swaptions . . . . . . . . . . . . . . . . . . . 127 10.6. Digital Options . . . . . . . . . . . . . . . . . . . . 128 10.7. Forwards . . . . . . . . . . . . . . . . . . . . . . . 129 11. Futures . . . . . . . . . . . . . . . . . . . . . . . . . 131 iii 11.1. Discrete Time . . . . . . . . . . . . . . . . . . . . 132 11.2. Continuous Time . . . . . . . . . . . . . . . . . . . 133 12. Swap Rate Models . . . . . . . . . . . . . . . . . . . . 137 12.1. Linear Swap Rate Model . . . . . . . . . . . . . . . . 137 12.2. Exponential Swap Rate Model . . . . . . . . . . . . . 138 12.3. Calibration . . . . . . . . . . . . . . . . . . . . . . 139 12.4. Convexity Corrections . . . . . . . . . . . . . . . . . 139 iv LITERATURE These notes are based on interpretations and unifications of the following works, where sometimes the interpretations are loose. The core terminology for these notes is taken from Hunt and Kennedy. DISCLAIMER: These notes are meant to be of help, but contrary to appearance, are not a solid text, but always in progress. Unfortunately there is no reliable textbook covering the complete material. [1] Baxter, M. and Rennie, A., (1996). Financial calculus. Cambridge University Press, Cambridge. [2] Chung, K.L. and Williams, R.J., (1990). Introduction to stochastic integration, second edition. Birkhäuser, London. [3] Hunt, P.J. and Kennedy, J.E., (1998). Financial Engineering. Wiley. [4] Jacod, J. and Shiryaev, A.N., (1987). Limit theorems for stochastic processes. Springer-Verlag, Berlin. [5] Kramkov, D.O., (1996). Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probability Theory and Related Fields 105, 459­479. [6] Kopp, P.E. and Elliott, R.J., (1999). Mathematics and financial markets. Springer-Verlag, New York. [7] Karatzas, I. and Shreve, S.E., (1988). Brownian motion and stochastic calculus. Springer-Verlag, Berlin. [8] Karatzas, I. and Shreve, S.E., (1998). Methods of mathematical finance. Springer-Verlag, Berlin. [9] Musiela, M. and Rutkowski, M., (1997). Martingale Methods in Financial Modelling. Springer-Verlag, Berlin. [10] Revuz, D. and Yor, M., (1994). Continuous martingales and Brownian motion. Springer, New York. [11] Rogers, L.C.G. and Williams, D., (2000). Diffusions, Markov Processes and Martingales, volumes 1 and 2. Cambridge University Press, Cambridge. [12] Schoutens, W., (2003). Lévy Processes in Finance. Wiley. 1 Black-Scholes A financial derivative is a contract that is based on the price of an underlying asset, such as a stock price or bond price. An "option", of which there are many different types, is an important example. A main focus of "financial engineering" is on finding the "fair price" of such a derivative. Following the work by Black and Scholes in the 1970s the prices of derivatives are found through the principle of "no arbitrage". The concepts of "completeness" and "arbitrage-freeness" of an economy are central to make this work. In this chapter we discuss, as an introduction, the pricing of the European call option in the original model used by Black and Scholes. Even though this model is unrealistically simple, the ensuing "Black-Scholes formula" is still the most frequently used tool in practice. We denote by St the price of a stock at time t 0, and assume that S satisfies the stochastic differential equation (1.1) dSt = St dt + St dWt. Here W is a Brownian motion process on a given filtered probability space (, F, {Ft}, P), and and > 0 are given numbers. The filtration {Ft} is the completed natural filtration generated by W, and it is assumed that S is continuous and adapted to this filtration. This simple stochastic differential equation satisfies the conditions of the Itô theorem and hence this theorem guarantees the existence of a unique solution S. Using Itô's formula it is straightforward to verify directly that the solution takes the form St = S0e(- 1 2 2 ) t+ Wt . In particular, the stock price is strictly positive if S0 > 0, as we shall assume. The number is called the volatility of the stock. It determines 2 1: Black-Scholes the variability of the stock over time. The number gives the drift of the stock. It is responsible for the exponential growth of a typical stock price. We assume that, besides in the stock S, we can invest in a "riskless asset" with a predetermined yield, much as putting money in a savings account with a fixed interest rate. We assume that the price Rt of this investment at time t satisfies the differential equation dRt = rRt dt, R0 = 1. Here the number r is called the interest rate. Of course, the differential equation can be solved explicitly to give Rt = ert . This is the "continuously compounded interest" over the interval [0, t]. An amount of money V invested in the asset R at time 0 grows with certainty to an amount V ert at time t. Thus the interest rate gives the "time value of money": one unit of money that we are promised to receive at time t is worth e-rt units of money at time 0. Dividing an amount V by the number ert to get the amount V e-rt is called discounting. The interest rate r should be interpreted as being corrected for inflation or other effects, so that it gives the "true" economic rate of growth of a risk-free asset. A portfolio or strategy (, ) is defined to be a pair of predictable processes and . The pair (t, t) gives the numbers of bonds and stocks owned at time t, giving the portfolio value (1.2) Vt = tRt + tSt. The predictable processes and can depend on the past until "just before t" and we may think of changes in the content of the portfolio as a reallocation of bonds and stocks that take place just before the new stock price is known. A portfolio is "self-financing" if such reshuffling can be carried out without import or export of money, whence changes in the value of the portfolio are due only to changes in the values of the underlying assets. More precisely, we call the strategy (, ) self-financing if (1.3) dVt = t dRt + t dSt. This is to be interpreted in the sense that V must be a semimartingale satisfying V = V0 + R + S. It is implicitly required that and are suitable integrands relative to R and S. In finance the reshuffling of a portfolio to attain a certain aim is called "hedging". Originally hedging was understood to be a strategy to control or limit risk, but following several crashes "hedging" now also has a connotation of being "risky" and "greedy" in everyday language. A contingent claim with expiry time T > 0 is defined to be an FT measurable random variable. It is interpreted as the "pay-off" at the expiry 1: Black-Scholes 3 time of a "derivative", a contract based on the stock. In this section we consider contracts that can only be exercised, or "paid out", at expiry time. The European call option is an important example. It is a contract that gives the right, but not the obligation, to buy a stock at a predetermined price K at a predetermined time T in the future. A main question we like to answer is: what is the fair price of a European option at time 0? Because an option gives a right and not an obligation, it must cost some money to acquire one. It is easy to determine the value of the European option at time T. If the stock price ST is higher than the "strike price" K (the "option is in the money"), then we will exercise our right, and buy the stock for the price K. If we wished we could sell the stock immediately after acquiring it, and thus make a net gain of ST - K. On the other hand, if the stock price ST would be below the strike price K ("the option is out of the money"), then the option is worthless, and we would not use our right. We can summarize the two cases by saying that the value of the option at the time T is equal to (ST - K)+ , where x+ = x 0 is the positive part of a number. At time 0 this value is still unknown and hence cannot be used as the price of the option. A first, naive idea would be to set the price of an option equal to the expected value E(ST -K)+ . This does not take the time value of money into account, and hence a better idea would be to set the option price equal to the expected discounted value Ee-rT (ST - K)+ . This is still too naive, as Black and Scholes first argued in the beginning 1970s. It turns out that the correct price is an expectation, but the expectation must be computed under a different probability measure. Before giving Black and Scholes' argument (in a martingale form introduced by Harrison and Pliska in the 1980s), we note that the same type of reasoning applies to many other contracts as well. The main points are that the contract can pay out at expiry time only and that the value of the contract at expiry time T can be expressed as an FT -measurable random variable, called a contingent claim. Some examples of contingent claims are: (i) European call option: (ST - K)+ . (ii) European put option: (K - ST )+ . (iii) Asian call option: T 0 St dt - K + . (iv) lookback call option: ST - min0tT St, (v) down and out barrier option: (ST - K)+ 1{min0tT St L}. The constants K and L and the expiry time T are fixed in the contract. There are many more possibilities; the more complicated contracts are referred to as exotic options. Note that in (iii)­(v) the claim depends on the history of the stock price throughout the period [0, T]. All contingent claims can be priced following the same no-arbitrage approach that we outline be- low. Because the claims we wish to evaluate always have a finite term T, 4 1: Black-Scholes all the processes in our model matter only on the interval [0, T]. We may or must understand the assumptions and assertions accordingly. The process ~S defined by ~St = R-1 t St gives the discounted price of the stock. It turns out that claims can be priced by reference to a "martingale measure", defined as the (unique) measure that turns the process ~S into a martingale. By Itô's formula and (1.1), (1.4) d ~St = e-rt dSt + St(-re-rt ) dt = - r e-rt St dt + e-rt St dWt. Under the measure P governing the Black-Scholes stochastic differential equation (1.1) the process W is a Brownian motion by assumption, and hence ~S is a local martingale if and only if its drift component vanishes, i.e. if r. This will rarely be the case in the real world. On the other hand, under a different probability measure the discounted stock process ~S may well be a martingale, i.e. it may be a martingale if viewed as a process on the filtered space (, F, {Ft}, ~P), equipped with a different probability measure ~P. Because the drift term in (1.4) causes the problem, we could first rewrite the equation as (1.5) d ~St = e-rt St d ~Wt, for the process ~W defined by ~Wt = Wt - t, = r - . (The number is called the market price of risk. If it is zero, then the real world is already "risk-neutral"; if not, then the number measures the deviation from a risk-neutral market relative to the volatility process.) If we could find a probability measure ~P such that the process ~W is a ~Pmartingale, then the process ~S would be a ~P-local martingale. Girsanov's theorem, which we shall obtain in Chapter 2, permits us to do this: it implies that the process ~W is a Brownian motion under the measure ~P with density d~P/dP = exp( WT - 1 2 2 T) relative to P. Presently, we do need to know the nature of ~P; all that is important is the existence of a probability measure ~P such that ~W is a Brownian motion. We claim that the "reasonable price" at time 0 for a contingent claim with pay-off X at time T is the expectation under the measure ~P of the discounted value of the claim at time T, i.e. V0 = ~ER-1 T X, where ~E denotes the expectation under ~P. This is a consequence of economic, no-arbitrage reasoning, based on the following theorem. 1: Black-Scholes 5 1.6 Theorem. Let X be a nonnegative contingent claim with ~ER-1 T |X| < . Then there exists a self-financing strategy with value process V as in (1.2) such that: (i) V 0 up to indistinguishability. (ii) VT = X almost surely. (iii) V0 = ~ER-1 T X. Proof. The process ~S = R-1 S is a continuous semimartingale under P and a continuous local martingale under ~P, in view of (1.5). Let ~V be a cadlag version of the ~P-martingale ~Vt = ~E R-1 T X| Ft . Suppose that there exists a predictable process such that d ~Vt = t d ~St. Then ~V is continuous, because ~S is continuous, and hence predictable. Define = ~V - ~S. Then is predictable, because ~V , and ~S are predictable. The value of the portfolio (, ) is given by V = R + S = ( ~V - ~S)R + S = ~V R and hence, by Itô's formula and (1.4), dVt = ~Vt dRt + Rt d ~Vt = (t + t ~St) dRt + Rtt d ~St = (t + tR-1 t St) dRt + Rtt -StR-2 t dRt + R-1 t dSt = t dRt + t dSt. Thus the portfolio (, ) is self-financing. Statements (i)­(iii) of the theorem are clear from the definition of ~V and the relation V = R ~V . We must still prove the existence of the process . In view of (1.5) we need to determine this process such that d ~Vt = tSte-rt d ~Wt. Because the process Ste-rt is strictly positive, it suffices to determine a predictable process H such that d ~Vt = Ht d ~Wt. The process ~W is a ~P-Brownian motion and ~V is a ~P-martingale. Furthermore, the filtrations generated by ~W and W are identical. Therefore, the existence of an appropriate process H follows from the representing theorem for Brownian local martingales. According to this theorem, proved in Chapter 3, any local martingale relative to the filtration Ft, which by assumption is the filtration generated by ~W, can be represented as a stochastic integral relative to ~W. 6 1: Black-Scholes We interpret the preceding theorem economically as saying that V0 = ~ER-1 T X is the just price at time 0 for the contingent claim X. We argue this by exhibiting "winning" strategies if the price is higher or lower than V0. Suppose that the price P0 of the claim at time 0 would be higher than V0. Then rather than buying the option we could buy the portfolio (0, 0) as in the theorem. This would save us the amount P0 - V0. We could next reshuffle our portfolio during the time interval [0, T] according to the strategy (, ) and hence, by the theorem, end up with a portfolio (T , T ) with value VT exactly equal to VT = (ST - K)+ , the value of the option at expiry time. Thus we have made a certain profit and we would never buy the option. On the other hand, if the price P0 of the option were lower than V0, then anybody in the possession of a portfolio (0, 0), worth V0, might sell this at time 0, buy an option for P0 and put money V0 - P0 aside. During the term of the option the portfolio (-0, -0), where the minus indicates "sold", could be reshuffled according to the inverse strategy (-t, -t), yielding a capital at time T of -VT = -(ST - K)+ . Besides we also have an option worth (ST - K)+ , and the money V0 - P0 set aside at time 0. Again this leads to a certain profit and hence nobody would keep stocks and savings. For a general claim it may or may not be easy to evaluate the expectation ~ER-1 T X explicitly, as the claim X may depend on the full history of the stock process. For pricing claims that depend on the final value ST of the stock only, it is straightforward calculus to obtain a concrete formula. For example, consider the price of a European call option. First we write the stock price in terms of the ~P-Brownian motion ~W as St = S0e(r- 1 2 2 ) t+ ~Wt . In particular, under ~P, the variable log(St/S0) is normally distributed with mean (r - 1 2 2 )t and variance 2 t. The price of a European call option can be written as, with Z a standard normal variable, e-rT ~E(ST - K)+ = e-rT E S0e(r- 1 2 2 )T + T Z - K + . It is straightforward calculus to evaluate this explicitly, giving the classical Black-Scholes formula S0 log(S0/K) + (r + 1 2 2 )T T - Ke-rT log(S0/K) + (r - 1 2 2 )T T . Remarkably, the drift coefficient does not make part of this equation: it plays no role in the pricing formula. Apparently, the systematic part of the stock price diffusion can be completely hedged away. 1: Black-Scholes 7 That ~ER-1 T X is the fair price of the claim X is not a mathematical theorem, but the outcome of economic reasoning. One may or may not find this reasoning completely convincing. A hole in the pricing argument concerns the uniqueness of the strategy (, ) in the theorem. If there were another self-financing strategy ( , ) with the same value VT at expiry time, but a different value V0 at time 0, then clearly the argument would not be tenable, as it would lead to two "fair" prices. If there existed two strategies with the same outcome at time T, but different values at time 0, then the difference strategy defined as (- , - ) would start with a nonzero value V0 -V0, but end with certainty with a zero value VT - VT . Thus this strategy, or its negative, would make money with certainty. This is referred to as an arbitrage. If an economy allows arbitrage, then the preceding economic argument does not make sense. We shall see later that the Black-Scholes economy is "arbitrage-free", provided that this concept is defined appropriately. The following example shows that this issue must be treated with care. It exhibits, within the Black-Scholes model, a self-financing strategy that can be initiated with value 0 at time 0, and leads with certainty to an arbitrarily large positive value at time T. The construction is comparable to the well-known "doubling scheme" in a fair betting game that pays out twice the stake, or nothing, both with probability 1/2. The doubling scheme consists of playing until we win the bet for the first time, doubling the stake at each time we loose. If we win after n + 1 bets, then our total gain is -1 - 2 - 4 - - 2n-1 + 2n = 1 > 0. Because it is certain that we win eventually, we gain with certainty. The reason that the doubling scheme does not work in practice, and it is the same with the investment strategy in the following example, is that we may need to play arbitrarily many times and our expected loss before we finally win is infinite. Thus we need an infinite capital to be certain that we can bring our strategy to an end as planned. Finance theory gets around this problem by excluding strategies as in the following example from consideration. Among the set of remaining "admissible strategies", suitably defined, no strategy allows for arbitrage. 1.7 Example (Arbitrage). Consider the special case of the Black-Scholes model, where = r = 0, = 1, and S0 = 1. We claim that for every constant > 0 there exists a self-financing strategy (, ) with value process V satisfying V0 = 0 and VT . In other words, the economy permits arbitrage of arbitrarily large size if we allow all self-financing trading strategies. We construct the strategy by stopping another strategy ( , ) as soon as the latter strategy's value process reaches the level . Define (t, t) = -1 T - t + t 0 1 Ss T - s dSs, 1 St T - t . In view of the fact that Rt = 1 by definition, for every t, the value process 8 1: Black-Scholes of this strategy is given by Vt = tRt + tSt = t 0 1 Ss T - s dSs. The equation shows that V0 = 0. Furthermore, the strategy ( , ) is selffinancing, as t dRt + t dSt = t dSt = dVt . For U equal to the stopping time U = inf{t > 0: Vt = }, define (, ) = ( , )1[0,U] + (, 0)1(U,T ]. This corresponds to waiting until the value of the portfolio under strategy ( , ) reaches the level and next investing all the money (i.e. the value ) in the risk-free asset R. It is intuitively clear that the new strategy is self-financing, as it is self-financing before U, at U, and clearly also after U. This can also be proved rigorously. (Cf. Exercise 4.3.) The value process of the strategy (, ) is equal to on the time interval [U, T] (as the interest rate is zero), and hence has value at time T whenever U < T. This is the case with probability one, as we shall now show. The process Yt = VT (1-e-t) can be shown to be a Brownian motion, and Vt = Y- log(1-t/T ). If t increases from 0 to T, then - log(1 - t/T) increases from 0 to . The properties of Brownian motion yield that lim supt Yt = almost surely. Therefore, the value process Vt reaches any level on the interval [0, T) with certainty. 1.8 EXERCISE. Verify that the process Y in the preceding example is a Brownian motion process. [Hint: compute its quadratic variation process and use Lévy's theorem, Theorem 2.6.] 2 Change of Measure Girsanov's theorem concerns the martingale property under a change of the probability measure on the underlying filtered space. Because densities often come as "exponential processes", we first recall the definition of such processes. 2.1 Exponential Processes The exponential process corresponding to a continuous semimartingale X is the process E(X) defined by E(X)t = eXt- 1 2 [X]t . The name "exponential process" would perhaps suggest the process eX rather than the process E(X) as defined here. The additional term 1 2 [X] in the exponent of E(X) is motivated by the extra term in the Itô formula. An application of this formula to the right side of the preceding display yields (2.1) dE(X)t = E(X)t dXt. (Cf. the proof of the following theorem.) If we consider the differential equation df(x) = f(x) dx as the true definition of the exponential function f(x) = ex , then E(X) is the "true" exponential process of X, not eX . Besides that, the exponentiation as defined here has the nice property of turning local martingales into local martingales. 2.2 Theorem. The exponential process E(X) of a continuous local martingale X with X0 = 0 is a local martingale. Furthermore, (i) If Ee 1 2 [X]t < for every t 0, then E(X) is a martingale. 10 2: Change of Measure (ii) If X is an L2-martingale and E t 0 E(X)2 s d[X]s < for every t 0, then E(X) is an L2-martingale. Proof. By Itô's formula applied to the function f(Xt, [X]t) = E(X)t, we find that dE(X)t = E(X)t dXt + 1 2 E(X)t d[X]t + E(X)t (-1 2 ) d[X]t. This simplifies to (2.1) and hence E(X) = 1+E(X)X is a stochastic integral relative to X. If X is a local martingale, then so is E(X). Furthermore, if X is an L2-martingale and 1[0,t]E(X)2 dX < for every t 0, then E(X) is an L2-martingale. This condition reduces to the condition in (ii). The proof of (i) should be skipped at first reading. If 0 Tn is a localizing sequence for E(X), then Fatou's lemma gives E E(X)t| Fs lim inf n E(E(X)tTn | Fs = lim inf n E(X)sTn = E(X)s. Therefore, the process E(X) is a supermartingale. It is a martingale if and only if its mean is constant, where the constant must be EE(X)0 = 1. By the representation theorem for Brownian martingales we may assume that the local martingale X takes the form Xt = B[X]t for a process B that is a Brownian motion relative to a certain filtration. For every fixed t the random variable [X]t is a stopping time relative to this filtration. We conclude that it suffices to prove: if B is a Brownian motion and T a stopping time with E exp(1 2 T) < , then E exp(BT - 1 2 T) = 1. Because 2Bs is normally distributed with mean zero and variance 4s, E t 0 E(B)2 s ds = t 0 Ee2Bs e-s ds = t 0 es ds < By (ii) it follows that E(B) is an L2-martingale. For given a < 0 define Sa = inf{t 0: Bt - t = a}. Then Sa is a stopping time, so that E(B)Sa is a martingale, whence EE(B)Sat = 1 for every t. It can be shown that Sa is finite almost surely and EE(B)Sa = EeBSa - 1 2 Sa = 1. (The distribution of Sa is known in closed form. See e.g. Rogers and Williams I.9, p18-19; because BSa = Sa + a, the right side is the expectation of exp(a + 1 2 Sa).) With the help of an integration lemma we conclude that E(B)Sat E(B)Sa in L1 as t , and hence E(B)Sa is uniformly integrable. By the optional stopping theorem, for any stopping time T, 1 = EE(B)Sa T = E1T 0 up to ~P-evanescence; and also up to P-evanescence if ~P and P are locally equivalent. (iv) There exists a stopping time T such that L > 0 on [0, T) and L = 0 on [T, ) up to P-evanescence. 14 2: Change of Measure Proof. (i). For every n N the optional stopping theorem applied to the uniformly integrable martingale Ln yields LT n = E(Ln| FT ), P-almost surely. For a given F FT the set F {T n} is contained in both FT and Fn. We conclude that ELT 1F 1T n = ELT n1F 1T n = ELn1F 1T n = ~E1F 1T n. Finally, we let n . (ii). Because T = lim Tn defines a stopping time, assertion (i) yields that ~P(T < ) = ELT 1T <. If P(T = ) = 1, then the right side is 0 and hence ~P(T = ) = 1. (iii). For n N define a stopping time by Tn = inf{t > 0: Lt < n-1 }. By right continuity LTn n-1 on the event Tn < . Consequently property (i) gives ~P(Tn < ) = ELTn 1Tn< n-1 . We conclude that ~P(inft Lt = 0) n-1 for every n, and hence inft Lt > 0 almost surely under ~P. Furthermore, Tn almost surely under ~P, and then by (ii) also under P if ~P and P are locally equivalent. This is equivalent to the sample paths of L being bounded away from 0 on compacta up to P-evanescence. (iv). The stopping times Tn defined in the proof of (iii) are strictly increasing and hence possess a limit T. By definition of Tn we have Lt n-1 on [0, Tn), whence Lt > 0 on [0, T). For any m the optional stopping theorem gives E(LT m| FTnm) = LTnm n-1 on the event Tn m. Because {Tn m} FTnm, we can conclude that ELT m1T m ELT m1Tnm n-1 for every m and n, and hence LT = 0 on the event T < . For any stopping time S T another application of the optional stopping theorem gives E(LSm| FT m) = LT m = 0 on the event T m. We conclude that ELSm1T m = 0 for every n and hence LS = 0 on the event S < . This is true in particular for S = inf{t > T: Lt > }, for any > 0, and hence L = 0 on (T, ). 2.11 EXERCISE. If L is the density process of ~P relative to P and V0 is a strictly positive F0-measurable random variable such that EPV0L0 = 1, then V0L is also a density process. Of which measure? 2.12 EXERCISE. If ~P and P are locally equivalent measures on a filtered space (, F, {Ft}) with density process L and Ft Ft is a sub-filtration, then there exists a cadlag version of the process t EP(Lt| Ft) and this is the density process of the restriction of ~P to F relative to the restriction of P to F. Show this. 2.13 EXERCISE. If P, Q and R are locally equivalent measures on a filtered space (, F, {Ft}), then R has density process KL with respect to P, for L the density process of R with respect to Q and and K the density process of Q relative to P. If M is a local martingale on the filtered space (, F, {Ft}, P), then it typically looses the local martingale property if we use another measure ~P. 2.3: Cameron-Martin-Girsanov Theorem 15 The Cameron-Martin-Girsanov theorem shows that M is still a semimartingale under ~P, and gives an explicit decomposition of M in its martingale and bounded variation parts. We start with a general lemma on the martingale property under a "change of measure". We refer to a process that is a local martingale under P as a P-local martingale. For simplicity we restrict ourselves to the case that ~P and P are locally equivalent, i.e. the restrictions ~Pt and Pt are locally absolutely continuous for every t. 2.14 Lemma. Let ~P and P be locally equivalent probability measures on (, F, {Ft}) and let L be the corresponding density process. Then a stochastic process M is a ~P-local martingale if and only if the process LM is a P-local martingale. Proof. We first prove the theorem without "local". If M is an adapted ~P-integrable process, then, for every s < t and F Fs, ~EMt1F = ELtMt1F , ~EMs1F = ELsMs1F , The two left sides are identical for every F Fs and s < t if and only if M is a ~P-martingale. Similarly, the two right sides are identical if and only if LM is a P-martingale. We conclude that M is a ~P-martingale if and only if LM is a P-martingale. If M is a ~P-local martingale and 0 Tn is a localizing sequence, then the preceding shows that the process LMTn is a P-martingale, for every n. Then so is the stopped process (LMTn )Tn = (LM)Tn . Because Tn is also a localizing sequence under P, we can conclude that LM is a P-local martingale. Because ~P and P are locally equivalent, we can select a version of L that is strictly positive, by Lemma 2.10(iii). Then dPt/d~Pt = L-1 t and we can use the argument of the preceding paragraph in the other direction to see that M = L-1 (LM) is a ~P-local martingale if LM is a P-local martingale. Warning. A sequence of stopping times is defined to be a "localizing sequence" if it is increasing everywhere and has almost sure limit . The latter "almost sure" depends on the underlying probability measure. Thus a localizing sequence for a measure P need not be localizing for a measure ~P. In view of Lemma 2.10(ii) this problem does not arise if the measures ~P and P are locally equivalent. In the preceding lemma the "local martingale part" can be false if ~P is locally absolutely continuous relative to P, but not the other way around. If M itself is a P-local martingale, then generally the process LM will not be a P-local martingale, and hence the process M will not be a ~P-local martingale. We can correct for this by subtracting an appropriate process. 16 2: Change of Measure We restrict ourselves to continuous local martingales M. Then a P-local martingale becomes a ~P local martingale plus a "drift" (L-1 - )[L, M], which is of locally bounded variation. 2.15 Theorem (Girsanov). Let ~P and P be locally equivalent probability measures on (, F, {Ft}) and let L be the density process of ~P relative to P. If M is a continuous P-local martingale, then M - L-1 - [L, M] is a ~P-local martingale. Proof. By Lemma 2.10(ii) the process L- is strictly positive under both ~P and P, whence the process L-1 - is well defined. Because it is left-continuous, it is locally bounded, so that the integral L-1 - [L, M] is well defined. We claim that the two processes LM - [L, M] L(L-1 - [L, M]) - [L, M] are both P-local martingales. Then, taking the difference, we see that the process L(M - L-1 - [L, M]) is a P-local martingale and hence the theorem is a consequence of Lemma 2.14. That the first process in the display is a P-local martingale is an immediate consequence of properties of the quadratic variation. For the second we apply the integration-by-parts (or Itô's) formula to see that d L(L-1 - [L, M]) = (L-1 - [L, M]) dL + L- d(L-1 - [L, M]). No "correction term" appears at the end of the display, because the quadratic covariation between the process L and the continuous process of locally bounded variation L-1 - [L, M] is zero. The integral of the first term on the right is a stochastic integral (of L-1 - [L, M]) relative to the P-martingale L and hence is a P-local martingale. The integral of the second term is [L, M]. It follows that the process L(L-1 - [L, M]) - [L, M] is a local martingale. * 2.16 EXERCISE. In the preceding theorem suppose that M is not necessarily continuous. Show that: (i) If L is continuous, then the theorem is true as stated. (ii) If the predictable quadratic covariation L, M is well defined, then the process M - L-1 - L, M is a ~P-local martingale, even if L and M are cadlag, but discontinuous. For cadlag local L2-martingales L and M the predictable quadratic covariation process L, M is defined as the unique predictable process of locally bounded variation such that LM - L, M is a local martingale, and can be shown to be equal to [L, M] - LM. More generally we can consider L, M well defined if there exists a predictable process of locally bounded variation such that [L, M] - L, M is a local martingale. This is certainly 2.3: Cameron-Martin-Girsanov Theorem 17 the case if the process [L, M] is locally integrable, so that it has a compensator by the Doob-Meyer decomposition. [Hint: (ii) can be proved following the proof of the preceding theorem, but substituting L, M for [L, M]. In the second part of the proof we do obtain a term d L, L-1 - L, M . This may be nonzero, but can be written as L-1 - L, M dL (Cf. Jacod and Shiryaev, I.4.49) and contributes another local martingale part. * 2.17 EXERCISE. Any semimartingale X can be written as X = X0 + M + A for a local martingale with bounded jumps M and a process of locally bounded variation A. It can be shown that [L, M] is locally integrable for any local martingale L and local martingale with bounded jumps M. (Localize by the minimum of Tn = inf{t > 0: t 0 |d[L, M]| > n} and a stopping time making the jumps of L integrable. See Jacod and Shriyaev, III.3.14.) Deduce from this and the (ii) of the preceding problem that any P-semimartingale is a ~P-semimartingale for any equivalent probability measure ~P. The quadratic covariation process [L, M] in the preceding theorem was meant to be the quadratic covariation process under the orginal measure P. Because ~P and P are assumed locally equivalent and a quadratic covariation process can be defined as a limit of inner products of increments, it is actually also the quadratic variation under ~P. Because L-1 - [L, M] is continuous and of locally bounded variation, the process M -L-1 - [L, M] possesses the same quadratic variation process [M] as M, where again it does not matter if we use P or ~P as the reference measure. Thus even after correcting the "drift" due to a change of measure, the quadratic variation remains the same. The latter remark is particularly interesting if M is a P-Brownian motion process. Then both M and M - L-1 - [L, M] possess quadratic variation process the identity. Because M -L-1 - [L, M] is a continuous local martingale under ~P, it is a Brownian motion under ~P by Lévy's theorem. This proves the following corollary. 2.18 Corollary. Let ~P and P be locally equivalent probability measures on (, F, {Ft}) and let L be the corresponding density process. If B is a P-Brownian motion, then B - L-1 - [L, B] is a ~P-Brownian motion. Many density processes L arise as exponential processes. In fact, given a strictly positive, continuous martingale L, the process X = L-1 - L is well defined and satisfies L- dX = dL. The exponential process is the unique solution to this equation, whence L = L0E(X). Girsanov's theorem takes a particularly simple form if formulated in terms of the process X. 2.19 Corollary. Let ~P and P be locally equivalent probability measures on (, F, {Ft}) and let the corresponding density process L take the form 18 2: Change of Measure L = L0E(X) for a continuous local martingale X, 0 at 0, and a strictly positive F0-measurable random variable L0. If M is a continuous P-local martingale, then M - [X, M] is a ~P-local martingale. Proof. The exponential process L = L0E(X) satisfies dL = L- dX, or equivalently, L = L0 + L- X. Hence L-1 - [L, M] = L-1 - [L- X, M] = [X, M]. The corollary follows from Theorem 2.15. A special case arises if L = E(Y B) for Y a predictable process and B a Brownian motion. Then (2.20) d~Pt dPt = E(Y B)t = e t 0 Ys dBs- 1 2 t 0 Y 2 s ds a.s.. By the preceding corollaries (with X = Y B and M = B) the process t Bt - t 0 Ys ds is a Brownian motion under ~P. This is the original form of Girsanov's theorem. The following exercise asks to derive the vector-valued form of this theorem. 2.21 EXERCISE (Vector-valued Girsanov). Let W = (W(1) , . . . , W(d) ) be a d-dimensional P-Brownian motion process and let Q be a probability measure that is absolutely continuous relative to P with density process of the form L = E( d j=1 (j) W(j) ) relative to P, for some predictable process ((1) , . . . , (d) ). Show that the process ~W with coordinates ~W(i) = W(i) - 0 (i) s ds is a d-dimensional Q-Brownian motion. It is a fair question why we would be interested in "changes of measure" of the form (2.20). We shall see some reasons when discussing stochastic differential equations or option pricing in later chapters. For now we can note that in the situation that the filtration is the completion of the filtration generated by a Brownian motion any change to an equivalent measure is of the form (2.20). * 2.22 Lemma. Let {Ft} be the completion of the natural filtration of a Brownian motion process B defined on (, F, P). If ~P is a probability measure on (, F) that is equivalent to P, then there exists a predictable process Y with t 0 Y 2 s ds < almost surely for every t 0 such that the restrictions ~Pt and Pt of ~P and P to Ft satisfy (2.20). Proof. The density process L is a P-martingale relative to the filtration {Ft}. Because this is a Brownian filtration, Theorem 3.2 and Example 3.3 imply that L permits a continuous version. Because L is positive, the process 2.3: Cameron-Martin-Girsanov Theorem 19 L-1 is well defined, predictable and locally bounded. Hence the stochastic integral Z = L-1 L is a well-defined local martingale, relative to the Brownian filtration {Ft}. By Example 3.3 it can be represented as Z = Y B for a predictable process Y as in the statement of the lemma. The definition Z = L-1 L implies dL = L dZ. Because F0 is trivial, the density at zero can be taken equal to L0 = 1. This pair of equations is solved uniquely by L = E(Z). (Cf. Exercise 2.3.) In many applications of Girsanov's theorem the process Y is actually given first, and the purpose is to "remove a drift" of the form t 0 Ys ds from a given Brownian motion B. Then we would like to construct the new measure ~P starting from P, B, and Y . From the preceding we see that this is achieved by constructing ~P so as to have density process L = E(Y B) relative to P. This requires the exponential process E(Y B) to be at least a martingale. By Novikov's theorem a sufficient condition for this is that, for every t > 0, Ee 1 2 t 0 Y 2 s ds < . If E(Y B) is a uniformly integrable martingale, then we can define d~P = E(Y B) dP, and we conclude that the process Bt - t 0 Ys ds is a Brownian motion. Under just Novikov's condition the exponential process E(Y B) is not necessarily uniformly integrable, but it is a martingale and hence uniformly integrable if restricted to a finite interval [0, T]. Consequently, the stopped exponential process E(Y B)T = E (Y 1[0,T ]) B is uniformly integrable. Then the corresponding density process is given by (2.20) with Y 1[0,T ] replacing Y . We can conclude that the process {Bt T t 0 Ys ds: t 0} is a Brownian motion under the measure ~P. In particular, the process Bt - t 0 Ys ds is a Brownian motion on the restricted time interval [0, T] relative to the measure ~PT with density E(Y B)T relative to P. If Novikov's condition is satisfied for every t > 0, then we can obtain this conclusion for every T > 0. The measures ~PT depend on T, but are clearly related. We conclude this section by a discussion of conditions under which they can be put together to a single measure. * 2.3.1 Local to Global If a probability measure ~P is locally absolutely continuous relative to a probability measure P, then the corresponding density process is a nonnegative P-martingale with mean 1. We may ask if, conversely, every nonnegative martingale L with mean 1 on a given filtered probability space (, F, {Ft}, P) arises as the density process of a measure ~P relative to P. In the introduction of this section we have seen that the answer to this question is positive if the martingale is uniformly integrable, but the answer is negative in general. 20 2: Change of Measure Given a martingale L and a measure P we can define for each t 0 a measure ~Pt on the -field Ft by d~Pt dPt = Lt. If the martingale is nonnegative with mean value 1, then this defines a probability measure for every t. The martingale property ensures that the collection of measures ~Pt is consistent in the sense that ~Ps is the restriction of ~Pt to Fs, for every s < t. The remaining question is whether we can find a measure ~P on F for which ~Pt is its restriction to Ft. Such a "projective limit" of the system (~Pt, Ft) does not necessarily exist under just the condition that the process L is a martingale. A sufficient condition is that the filtration be generated by some appropriate process. Then we can essentially use Kolmogorov's consistency theorem to construct ~P. 2.23 Theorem. Let L be a nonnegative martingale with mean 1 on the filtered space (, F, {Ft}, P). If Ft is the filtration (Zs: s t) generated by some stochastic process Z on (, F) with values in a Polish space, then there exists a probability measure ~P on F whose restriction to Ft possesses density Lt relative to P for every t > 0. Proof. Define a probability measure ~Pt on F by its density Lt relative to P, as before. For 0 t1 < t2 < < tk let Rt1,...,tk be the distribution of the vector (Zt1 , . . . , Ztk ) on the Borel -field Dk of the space Dk if (, F) is equipped with ~Ptk . This system of distributions is consistent in the sense of Kolmogorov and hence there exists a probability measure R on the space (D[0,) , D[0,) ) whose marginal distributions are equal to the measures Rt1,...,tk . For a measurable set B D[0,) now define ~P Z-1 (B) = R(B). If this is well defined, then it is not difficult to verify that ~P is a probability measure on F = Z-1 (D[0,) ) with the desired properties. The definition of ~P is well posed if Z-1 (B) = Z-1 (B ) for a pair of sets B, B D[0,) implies that R(B) = R(B ). Actually, it suffices to show that this is true for every pair of sets B, B in the union A of all cylinder -fields in D[0,) (the collection of all measurable sets depending on only finitely many coordinates). Then ~P is well defined and -additive on tFt = Z-1 (A), which is an algeba, and hence possesses a unique extension to the -field F, by Carathéodory's theorem. The algebra A consists of all sets B of the form B = z D[0,) : (zt1 , . . . , ztk ) Bk for a Borel set Bk in Rk . If Z-1 (B) = Z-1 (B ) for sets B, B A, then there exist k, coordinates t1, . . . , tk, and Borel sets Bk, Bk such that {(Zt1 , . . . , Ztk ) Bk} = {(Zt1 , . . . , Ztk ) Bk} and hence Rt1,...,tk (Bk) = Rt1,...,tk (Bk), by the definition of Rt1,...,tk . 2.3: Cameron-Martin-Girsanov Theorem 21 The condition of the preceding lemma that the filtration be the natural filtration generated by a process Z does not permit that the filtration is complete under P. In fact, completion may cause problems, because, in general, the measure ~P will not be absolutely continuous relative to P. This is illustrated in the following simple problem. * 2.24 Example (Brownian motion with linear drift). Let B be a Brownian motion on the filtered space (, F, {Ft}, P), which is assumed to satisfy the usual conditions. For a given constant > 0 consider the process L defined by Lt = eBt- 1 2 2 t . The process L can be seen to be a P-martingale, either by direct calculation or by Novikov's condition, and it is nonnegative with mean 1. Therefore, for every t 0 we can define a probability measure ~Pt on Ft by d~Pt = Lt dP. Because by assumption the Brownian motion B is adapted to the given filtration, the natural filtration Fo t generated by B is contained in the filtration Ft. The measures ~Pt are also defined on the filtration Fo t . By the preceding lemma there exists a probability measure ~P on (, Fo ) whose restriction to Fo t is ~Pt, for every t. We shall now show that: (i) There is no probability measure ~P on (, F) whose restriction to Ft is equal to ~Pt. (ii) The process Bt - t is a Brownian motion on (, Fo , {Fo t }, ~P) (and hence also on the completion of this filtered space). It was argued previously using Girsanov's theorem that the process {Bt -t: 0 t T} is a Brownian motion on the "truncated" filtered space (, FT , {Ft FT }, ~PT ), for every T > 0. Because the process is adapted to the smaller filtration Fo t , it is also a Brownian motion on the space (, Fo T , {Fo t Fo T }, ~PT ). This being true for every T > 0 implies (ii). If there were a probability measure ~P on (, F) as in (i), then the process Bt - t would be a Brownian motion on the filtered space (, F, {Ft}, ~P), by Girsanov's theorem. We shall show that this leads to a contradiction. For n R define the event F = : lim t Bt() t = . Then F Fo and F F = for = . Furthermore, by the ergodic theorem for Brownian motion, P(F0) = 1 and hence P(F) = 0. Because Bt -t is a Brownian motion under ~P, also ~P(F) = 1 and hence ~P(F0) = 0. Every subset F of F possesses P(F) = 0 and hence is contained in F0, by the (assumed) completeness of the filtration {Ft}. If Bt - t would be a Brownian motion on (, F, {Ft}, ~P), then Bt - t would be independent (relative to ~P) of F0. In particular, Bt would be independent of the event {Bt C} F for every Borel set C. Because ~P(F) = 1, the variable Bt 22 2: Change of Measure would also be independent of the event {Bt C}. This is only possible if Bt is degenerate, which contradicts the fact that Bt -t possesses a normal distribution with positive variance. We conclude that ~P does not exist on F. The problem (i) in this example is caused by the fact that the projective limit of the measures ~Pt, which exists on the smaller -field Fo , is orthogonal to the measure P. In such a situation completion of a filtration under one of the two measures effectively adds all events that are nontrivial under the other measure to the filtration at time zero. This is clearly undesirable if we wish to study a process under both probability measures. 3 Martingale Representation The pricing theory for financial derivatives is based on "replicating strategies". At an abstract level such strategies are representers in "martingale representation theorems". In this chapter we discuss a general, abstract theorem and some examples. 3.1 Representations We shall say that a local martingale M on a given filtered probability space (, F, {Ft}, P) has the representing property if every cadlag local martingale N on this filtered space can be written as a stochastic integral N = N0 + H M relative to M, for some predictable process H. Thus the infinitesimal increment dNt of an arbitrary cadlag local martingale N is a multiple Ht dMt of the corresponding increment of M. Intuitively, the predictability of H means that the quantity Ht is known "just before t", and hence "all the randomness in dNt is contained in dMt". A constructive interpretation of the representation N = N0+HM is to view a sample path of N as evolving through extending it at every time t by a multiple Ht dMt of the increment dMt, where the multiplication factor Ht can be considered a constant and the increment dMt is a random variable "generated" at time t. Obviously, the representing property is a very special property. Its validity depends on both the local martingale M and the filtration. The classical example is given by the pair of a Brownian motion and its augmented natural filtration. In this chapter we deduce this result from a characterization of the representing property through the uniqueness of a "martingale measure" and Girsanov's theorem. This characterization also implies the representing property of the continuous martingale part of a solution to a 24 3: Martingale Representation stochastic differential equation, if this is weakly unique. 3.1 Definition. A (local) martingale measure corresponding to a given (local) martingale M on a given filtered probability space (, F, {Ft}, P) is a probability measure Q on (, F) such that (i) Q P. (ii) Q0 = P0. (iii) M is a Q-(local) martingale. By assumption the original measure P is a martingale measure. The following theorem characterizes the representing property of M through the uniqueness of P as a martingale measure. Warning. Other authors say that a local martingale N possesses the representation property relative to M if N can be represented as a stochastic integral N = N0 + H M. We shall not use this phrase, but note that M possesses the "representing property" if all local martingales N possess the "representation property" relative to M. Warning. Other authors (e.g. Jacod and Shiryaev) define the representing property of a process M with jumps through representation of each local martingale N as N = N0 + G Mc + H (M - M ), where Mc is the continuous, local martingale part of M, M -M is its "compensated jump measure", and (G, H) are suitable predictable processes. This type of representation permits more flexibility in using the jumps than representations of the form N = N0 + H M. Lévy processes do for instance possess the representing property in this extended sense, whereas most Lévy processes with jumps do not possess the representing property as considered in this chapter. Within the financial context representation through a stochastic integral H M is more appropriate, as this corresponds to the gain process of a trading strategy. 3.2 Theorem. A continuous (local) martingale M on a given filtered probability space (, F, {Ft}, P) possesses the representing property if and only if P is the unique (local) martingale measure on F corresponding to M and (, F, {Ft}, P). The proof of this theorem is given in a series of steps in Section 3.4. The theorem applies both to martingales and local martingales M. If M is a martingale, then a "martingale measure" may be understood to be a measure satisfying (i)­(ii) of the preceding definition under which M is a martingale, not a local martingale. Warning. Some authors do not include property (i) in the definition of a martingale measure. In that case the correct characterization is that P is an extreme point in the set of all martingale measures. If (i) is included in the definition of a martingale measure (as is done here), then the latter description is also correct, but the "extremeness" of P is a trivial consequence of the fact that it is the only martingale measure. 3.1: Representations 25 That the representing property can be characterized by uniqueness of a martingale measure is surprising, but can be explained informally from Girsanov's theorem and the fact that a density process is a local martingale. If M is a continuous P-local martingale and Q an equivalent probability measure with density process L relative to P, then the process M - L-1 - [L, M] is a Q-local martingale, by Girsanov's theorem. Hence M is a Qlocal martingale, and Q a martingale measure, if only if [L, M] = 0. This expresses some sort of "orthogonality" of L and M. We shall see that the process M possesses the representing property if and only there exists no nontrivial local martingale L that is orthogonal to M in this sense. Because a density process is a local martingale, this translates into uniqueness of P as a martingale measure. The theorem is true for vector-valued local martingales M, provided that the stochastic integrals HM are interpreted appropriately. The correct interpretation is not entirely trivial. For a one-dimensional, continuous local martingale the integral H M is defined for any predictable process H such that t 0 H2 s d[M]s < almost surely for every t. Consequently, if H = (H(1) , . . . , H(d) ) and M = (M(1) , . . . , M(d) ) are vectors of predictable processes and continuous local martingales such that (H (i) s )2 d[M(i) ]s < almost surely for every i, then every of the stochastic integrals H(i) M(i) is well defined and we define H M = n i=1 H(i) M(i) . If the (d × d)-matrix [M] of quadratic variations [M(i) , M(j) ] is "uniformly nonsingular", then this set of predictable integrands is appropriate and the representing theorem is true as stated. For instance, this is the case for M equal to multivariate Brownian motion, when [M]t is t times the identity matrix. However, in general the set of processes H M obtained in this way is not large enough to make the representing theorem true. In Section 3.5 we define the stochastic integral HM for a larger class of predictable processes H, essentially through a "closure operation". With this extended definition Theorem 3.2 is correct for multivariate local martingales, as stated. The classical example of a martingale with the representing property is Brownian motion. There are nice direct proofs of this fact, but in the following example we deduce it from the preceding theorem. Under the latter condition the processes H and M can be localized by the stopping times Tn = inf{t > 0: |Mt| > n, t 0 H 2 s d[M]s > n}. The process H1[0,Tn] is in the space L2([0, ) × , P, MTn ) for every n (where P is the predictable -field and M the Doléans measure of an L2-martingale M), and hence H1[0,Tn] MTn is well defined; its almost sure limit as n exists and is the stochastic integral H M. 26 3: Martingale Representation 3.3 Example (Brownian motion). Let M be Brownian motion on the filtered probability space (, F, {Ft}, P), with Ft equal to the augmented natural filtration generated by M. In this setting Brownian motion possesses the representing property. To prove this it suffices to establish uniqueness of P as a martingale measure. If Q is another martingale measure, then the quadratic variation of M under Q is the same as under P, because by definition Q and P are equivalent. Because M is a P-Brownian motion, this quadratic variation is the identity function. Hence, by Lévy's theorem, M is also a Q-Brownian motion. We conclude that the induced law of M on R[0,) is the same under both P and Q, or, equivalently, the measures P and Q are the same on the -field Fo : = (Mt: t 0). Then they also agree on the completion of this -field, which is F, by assumption. The preceding theorem extends to local martingales with jumps, at least to locally bounded ones and for representing nonnegative local mar- tingales. 3.4 Theorem. If P is a unique local martingale measure for the locally bounded, cadlag, local martingale M on the filtered probability space (, F, {Ft}, P), then every nonnegative local martingale N on (, F, {Ft}, P) can be written as N = N0 + H M for some predictable process H. The stochastic integral H M must be interpreted appropriately, and we must allow a "maximal" set of possible integrands H to make the theorem true. Vectors of locally bounded predictable processes are of course valid integrands, but the process H in the theorem is only restricted to be "M-integrable". The "maximal" extension of the domain of the stochastic integral for processes M with jumps is even more technical than for continuous processes. See Section 3.5 for a discussion. The preceding theorem can be derived from the following theorem, which extends the representation to supermartingales. Because a nonnegative stochastic integral relative to a locally martingale is automatically a local martingale, a "true" nonnegative supermartingale N cannot be represented as N = N0 + H M. However, the following theorem shows that there is a process H such that the difference C = N - N0 - H M has nondecreasing sample paths. Thus the decreasing nature (in the mean) of the supermartingale N is captured by the (pointwise) increasing process C. The theorem also replaces the condition that the martingale measure is unique by the assumption that the process N is a supermartingale under every local martingale measure. 3.5 Theorem. Let M be a locally bounded, cadlag, local martingale on the filtered probability space (, F, {Ft}, P). If the nonnegative, cadlag process 3.2: Stability 27 N is a Q-supermartingale for every local martingale measure Q for M, then there exist a predictable process H and an adapted, nondecreasing process C such that N = N0 + H M - C. Proof. For a proof of Theorem 3.5 see D.O. Kramkov, Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets, Probability Theory and Related Fields 105, 1996, 459­ 479. The definition of a martingale measure in this paper does not include (ii) of Definition 3.1. However, if N is a (super)martingale relative to every martingale measure satisfying (i)+(ii)+(iii), then N is automatically a (super)martingale relative to every measure Q that satisfies only (i)+(iii). Indeed, given Q satisfying (i)+(iii) the measure ~Q with density dP0/dQ0 relative to Q satisfies (i)+(ii), and also (iii) because LM is a Q-local martingale for L = dP0/dQ0 (constant in time) the density process of ~Q relative to Q. Because ~Q is a martingale measure in the sense of Definition 3.1, the process N is a ~Q-supermartingale by assumption, which implies that it is a Q-supermartingale. Theorem 3.4 can be derived from Theorem 3.5 by first noting that a nonnegative local martingale is a supermartingale. Therefore, according to Theorem 3.5 the local martingale N of Theorem 3.4 can be written N = N0 + H M - C for a nondecreasing process C. The process C is the difference of two local martingales and hence a local martingale itself. (We use that the stochastic integral H M is a local martingale, which is not automatic with the extended definition of stochastic integral, but true in this case as the process is nonnegative.) If it is localized by stopping times Tn, then CTn is a martingale and hence ECTn t = EC0 = 0. By nonnegativity CTn t = 0 almost surely, whence C = 0. * 3.6 EXERCISE. Suppose that the process N in Theorem 3.5 is a nonnegative, cadlag process which is a Q-supermartingale for every martingale measure Q for M. Investigate whether the assertion of the theorem is still true. [Can the class of martingale measures be smaller than the class of local martingale measures?] 3.2 Stability In this section we discuss two situations in which the representing property of a given local martingale is inherited by another local martingale constructed from it. The first situation concerns a change of measure. If M is a continuous P-local martingale and ~P an equivalent probability measure with density process L relative to P, then the process ~M = M - L-1 - [L, M] is 28 3: Martingale Representation a ~P-martingale, by Girsanov's theorem. Because M and ~M differ only by a process of bounded variation, and this process is chosen to retain the martingale property, we should expect that ~M possesses the representing property for ~P-local martingales if M possesses the representing property for P-local martingales. The following lemma shows that this expectation is justified. 3.7 Lemma. Let M be a continuous P-local martingale on the filtered probability space (, F, {Ft}, P) and let ~P be a probability measure that is equivalent to P with density process L. Then M possesses the representing property on (, F, {Ft}, P) if and only if the continuous ~P-local martingale ~M = M-L-1 - [L, M] possesses the representing property on (, F, {Ft}, ~P). Proof. Suppose that ~M possesses the representing property on the filtered space (, F, {Ft}, ~P), and let N be a cadlag P-local martingale. The process 1/L is the density process of P relative to ~P, and hence is a ~P-martingale. Thus it can be represented as a stochastic integral relative to ~M and hence is continuous. The process ~N = N - L-1 - [L, N] is a ~P-local martingale, by Girsanov's theorem, and hence there exists a predictable process such that ~N = ~N0 + ~M. This implies that N - N0 - M = L-1 - [L, N] - L-1 - [L, M]. The left side is a cadlag P-local martingale, 0 at 0, and the right side is a continuous process of bounded variation. It follows that both sides are identically zero, whence N = N0 + M. The converse is proved similarly. Next suppose that N = M for a predictable process and a local martingale M that possesses the representing property on the filtered probability space (, F, {Ft}, P). If is never zero, then we can write H M = (H/) N and hence every stochastic integral relative to M can be expressed as a stochastic integral relative to N. Thus N inherits the representing property from M in this case. 3.8 EXERCISE. Verify that H/ is a good integrand for N = M whenever H is a good integrand for M, where "good" in the second case means that t 0 H2 s d[M]s < almost surely, for every t > 0. This remains true if the local martingale M is vector-valued and is a matrix-valued predictable process. For simplicity suppose that the local martingale M is d-dimensional and that takes its values in the set of d × d-matrices, so that N is d-dimensional as well. As intuition suggests, the correct condition is that t is invertible for every t. To make this true we need the extended definition of a stochastic integral relative to a multivariate local martingale, discussed in Section 3.5. It is implicitly assumed in the following lemma that each of the stochastic integrals (i,) M, with (i,) the ith row of the matrix , is well defined in the sense discussed in Section 3.5. 3.3: Stochastic Differential Equations 29 3.9 Lemma. Suppose that the continuous, local martingale M possesses the representing property on the filtered probability space (, F, {Ft}, P). If N = M for a predictable process taking its values in the invertible matrices, then N also possesses the representing property on the filtered probability space (, F, {Ft}, P). Proof. Let [M]t = t 0 Cs d|[M]|s be a representation of [M] as used in Section 3.5. Then by assumption, for every i, t 0 k l (i,k) s (i,l) s C(k,l) s d|[M]|s < , a.s.. This is exactly the ith diagonal element of the analogous representation of [N], which is given by [N]t = t 0 sCsT s d|[M]|s. It follows readily that the process (T )-1 H is a good integrand for N if and only if the process H is a good integrand for M, i.e. t 0 (T s )-1 Hs 2 d[N]s < , iff t 0 H2 s d[M]s < , a.s.. Furthermore, it can be verified that H ( M) = (T H) M, so that H M = (T )-1 H N, whenever these integrals are well defined. 3.3 Stochastic Differential Equations Consider a stochastic differential equation of the form (3.10) dXt = (t, Xt) dt + (t, Xt) dWt. Here and are given vector-valued and matrix-valued, measurable functions and W is a vector-valued Brownian motion process on a given filtered space (, F, {Ft}, P). By definition a solution of this equation is an adapted vector-valued process X with continuous sample paths such that (3.11) Xt = X0 + t 0 (s, Xs) ds + t 0 (s, Xs) dWs. Here it is implicitly required that the two integrals on the right, a Lebesgue integral and a stochastic integral, respectively, are well defined. This is the case if, almost surely, for every t > 0, t 0 (s, Xs) ds < , t 0 (s, Xs) 2 ds < . 30 3: Martingale Representation Ostensibly, under these conditions, the two integrals in (3.11) give the decomposition of X into its bounded variation part and its local martingale part. In this section we show that the local martingale part possesses the representing property for all FX t -adapted martingales if the solution to the stochastic differential equation is "weakly unique". As is customary we can discern "strong" and "weak solutions" of the stochastic differential equation (3.10). For a strong solution the filtered space (, F, {Ft}, P) is given together with a pair (W, ) of a Brownian motion W and an F0-measurable random variable defined on it, and the solution X is a continuous, adapted process defined on the same filtered space satisfying (3.11) and X0 = . Often the precise definition also includes further measurability requirements on X, such as adaptation to the augmented filtration generated by (W, ). For a weak solution only the functions and are given a-priori, and the solution consists of a filtered space (, F, {Ft}, P) and a Brownian motion W and a process X defined on it satisfying (3.11) and with X0 having a predescribed law. The conceptually simplest sufficient conditions for weak uniqueness are the "Itô Lipschitz conditions", which also guarantee existence of a strong solution. Alternatively, there are sufficient conditions in terms of the generator of the diffusion equation. (See e.g. Karatzas and Shreve, Theorem 4.28.) To the notion of a weak solution corresponds a notion of "weak uniqueness", which is also referred to as "uniqueness-in-law". The "law" of a solution X may be understood to be the set of distributions of all vectors (Xt1 , . . . , Xtk ) for t1, . . . , tk varying over Rk , and its law at time 0 is the law of X0. The stochastic differential equation (3.11) allows a weakly unique solution if any two solutions X and ~X with the same law at time 0, possibly defined relatively to different Brownian motions on different filtered probability spaces, possess the same laws. Equivalently, the solution is weakly unique if for any two solutions X and ~X such that X0 and ~X0 are equal in distribution, the distributions of the vectors (Xt1 , . . . , Xtk ) and ( ~Xt1 , . . . , ~Xtk ) on Rk are equal for every k and every set of time points 0 t1 tk. (We may think of the law of a solution X as the probability distribution induced by the map X: C[0, )n on the Borel sets of the metric space C[0, )n equipped with the topology of uniform convergence on compacta. Then "uniqueness-in-law" means that every weak solution induces the same law on the "canonical" space C[0, )n .) By definition a solution X defined on the filtered space (, F, {Ft}, P) is adapted to the given filtration Ft, but it is of course also adapted to its natural filtration (FX t )o . Let FX t be the completion of this filtration, and assume that it is right-continuous. The process At = t 0 (s, Xs) ds is adapted to FX t by Fubini's theorem, and hence so is the process Xc t = t 0 (s, Xs) dWs, because it is the difference of X and A. Because Xc is a continuous Ft-local martingale, it is also an FX t -local martingale (Exercise 3.12) and hence Xc is also the continuous martingale part of X 3.3: Stochastic Differential Equations 31 relative to the filtration FX t . We pose the martingale representing problem for Xc relative to the filtered space (, F, {FX t }, P). 3.12 EXERCISE. If X is a continuous local martingale on the filtered space (, F, {Ft}, P) that is adapted to the smaller filtration Gt Ft, then X is also a Gt-local martingale. [Hint: verify this first without "local". Next show that a localizing sequence can be chosen Gt-adapted.] ** 3.13 EXERCISE. Is this still true without "continuous"? 3.14 Theorem. If the solution to the stochastic differential equation (3.10) defined by and is weakly unique, then for any solution X on a given filtered space (, F, {Ft}, P) equipped with a given Brownian motion W the local martingale part Xc possesses the representing property relative to the filtered space (, F, {FX t }, P). Proof. It suffices to show that P is a unique martingale measure for Xc on (, FX ). We give two proofs, a short indirect one and a longer direct proof. The first proof proceeds by showing that for any martingale measure Q the process X is a solution to the "martingale problem" on (, F, {FX t }, Q), i.e. for any twice continuously differentiable function f with compact support the process f(Xt) - f(X0) - t 0 f (Xs)(s, Xs) + 1 2 f (Xs)T (s, Xs) ds is a Q-martingale relative to FX t . If this is the case, then there exists an extension of the filtered space (, F, {FX t }, Q) with a Brownian motion ~W defined on it that together with the extension ~X of X provides a weak solution to the stochastic differential equation (3.11) (see e.g. Karatzas and Shreve, Proposition 4.6) and the law of ~X coincides with the law of X under Q (Karatzas and Shreve, Corolllary 4.8). For a martingale measure Q for Xc on (, F, {FX t }) the measures Q and P are the same on FX 0 by assumption and hence the initial law of X0 is identical under P and Q. By the assumed weak uniqueness the laws of the process X under the measures P and Q must agree. Equivalently, the measures P and Q agree on the field (Xt: t 0) = (FX )o , and hence also on the completion FX , because P and Q are assumed equivalent. To prove that X solves the martingale problem on (, F, {FX t }, Q), we use Itô's formula to see that df(Xt) = f (Xt) dXt + 1 2 f (Xt) d[X]t = f (Xt)(t, Xt) dWt + f (Xt)(t, Xt) + 1 2 f (Xt)T (t, Xt) dt . The assumption that Q is a martingale measure entails that Xc t = t 0 (s, Xs) dWs is a Q-local martingale relative to FX t , and hence so is t 0 f (Xs)(s, Xs) dWs = t 0 f (Xs) dXc s . 32 3: Martingale Representation Rather than referring to the general results on the martingale problem we can also give a direct proof. First consider the case that X is onedimensional. Then is a real-valued function and we can define a process Wt = t 0 1(s,Xs)>0 ds = t 0 1 (s, Xs) 1(s,Xs)>0 dXc s . The second representation shows that W is adapted to FX t . Furthermore, if Q is a martingale measure, then Xc is a Q-local martingale relative to FX t and hence so is W. Suppose that we can define a Brownian motion W on the filtered space (, F, {FX t }, Q) that is independent of W, and set ~Wt = Wt + t 0 1(s,Xs)=0 d Ws. Then ~W is a continuous local martingale with quadratic variation process equal to the identity (check!) and hence ~W is a Brownian motion, by Lévy's theorem. Because (t, Xt) dWt and (t, Xt) d ~Wt are identical, the stochastic differential equation (3.11) holds with W replaced by ~W and hence the process X together with the Brownian motion ~W provide a weak solution to (3.11). By the assumed weak uniqueness we then obtain that P = Q on FX , as before. It may not be possible to construct an independent Brownian motion W on the original filtered space (, F, {FX t }, Q), and then the preceding construction is not possible. However, we may always replace this filtered space by (the completion of) a space of the form ( × , F × F, {FX t } × Ft, Q× Q), with a Brownian motion W defined on (, F, { Ft}). We can view (X, W) and W as processes defined on this product space (depending only on the first and second coordinates of (, ) respectively), and then have a weak solution in this extended setting. The proof can then be completed as before. In the multi-dimensional case we follow a similar argument, but the process ~W must be constructed with more care. The Brownian motion W is d-dimensional and the process takes its values in the (n × d)-matrices. There exist matrix-valued, continuous, FX t -adapted process O, , and U such that, for every t, (cf. Karatzas and Shreve ???) (i) t: = (t, Xt) = UtOt tOT t . (ii) Ot is a (d × d)-orthogonal matrix. (iii) t is a (d×d) diagonal (matrix with the eigenvalues of the matrix tT t on the diagonal). (iv) Ut is an (n × d) matrix with UT t Ut = I on the range of the matrix tT t . Given a d-dimensional Brownian motion W, defined on an extension of the filtered space (, F, {FX t }, Q) and independent of (X, W), define ~Wt = t 0 Os1s>0OT s dWs + t 0 Os1s=0OT s d Ws. 3.4: Proof of Theorem 3.2 33 Here 1t>0 is the diagonal matrix with 1 or 0 on the diagonal if the corresponding diagonal element of t is positive or 0, and 1t=0 = I - 1t>0. Then (t, Xt) d ~Wt = UtOt tOT t Ot (1t>0OT t dWt + 1t=0OT t d Wt) = (t, Xt) dWt. It follows that X satisfies the stochastic differential equation (3.11) with W replaced by ~W. By a similar calculation, with -1/2 t 1t>0 the diagonal matrix with diagonal entries 0 or -1/2 if the corresponding entropy of t is , -1/2 t 1t>0OT t UT t dXc t = -1/2 t 1t>0OT t UT t t dWt = 1t>0OT t dWt. It follows that the process t 0 Os1s>0OT s dWs, and hence also the process ~W, is adapted to FX t . Because the quadratic variation can be computed to be [ ~W]t = tI, it follows by Lévy's theorem that ~W is a Q-Brownian motion. Thus X together with ~W and the (possibly) extended filtered probability space is a weak solution to the stochastic differential equation (3.11). We can finish the proof as before. 3.15 EXERCISE. Compare the assertion of Theorem 3.14 with the assertion that can be obtained from Lemma 3.9. Do both approaches have advantages? * 3.4 Proof of Theorem 3.2 Given a filtered probability space (, F, {Ft}, P), let M2 be the set of all cadlag L2-bounded martingales on this space, and let M2 0 be the subset of such martingales that are 0 at 0. Every L2-bounded martingale M possesses an almost sure limit M as t , which is also an L2-limit. The space M2 can be alternatively described as the set of all martingales M with sup t EM2 t < , EM2 < , E sup t M2 t < . By the maximal inequalities for cadlag martingales the three finiteness conditions in this display are equivalent. The set M2 is a Hilbert space relative to the inner product (M, N) = EMN. Let M be a given continuous, local martingale on the filtered space (, F, {Ft}, P). The Doléans measure of M is defined by dM (t, ) = d[M]t() dP(). How much of the special structure of (3.11) is actually needed? 34 3: Martingale Representation Let L2(M) denote the set of predictable stochastic processes H: [0, ) × R such that H2 dM < . By the isometry defining the stochastic integral, H2 dM = E(H M)2 . Thus the set L2(M) consists exactly of all predictable processes H such that the stochastic integral H M is an L2-bounded martingale, and the map H H M from L2(M) to M2 0 is an isometry. The local martingale M can "represent" all L2-bounded cadlag martingales if this isometry is onto. In that case, for every cadlag L2-bounded martingale N there exists a process H in L2(M) such that N -N0 = H M. Because the map H H M is an isometry, it possesses a closed range space, given by I2 (M): = N M2 0: N = H M, for some H L2(M) . By the projection theorem for Hilbert spaces, the Hilbert space M2 0 can be decomposed as M2 0 = I2 (M) I2 (M) , and I2 (M) is equal to M2 0 if and only if its orthocomplement I2 (M) is zero. In other words, the local martingale M can represent all cadlag L2-bounded martingales if and only if every cadlag L2-bounded martingale N with N0 = 0 and (N, M) = 0 is identically zero. The following lemmas translate this orthogonality into a martingale property. 3.16 Lemma. The spaces I2 (M) and I2 (M) are closed under stopping: if T is a stopping time and N I2 (M), then NT I2 (M), and similarly for I2 (M) . Proof. For the space I2 (M) the assertion is immediate from the representation NT = (H1[0,T ]) M if N = H M. For the space I2 (M) we deduce the assertion from the equalities (NT , H M) = E(NT )(H M) = ENT (H M) = ENT (H M)T = EN(H M)T = ENT (H M)T = 0, because we already noted that (H M)T is contained in I2 (M). 3.17 Lemma. For every cadlag L2-bounded martingale N we have N I2 (M) if and only if NM is a local martingale. In this case, if N is bounded and M is a martingale, then NM is also a martingale. Proof. Suppose that N I2 (M). Then NT I2 (M) by Lemma 3.16, for every stopping time T. There exists a localizing sequence Tn such that MTn is a bounded martingale for every fixed n. Because MTn = 1[0,Tn] M, the 3.4: Proof of Theorem 3.2 35 process MTn is contained in I2 (M), and hence also MTnT , by Lemma 3.16, for any stopping time T. We conclude that ENT (MTn )T = E(NT )(MTnT ) = (NT , MTnT ) = 0. Because |NT (MTn )T | supt |Nt| supt |MTn t | is contained in L2, as N is L2-bounded and M is bounded, these expectations indeed exist and the variables NT (MTn )T are integrable. Because this is true for every stopping time T, the process NMTn is a uniformly integrable martingale. Consequently, the process (NM)Tn = (NMTn )Tn is a martingale, whence the process NM is a local martingale. Conversely, if NM is a local martingale, then [N, M] = 0, by the uniqueness of the Doob-Meyer decomposition. Consequently, for every predictable process H such that H M is well defined [N, H M] = H [N, M] = 0, which implies that N(H M) is a local martingale. Because N(H M) (N2 + (H M)2 )/2, it is dominated, and hence also a uniformly integrable martingale. By the martingale property (N, H M) = EN(H M) = EN0(H M)0 = 0. This concludes the proof of the first assertion. For the second assertion it suffices to show that the stopped process (NM)n is a (uniformly integrable) martingale for every n N. We have already shown that NM is a local martingale. If N is bounded by the constant C, then |(NM)n | C|Mn |. Because Mn is a uniformly integrable martingale, it is of class D, whence the process (NM)n is also of class D, and hence (NM)n is a martingale. (See the following exercise.) 3.18 EXERCISE. A process M is said to be of class D if the collection of random variables {MT : T finite stopping time} is uniformly integrable. Show that: (i) If M is a local martingale of class D, then M is a uniformly integrable martingale. (ii) A uniformly integrable martingale is of class D. (iii) If M is of class D and |N| |M|, then N is of class D. 3.19 Lemma. If P is a unique martingale measure for M, then I2 (M) = M2 0. Proof. If N M2 0 is bounded in absolute value by 1/2, then we can define a measure Q through dQ = (1+N) dP. By the martingale property EPN = EPN0 = 0, and hence Q is a probability measure. The density process of Q relative to P is 1 + N, which is positive and equal to 1 at zero. Hence Q and P are equivalent and Q0 = P0. If N I2 (M), then NM is a P-(local) martingale by Lemma 3.17, and hence so is the process (1 + N)M = M + NM. We conclude that M is a Q-(local) martingale, and hence Q is a (local) martingale measure. By the uniqueness of P, it follows that Q = P, or, equivalently, 1 + N = 1. Thus N = 0. 36 3: Martingale Representation A minor extension of this argument shows that there exists no nonzero bounded, cadlag martingale N with N0 = 0 that is orthogonal to I2 (M). It suffices to multiply N by a suitably small constant and apply the preceding argument. For a given nontrivial martingale N M2 with uniformly bounded jumps and T = inf{t 0: |Nt| > 1}, the stopped process NT is a bounded martingale, which is nontrivial, as |NT | 1. If N is orthogonal to I2 (M), then so is NT , by Lemma 3.16. Thus no nontrivial local martingale with uniformly bounded jumps is orthogonal to I2 (M). In particular, no continuous martingale is orthogonal to I2 (M). The set M2,c 0 of all L2-bounded, continuous martingales, 0 at 0, is a closed subspace of M2 0, and contains I2 (M). It follows that M2,c 0 = I2 (M). The elements of the orthocomplement of M2,c 0 in M2 0 are by definition the L2-bounded, "purely discontinuous martingales", 0 at 0. Every such martingale N can be orthogonally decomposed as a series n(Nn - An) of "compensated jumps", where each Nn can be taken of the form Nn = NTn 1[Tn,) for Tn a stopping time, |NTn | bounded, and the compensator An a continuous process. Consequently, if there exists a nontrivial purely discontinuous martingale in M2 0, then there also exists such a process with bounded jumps. As in the present case there exists no nontrivial martingale with bounded jumps orthogonal to I2 (M), it follows that the orthocomplement of M2,c 0 in M2 0 is 0, and hence I2 (M) = M2 0. Proof of Theorem 3.2. Suppose first that P is a unique (local) martingale measure. Then, by the preceding lemma, every cadlag L2-bounded martingale on (, F, {Ft}, P) is continuous. We first extend this to cadlag local martingales. By localization it suffices to show that every cadlag uniformly integrable martingale is continuous. Given a uniformly integrable cadlag martingale N, let Nn be the random variable N truncated to the interval [-n, n], and let Nn be the cadlag version of the process E(Nn | Ft). Because this is a bounded martingale, it is contained in M2 , and hence it is continuous, by Lemma 3.19. By dominated convergence Nn N in L1 and hence supt |Nn t - Nt| 0 in probability by the L1-maximal inequality for submartingales. It follows that N is continuous as well. For a given continuous, local martingale N there exists a localizing sequence 0 Tn such that NTn is a bounded martingale, for every n. By the preceding lemma there exists Hn L2(M) such that NTn = N0 + Hn M, for every n. Because NTm and NTn agree on [0, Tm] for m n, so do the stochastic integrals Hm M and Hn M. This implies that E (Hn -Hm)2 1[0,Tm] d[M] = 0, or equivalently that Hn = Hm almost surely on [0, Tm] under the Doléans measure M . This shows that H defined by H = Hn on (Tn-1, Tn] coincides up to an M -null set with Hn on (0, Tn], whence H M = Hn M = N - N0 on [0, Tn]. Conversely, suppose that M has the representing property, and let Q be a martingale measure. The density process L of Q relative to P is a 3.5: Multivariate Stochastic Integrals 37 P-martingale and hence can be written as as a stochastic integral L = L0 + H M. Because M is a Q-(local) martingale, the process LM is a P-(local) martingale by Lemma 3.17, and hence so is the process (L L0)M = LM - L0M. This implies that 0 = [L - L0, M] = H [M] almost surely. Consequently, the process H is zero almost surely under the Doléans measure of M, and hence L = L0 + H M = L0. The variable L0 is the density of Q0 relative to P0 and hence is 1 almost surely by the assumption that Q0 = P0. Thus L = 1 and Q = P on F. * 3.5 Multivariate Stochastic Integrals In this section we give the proper definition of the stochastic integral H X for vector-valued predictable processes H = (H(1) , . . . , H(d) ) and a vectorvalued semimartingale X = (X(1) , . . . , X(d) ). If each coordinate process H(i) is locally bounded, then the stochastic integral H X is just the sum H X: = d i=1 H(i) X(i) of the integrals of the coordinates. For the purpose of the representation theorem this definition must be extended to a larger class of integrands H. This involves both dropping the local boundedness and taking care of interactions between the coordinate integrals. We present the extension first for continuous local martingales X and next for general semimartingales. 3.5.1 Continuous Local Martingales In this section we define the stochastic integral HM of a vector-valued predictable process H and a vector-valued local martingale M. We first define the integral for suitably integrable processes H through an L2-isometry, and next extend by localization. Recall that the "extension" is null if M is a multivariate Brownian motion, so that for most purposes this section is not needed. Suppose M = (M(1) , . . . , M(d) ) is a vector-valued, continuous martingale defined on the filtered probability space (, F, {Ft}, P). For each coordinate M(i) define L2(M(i) ) as the set of predictable processes H(i) such that E 0 (H (i) s )2 d[M]s < (as in Section 3.4 except that presently we do not assume that M is L2-bounded). If H is a vector-valued predictable process, such that H(i) L2(M(i) ) for every i, then every of the stochastic integrals H(i) M(i) is a well-defined L2-bounded martingale and 38 3: Martingale Representation hence we can define another L2-bounded martingale by H M: = d i=1 H(i) M(i) . This defines a map H H M from the product space L2(M(1) ) × × L2(M(d) ) to the set M2 of L2-bounded martingales. A main reason that we would like to extend the definition of the stochastic integral H M to a wider class of predictable processes H is that the range of this map is not necessarily closed. Even though each of the classes of processes H(i) M(i) , when H(i) ranges over L2(M(i) ), is closed in M2 , being the image of the Hilbert space L2(M(i) ) under an isometry, their sumspace is not necessarily closed. If M is to have the representing property, then we must add the missing elements. If H(i) L2(M(i) ) for every i, then the process (H M)2 - [H M] is a uniformly integrable martingale, zero-at-zero, from which we can infer that E(H M)2 = E[H M]2 . By the bilinearity of the quadratic variation and the rule d[H(i) M(i) , H(j) M(j) ] = H(i) H(j) d[M(i) , M(j) ], this can be written in the form E(H M)2 = i j E 0 H(i) s H(j) s d[M(i) , M(j) ]s. All individual terms of the double sum on the right side are finite, and we can interchange the order of double sum and expectation and integration. The potential for extension of the stochastic integral HM is that the double sum may possess a finite integral even if not all individual terms possess a finite integral. To operationalize this we view the multivariate quadratic variation process t [M]t = ([M(i) , M(j) ]t) as a matrix of distribution functions of (random) signed measures, and write it as an integral [M]t = t 0 Cs d|[M]|s of a predictable process C with values in the set of nonnegative-definite, symmetric matrices relative to a univariate continuous, adapted increasing process |[M]|. (This is always possible. See e.g. Jacod and Shiryaev, II.2.9. We can choose |[M]| equal to the sum of the absolute variations of all components in the matrix, as suggested by our notation. The components [M(i) , M(j) ] are clearly absolutely continuous and hence have densities C(i,j) . That C can be chosen predictable requires proof.) Then the right side of the preceding display can be written as E 0 HT s CsHs d|[M]|s. We now define the space L2(M) as the set of all predictable processes H with values in Rd for which E 0 HT s CsHs d|[M]|s < . This is a linear space, which can be equipped with the inner product, for G, H L2(M), (G, H) = E 0 GT s CsHs d|[M]|s. 3.5: Multivariate Stochastic Integrals 39 The space L2(M) is complete for the corresponding norm and hence is a semi-Hilbert space. The product space L2(M(1) ) × × L2(M(d) ) forms a dense subspace of L2(M), and the map H H M is an isometry of this product space into M2 . We define the stochastic integral H M for every H L2(M) by the unique continuous extension of the map H H M to L2(M). The stochastic integral can be further extended to sufficiently integrable vector-valued predictable processes H by localization. If 0 T1 T2 is a sequence of stopping times such that Tn almost surely, and H1[0,Tn] L2(M) for every n, then the integral (H1[0,Tn]) M is well defined for every n. We define H M to be the almost limit of the sequence (H1[0,Tn]) M as n . It can be shown in the usual way that for m n the processes (H1[0,Tm]) M and (H1[0,Tn]) M are identical up to evanescence on [0, Tm] and hence that this limit exists. Furthermore, it can be shown that the definition is independent of the choice of the localizing sequence. An appropriate sequence of stopping times exists provided, for every t > 0, t 0 HT s CsHs d|[M]|s < , a.s.. In that case the stopping times Tn = inf{t > 0: t 0 HT s CsHs d|[M]|s > n} form a localizing sequence. The integrability condition is certainly satisfied if t 0 (H (i) s )2 d[M(i) ]s < for every i, in which case the stochastic integrals H(i) M(i) are well defined and the extended integral H M coincides with d i=1 H(i) M(i) . We collect this fact in the following lemmas, together with some other properties of the extended stochastic integral. For simplicity of notation set, with C and |[M]| as before, t 0 H2 s d[M]s = t 0 HT s CsHs d|[M]|s. 3.20 Lemma. Let M be a continuous, vector-valued local martingale and H a vector-valued, predictable process with H2 s d[M]s < almost surely. (i) H M = d i=1 H(i) M(i) if all the integrals on the right side are well defined. (ii) H M is a continuous, local martingale. (iii) [H M]t = t 0 H2 s d[M]s. (iv) [N, H M] = H [N, M] for every local L2-martingale N. (v) G (H M) = (GH) M for every predictable process G and vectorvalued predictable process H for which these integrals are well defined. (vi) (H1[0,T ]) H = (H M)T for every stopping time T. (vii) (H M) = HM. 40 3: Martingale Representation 3.21 Lemma (Dominated convergence). Let M be a continuous, vector-valued local martingale and H(n) a sequence of vector-valued predictable processes that converges pointwise on [0, ) × to 0. If t 0 (H (n) s )T CsH (n) s d|[M]|s 0 in probability for every t and H(n) H for a predictable process H such that t 0 H2 s Cs d|[M]|s < almost surely for every t and n, then supst |H(n) Ms| 0 in probability for every t. Proofs. See Jacod and Shiryaev, Section III.4.a. Note that Jacod and Shiryaev allow a general predictable process for |[M]|, where we use a continuous, adapted process. Their condition 4.3 that the process (H C H) |[M]| is locally integrable is equivalent to our condition of finiteness of t 0 HT s CsHs d|[M]|s, as a continuous, nonnegative increasing process is automatically locally integrable (and even locally bounded). The right side of (iv) must be read as t 0 i H (i) s D (i) s d|[M]|s for [Y, X(i) ] = D(i) |[M]|. (The sum may be integrable, even if the individual terms are not.) That the processes (H1[0,Tm]) M and (H1[0,Tn]) M are identical up to evanescence on [0, Tm] for stopping times Tm Tn follows from the identities (H1[0,Tm])M = (H1[0,Tm]1[0,Tn])M = (H1[0,Tn])M Tm , where the last equality follows from (vi). Jacod and Shiryaev state (v) for locally bounded predictable G, but it can be extended by approximation. Because H M is a continuous local martingale with quadratic variation t 0 HT s CsHs d|[M]|s, by (iii) the left side G (H M) of (iv) is well defined if t 0 G2 sHT s CsHs d|[M]|s is finite almost surely for every t. The right side is defined under the condition that t 0 (GsHs)T Cs(GsHs) d|[M]|s is finite almost surely for every t, which is equivalent. Equality (v) can be extended from locally bounded predictable processes G, H to the general case by approximation, using the second lemma. For the proof of the second lemma let Tm be a localizing sequence such that E 0 H2 s 1[0,Tm] Cs d|[M]|s < for every m. Because H(n) H it follows that E 0 (H (n) s 1[0,Tm])T CsH (n) s 1[0,Tm] d|[M]|s < for all m, whence (H(n) 1[0,Tm]) M is a well-defined L2-martingale. By Doob's in- equality E sup st (H(n) 1[0,Tm]) Ms 2 4E (H(n) 1[0,Tm]) Mt 2 = 4E Tm 0 (H(n) s )T CsH(n) s d|[M]|s. The integrand in the last double integral tends to zero pointwise, and is bounded by the integrable function H2 C 1[0,Tm]. By the dominated convergence theorem the double integral tends to zero. This shows that 3.5: Multivariate Stochastic Integrals 41 supst |(H(n) M)Tm s = supst |(H(n) 1[0,Tm] M)s| tends to zero in probability, for every m. If |Y (n) tTm | = |(Y (n) )Tm t | tends to zero for every m, then |Y (n) t | tends to zero. 3.22 Example (Brownian motion). The quadratic variation matrix [B] of a multivariate Brownian motion is the diagonal matrix with the identity function on the diagonal. It follows that in the preceding discussion we can set Ct equal to the identity matrix and |[M]| equal to the identity. The condition for existence of the integral H B becomes finiteness of the integrals t 0 HT s Hs ds, which is the same as finiteness of the integrals corresponding to each of the components. In this case the "extension" of the multivariate integral discussed in this section is unnecessary and does not yield anything new. All stochastic integrals H B are of the form d i=1 H(i) B(i) . 3.23 Example. Let M = B for B a e-dimensional Brownian motion and a (d × e)-matrix-valued predictable process with t 0 s 2 ds < . The integral B is understood as M(i) = e j=1 (i,j) B(j) , where the stochastic integrals on the right are well defined in the ordinary sense by the integrability condition on . Then [M]t = t 0 sT s ds and hence we can take C = T and |[M]| equal to the identity. The condition for existence of H M reduces to finiteness of the process t 0 T s Hs 2 ds. If the process is uniformly bounded away from infinity and singularity, then this reduces to finiteness of t 0 Hs 2 ds. The condition t 0 T s Hs 2 ds < is the natural one if we think of H M as (T H) B and apply Example 3.22. It may be weaker than the condition that Hs 2 d[M]s < , as shown in the exercise below. 3.24 EXERCISE. In Example 3.23 let = 1 0 K 1 - K for a predictable process K. (i) Show that we can take C = 1 K K K2 + (1 - K)2 and |[M]| equal to the identity. (ii) Show that H = (-K, 1)/(1 - K) is contained in L2(M) for any choice of K, but H(1) / L2(M(1) ) for some K. 3.5.2 Semimartingales In this section we define the integral H X for a vector-valued predictable process H and a cadlag semimartingale X. If X is a continuous local martingale, then the definition agrees with the definition in the preceding section. Proofs and further discussion of the results in this section can be found in the papers: C.S. 42 3: Martingale Representation By definition there exists a decomposition X = X0 + M + A of X into a local martingale M and a process of locally bounded variation A. The predictable process H = (H(1) , . . . , H(d) ) is called X-integrable if there exists such a decomposition such that: (i) H M exists as a stochastic integral; (ii) H A exists as a Lebesgue-Stieltjes integral. The integral H X is then defined as H X = H M + H A. Here it is understood that we use cadlag versions so that the integral H X is a cadlag semimartingale. Warning. This definition can be shown to be well posed: the sum HM+HA does indeed not depend on the decomposition X = X0+M+A. However, the decomposition itself is allowed to depend on the process H, and different H may indeed need different decompositions. It can be shown that a decomposition such that A contains all jumps of X with |X| > 1 or |H X| > 1 can be used without loss of generality. In particular, if X is continuous, then we can always use X = M + A with M continuous. Warning. If X is a local martingale, then we cannot necessarily use the decomposition X = X0 + (X - X0) + 0. Given H there may be a decomposition X = X0 + M + A such that (i)-(ii) hold, giving an integral H X, which is the sum of a stochastic and a Lebesgue-Stieltjes integral, whereas H X may not exist as a stochastic integral. Similarly, if X is of locally bounded variation, then we cannot necessarily use X = X0 + 0 + (X - X0). Warning. If X is a local martingale, then H X is not necessarily a local martingale. This is because the Lebesgue-Stieltjes integral H A for a local martingale of locally bounded variation A (as in (ii)) is not necessarily a local martingale. The Lebesgue-Stieltjes integral is defined pathwise, whereas the (local) martingale property requires some integrability. The stochastic integral H M in (i) is always a local martingale. If X is a local martingale and H X is bounded below by a constant, then H X is a local martingale. Warning. If Gt is a filtration with Ft Gt, X is a semimartingale relative to Ft (and hence Gt) and H is predictable relative to Ft, then it may happen that HX exists relative to Ft, but not relative to the filtration Gt. (Conversely, if it exists for Gt, then also for Ft.) Chou, P.A. Meyer, C. Stricker: Sur les intégrales stochastiques de processus prévisibles non bornés, Lecture Notes in Mathematics 784, 1980, 128­139; and J. Jacod: Intégrales stochastiques par rapport `a une semimartingale vectorielle et changements de filtration, Lecture Notes in Mathematics 784, 1980, 161­172. If H is X-integrable, then both X and H X are cadlag semimartingales, and hence each sample path of H or H X has at most finitely many jumps of absolute value bigger than 1; the jumps of H X are H X. It follows that the process of "big jumps" Xt = stXs1|Xs|>1 or |Hs Xs|>1 is well defined. Then X - X is a semimartingale, that can be decomposed as X - X = X0 + M + ~A, so that X = X0 + M + A with A = ~A + X. 3.5: Multivariate Stochastic Integrals 43 The "existence" of the integrals in (i) and (ii) means the following: (i) A stochastic integral H M of a predictable process H relative to a local martingale M exists if and only if the process t t 0 H2 s d[M]s is locally integrable. (ii) A Lebesgue-Stieltjes integral H A of a predictable process relative to a process of locally bounded variation exists if t 0 |Hs| d|A|s < almost surely for every t. If M is continuous or locally bounded, then its jump process is locally bounded, and hence the process t t 0 H2 s d[M]s is locally bounded and is certainly locally integrable to any order, as soon as it is finite. An alternative (equivalent) definition of the stochastic integral is through a limit of integrals of locally bounded processes. A predictable process H is said to be X-integrable if the sequence of processes (H1|H|n) X converges as n to a limit Y , which is denoted H X, in the sense that, for every t, sup |G|1 E (G (H1|H|n) X)t - (G Y )t 1 0. Here the supremum is taken over all predictable processes G with values in the interval [-1, 1]. Because we can choose G = 1[0,t], this implies that (H1|H|n) X t (H X)t in probability, for every t. Thus the stochastic integral is a limit relative to the collection of semi- metrics dt(X, Y ) = sup |G|1 E (G X)t - (G Y )t 1, t > 0. The topology generated by these metrics on the class of semimartingales is called the semimartingale topology. This topology is metrizable (restrict t to the natural numbers to reduce to a countable collection), and the class of semimartingales can be shown to form a complete metric space. From the definition of H X through the semimartingale topology it is clear that the stochastic integral is invariant under an equivalent change of measure: if Q is equivalent to P then the class of X-integrable processes H on (, F, Ft, Q) and (, F, Ft, P) are the same, and so is the stochastic integral H X. 3.25 Lemma. Let G, H be vector-valued predictable processes and X and Y be cadlag semimartingales. (i) If H is both X- and Y -integrable, then H is X + Y -integrable and H (X + Y ) = H X + H Y . (ii) If both G and H are X-integrable, then so is (G+H) and (G+H)X = G X + H X. (iii) If H is X-integrable, then G is H X-integrable if and only if GH is X-integrable, and in that case G (H X) = (GH) X. 44 3: Martingale Representation (iv) If H is X-integrable, then t 0 |Hs| |d[X, Y ]|s is finite almost surely and [H X, Y ] = H [X, Y ]. (v) If H is X-integrable, then (H X) = H X. 3.26 Lemma (Dominated convergence). Let X be a semimartingale and Hn a sequence of predictable processes Hn that converges pointwise to a process H. If |Hn| K for an X-integrable process K, then Hn X tends to H X in the semimartingale topology. The notation in the preceding suggests one-dimensional processes H and X rather than vector-valued processes. The results can be interpreted in a vector-valued sense after making the following notational conventions. The vector-valued semimartingale X can be decomposed coordinatewise as X = X0+M+A for a vector of local martingales M and a vector of processes of locally bounded varation A. The matrix-valued quadratic variation [M] of M can be written as [M] = C |[M]| for a process C with values in the symmetric, nonnegative-definite matrices and a nondecreasing, realvalued, adapted process |[M]|. The vector-valued process A can be written A = D |A| for C a vector-valued optional process and |A| a nondecreasing, adapted process. We then say that H M and H A exist if: (i) The process (HT CH |[M]|)1/2 is locally integrable. (ii) The Lebesgue-Stieltjes integral |HT D| |A| is finite. The stochastic integral H M is by definition the unique local martingale such that [H M, N] = (HT K) |[M]| for any local martingale N such that [M(i) , N] = K(i) |[M]|. The integral H A is by definition the LebesgueStieltjes integral HT D A. 4 Finite Economies In this chapter we consider an "economy" consisting of a vector A = (A(1) , . . . , A(n) ) of n "asset price processes". Throughout the chapter we assume that these processes are semimartingales defined on a given filtered space (, F, {Ft}, P) that satisfies the usual conditions. The -field Ft models our relevant knowledge at time t. In many applications this filtration is taken to be the augmented natural filtration of the process A, denoted by FA t , meaning that we know the past evolution of all the asset prices (and no more). It is often assumed that the -field F0 is trivial (up to null sets). For the general theory it is not necessary to make further assumptions on the asset price processes or the underlying filtered space. One typical more concrete specification would be that the assets satisfy a stochastic differential equation of the type, for given measurable functions and , dAt = (t, At) dt + (t, At) dWt, where W is a multi-dimensional Brownian motion. Under reasonable conditions (e.g. the Itô conditions) the process A will then be adapted to the filtration generated by W. We can then take the filtration Ft equal to the natural filtration generated by A, the natural filtration generated by W, or possibly a still bigger filtration. The choice of filtration is only essential for Theorem 4.31, the other results in this chapter being at a more abstract level and being true for a general filtration. A single filtration Ft is fixed throughout the chapter. Throughout the chapter we work with a finite time horizon, meaning that all processes need to be defined on a finite interval [0, T] only. Properties of processes should be interpreted to refer to the time interval [0, T] only. Alternatively, we may think of all processes being stopped at time T. 46 4: Finite Economies 4.1 Strategies and Numeraires At any point in time we invest in the assets A. The number of assets of every type as a process over time is a "trading strategy" or simply "strategy". 4.1 Definition. (i) A strategy is a predictable process with values in Rn such that the stochastic integral A is well defined. (ii) A strategy is self-financing if A = 0A0 + A. Products between vectors of the type A are to be understood as inner products; for instance tAt = n i=1 (i) t A (i) t . The dot-notation is reserved for the stochastic integral, where A is also to be understood as a linear combination n i=1(i) A(i) of stochastic integrals if each of the stochastic integrals (i) A(i) is well defined. The stochastic integral A of a locally bounded, predictable process is well defined relative to any semimartingale A. Thus locally bounded, predictable processes are always strategies. For some purposes it is necessary to allow a larger set of strategies, which may depend on A. In particular, for results on completeness involving the representing theorem for martingales, the stochastic integral A must be understood in the extended sense discussed in Section 3.5, which allows strategies that are not locally bounded. We say that a predictable process is A-integrable if the stochastic integral A is well defined. Because the theory employs also other probability measures besides the "true world measure" P, it is important to note that stochastic integrals (and hence the set of strategies) do not depend on the underlying measure, as long we use equivalent measures only. That this is true follows from the fact that the stochastic integral can always be written as a limit of integrals of simple integrands, which are Riemann sums and hence independent of any measure. We interpret the strategy t as the numbers of units of assets kept in an "investment portfolio" at time t. Thus the value of the portfolio at time t is given by Vt = tAt. A strategy is not restricted to be nonnegative. Owning a negative amount of an asset is referred to as "taking a short position" in that asset. This is possible in real markets, up to some limitations. For instance, you can borrow money from the bank, as long as the bank is confident that you will be able to pay the interest and/or return the money eventually. The predictability of a strategy can be interpreted as meaning that, for each t, the content of the portfolio at the time t is determined based 4.1: Strategies and Numeraires 47 on knowledge of the development of the asset prices before t only. (This intuitive interpretation should not be taken too seriously. For instance, not much would change if we would allow more general adapted processes in the case that the process A is continuous.) The self-financing property of the strategy can be more concisely written in differential notation as: d(tAt) = tdAt. The self-financing property ensures that the reshuffling of the contents of the portfolio over time is carried out without "money import". The relation in the preceding display requires that "a change in the value A of the portfolio is solely due to changes dAt in the values of the underlying assets". Thus we reconstitute the portfolio "just before time t" using the capital Vt- of the portfolio at that time. Next the value may change due to changes in value of the underlying assets. The resulting gain process A has "increments" t dAt and gives the cumulative increase or decrease of the portfolio value. 4.2 EXERCISE. For given stopping times T0 = 0 < T1 < < Tk = T and FTi -measurable random variables i consider the process = -11{0} + i i1(Ti,Ti+1]. Show that is a strategy, determine its value process, and show that is self-financing if and only if i-1ATi = iATi for every i 0. Interpret this intuitively! 4.3 EXERCISE. Let and be self-financing strategies and S a stopping time such that SAS = SAS. Show that the strategy 1[0,S) + 1[S,T ] is self-financing. 4.4 Definition. A numeraire is a strictly positive semimartingale of the form N = 0A0 + A for some self-financing strategy . A numeraire is special if the strategy can be chosen locally bounded. 4.5 EXERCISE. Show that N = A(1) is a special numeraire provided that it is strictly positive. Numeraires turn out to play an essential role in financial analysis. They will be used to write down the "fair" prices of options. Furthermore, "completeness" of an economy will be characterized by the existence of a numeraire of locally bounded variation. For now we may just think of numeraires as special units to measure our wealth. Rather than in absolute units such as euros or guilders, we can express asset prices and our portfolio relative to the value of the numeraire. If the asset prices are At in absolute units, then they are AN t : = At/Nt if quoted relative to the numeraire N. The following lemma states the intuitively obvious fact that the self-financing property of a strategy is retained if the value process is quoted in different units. 48 4: Finite Economies 4.6 Lemma (Unit invariance). For any strategy we have d(tAt) = t dAt if and only if d(tAN t ) = t dAN t for every numeraire N. Proof. By two applications of the partial integration formula, (4.7) d A 1 N = 1 Nd(A) + (A)- d 1 N + d A, 1 N , d A 1 N = 1 NdA + A- d 1 N + d A, 1 N . The self-financing property d(A) = dA implies that d[A, M] = d[A, M] for every semimartingale M. Therefore, if is self-financing, then the difference of the right sides of the display is equal to (A)- - A- d 1 N = - (A) - A d 1 N . This is zero, as the self-financing property also implies that the processes A and A have the same jumps. The preceding argument uses that N is a strictly positive semimartingale, but no other property of a numeraire. We can change back to absolute units by repeating the argument with AN and 1/N instead of A and N. We complete the proof by showing that the preceding manipulations are indeed justified. A strategy is by definition an A-integrable predictable process. We shall show that a self-financing strategy is also automatically AN -integrable. In view of the second line of (4.7) it suffices to show that is (1/N-)A-integrable, A- (1/N) integrable and [A, 1/N]-integrable. The first follows because the process 1/N- is locally bounded, and the third because A-integrability implies [A, X]-integrability for any semimartingale X. (See Lemma 3.25(iv).) If is self-financing, then A- = (A)- by the preceding display. This shows that the process A- is left-continuous, whence locally bounded and 1/N-integrable. Consequently A- is A- (1/N)-integrable. (See Lemma 3.25(iii).) We can quote the value of our portfolio in arbitrary units. Using a numeraire, a special type of unit, has the great advantage that we need worry less about the self-financing property: for any strategy there exists a self-financing strategy with the same gain process if the gain is measured in the numeraire. 4.8 Lemma. For any F0-measurable variable V0, any numeraire N and any AN -integrable predictable process there exists a self-financing strategy such that 0AN 0 = V0 and AN = AN . Proof. Suppose that N = 0A0 + A = A for some self-financing strategy . Then 1 = N/N = AN and hence 0 = d(AN ) = dAN , by the self-financing property of and unit invariance. Set = + , = V0 + AN - AN . 4.1: Strategies and Numeraires 49 Because AN - AN = ( AN )- - (AN )-, and is predictable, the process is predictable. Because ()AN = (AN ) = 0 = 0, it follows that AN = AN and hence dAN = dAN . Furthermore AN = AN + AN = AN + = V0 + AN , by the definition of . In particular (AN )0 = V0 and d(AN ) = dAN . Combining the preceding we see that dAN = d(AN ) and hence is self-financing relative to the numeraire N. By unit invariance is self-financing relative to any unit. 4.9 Definition. A numeraire pair (N, N) consists of a probability measure N on (, F) that is equivalent to P and a numeraire N such that AN is an N-local martingale. Warning. Hunt and Kennedy require that AN is an N-martingale. We shall call the numeraire pair a martingale numeraire pair in that case. The measure N in a numeraire pair is also referred to as a (local) martingale measure. It is a different type of martingale measure as considered before. Presently the processes A or its "rebased" version AN = A/N are typically not martingales under the initial measure P. The change to the martingale measure N ensures that the process AN is a local martingale. Not every economy admits a numeraire pair. In Lemma 4.16 we shall see that existence of a numeraire pair precludes the possibility of riskless gains (arbitrage). Conversely, an appropriate form of absence of no arbitrage implies existence of a numeraire pair. (This is called the "fundamental theorem of asset pricing".) In this chapter existence of a numeraire pair is always assumed. Then there are automatically many numeraire pairs. We shall see that uniqueness of the martingale measure N going with a given numeraire N is equivalent to "completeness of the economy". 4.10 Example (Black-Scholes). The classical Black-Scholes economy consists of two assets, which for simplicity of notation we shall write as A = (Rt, St). The process St corresponds to a risky asset, such as a stock, and is assumed to satisfy the differential equation, for a given Brownian motion W, dSt = St dt + St dWt. The parameter is assumed to be positive. The asset Rt is risk-free and satisfies the equations dRt = rRt dt, R0 = 1. The last equation can be solved to give Rt = ert . Similarly, the stochastic differential equation can be solved explicitly to give St = S0e(- 1 2 2 )t+Wt . 50 4: Finite Economies We shall show that the process Nt = ert is a numeraire, and shall find a corresponding martingale measure. That N is positive is clear. Furthermore, it is equal to one of the two processes in the economy and hence is the value process of a self-financing strategy. (Cf. Exercise 4.5). Thus N is a numeraire. By the partial integration formula (4.11) d e-rt St = e-rt dSt + St(-re-rt ) dt + 0 = e-rt St d ~Wt, for the process ~W defined by ~Wt = Wt - (r - )t/. Thus to ensure that the process AN = (1, SN ) is an N-local martingale, it suffices to determine the measure N such that the process ~W is an N-local martingale. By Novikov's condition, or direct calculation, the exponential process E(W) for t = (r-)/ is a P-martingale, with mean 1. Therefore, we can define a probability measure N by dN = E( W)T dP. By Girsanov's theorem the process ~W is then an N-Brownian motion (for the time parameter restricted to [0, T]), and hence an N-martingale. Thus we have shown that the process AN = (1, SN ) is an N-local martingale. From the explicit expression SN t = S0 exp( ~Wt-1 2 2 t) it follows that it is actually also an N-martingale. In the preceding we have not made the filtration that we work with explicit. An "initial" filtration Gt is implicit in the assumption that W is a Brownian motion. The present numeraire N is the value process of a deterministic strategy, which is predictable relative to any filtration. In view of the invertible relationship between A and W the augmented natural filtration FA t generated by A is equal to the augmented filtration FW t of the driving Brownian motion W, which may be smaller than Gt. The process AN is an N-martingale relative to the bigger filtration Gt, where by definition of a numeraire pair it suffices that it is a martingale relative to the filtration Ft we work with. Thus the preceding shows the existence of a numeraire pair relative to any filtration Ft that is sandwiched between FA t and the initial filtration Gt. The standard choice for Ft is FA t , for which the Black-Scholes economy is "complete", as we shall see. 4.12 EXERCISE. Extend the preceding example to the situation that the stock price process S satisfies a stochastic differential equation of the form dSt = (t, St)St dt+(t, St)St dWt. Find regularity conditions (e.g. boundedness) on the functions and that ensure the existence of a numeraire pair. * 4.13 Example (Lévy processes). A Lévy process X is a cadlag stochastic process with stationary and independent increments with initial value X0 = 0. We interpret the independence of the increments relative to a general filtration Ft. Thus for each s < t the increment Xt - Xs is independent 4.2: Arbitrage and Pricing 51 of Fs and possesses the same distribution as Xt-s. Consider an economy consisting of the asset processes Rt = ert and St = et+Xt for some constants r, , . Because Brownian motion is a Lévy process, the Black-Scholes economy is a special example. Brownian motion is the only Lévy process with continuous sample paths. More general Lévy processes have been introduced to introduce jumps in the asset processes. We shall exhibit a martingale measure R with the numeraire R under the condition that there exists a solution v to the equation (v + )/(v) = e(r-)v , for (u) = EeuX1 the Laplace transform of X1 (assumed to be finite in an interval). The key observation is that the process t evXt /(v)t is a Pmartingale for any v such that (v) < . Indeed, the stationarity and independence of the increments implies that EeuXt = (u)t and hence, for s < t, E evXt (v)t | Fs = E ev(Xt-Xs) (v)t-s | Fs evXs (v)s = evXs (v)s , where we use the independence of Xt - Xs from Fs, and the fact that Xt - Xs is distributed as Xt-s. For any v the P-martingale Lt: = evXt /(v)t is positive with mean 1 and hence defines a density process of a measure N relative to P. By Lemma 2.14 the discounted process S/R is an N-martingale if LS/R is a P-martingale. Because Lt St Rt = e(v+)Xt e(-r)t (v)t , this process is a P-martingale if e(-r)t (v)-t = (v + )-t . We shall see later that for most Lévy processes there exist infinitely many other martingale measures with the numeraire R. 4.14 EXERCISE. In the preceding example show that the process X is also a Lévy process under the measure N. 4.2 Arbitrage and Pricing It was seen in Example 1.7 that without some further restriction on the set of strategies it will be possible to make certain profits. Thus we allow only "admissible" strategies. The condition of existence of a solution v to the equation (v + )/(v) = e(r-)v is not automatic. 52 4: Finite Economies 4.15 Definition. A strategy is admissible if for any numeraire pair (N, N) the process AN is an N-martingale. If there exists no numeraire pair, then the preceding definition does not make sense. We are not interested in this situation, but for consistency we define the collection of admissible strategies to be empty if no numeraire pair exists. By the definition of a numeraire pair the process AN is an N-local martingale, and hence the process AN is typically an N-local martingale for any self-financing strategy . The special feature of an admissible strategy is that it is an N-martingale. Thus admissibility adds integrability properties to the gain processes AN . It prevents them from becoming too extreme. Warning. Definition 4.15 follows Hunt and Kennedy. Most authors define a strategy to be admissible if its value process is lower bounded by 0. Lemma 4.16 remains valid under the latter definition of admissible. In fact, it is valid as soon as the process AN is a supermartingale. Any local martingale that is lower bounded by a martingale, in particular a nonnegative local martingale, is a supermartingale. One might think of the preceding definition as giving a two-sided sense of admissibility, giving control for both buyer and seller. The following lemmas show that admissible strategies never yield arbitrage. Furthermore, if they lead to the same value at time T, then they have identical value processes throughout [0, T]. 4.16 Lemma (No arbitrage). If is an admissible, self-financing strategy, then it cannot happen that: (i) 0A0 < 0, but T AT 0 almost surely, or: (ii) 0A0 = 0, but T AT 0 almost surely and P(T AT > 0) > 0. 4.17 Lemma (Unique value). If and are admissible, self-financing strategies with T AT = T AT , then tAt = tAt for every t [0, T]. Proofs. By convention the existence of an admissible strategy implies the existence of a numeraire pair (N, N). The self-financing property of a strategy is, by unit invariance, equivalent to the identity AN = 0AN 0 +AN . If is admissible, then this is an N-martingale and taking expectations left and right under the martingale measure yields 0 A0 N0 = EN(0AN 0 ) = EN(T AN T ) = EN T AT NT . If the strategy would satisfy (i), then the left and right sides of this identity would be negative and nonnegative, respectively, which is impossible. The local martingale property is automatic if AN is continuous, but may fail if AN or AN possess too big jumps. See Section 3.5.2. 4.2: Arbitrage and Pricing 53 Similarly, if would be as in (ii), then the two sides would be zero and strictly positive, respectively. This concludes the proof of the first lemma. Under the conditions of the second lemma the process (-)AN is an N-martingale. The self-financing property yields, in view of unit invariance, that ( - )AN = (0 - 0)AN 0 + ( - ) AN , and hence the process (-)AN is an N-martingale. By assumption it vanishes at time T, whence it is identically zero throughout [0, T]. The first lemma is interpreted as saying that "the economy is arbitragefree": no admissible, self-financing strategy leads to sure profit. With the present definitions this is true without conditions. This is somewhat at odds with the literature, but is due to the convention to define the set of admissible strategies to be empty if there exists no numeraire pair. The second lemma shows that the value of an admissible, self-financing portfolio during the interval [0, T] is uniquely determined by its terminal value. This property is the justification for the no-arbitrage pricing principle. Consider an FT -measurable random variable X, interpreted as the value of a derivative contract at expiration time T, for which there exists a replicating strategy: an admissible, self-financing strategy such that X = T AT . Then the no-arbitrage principle leads us to define the "just price" for the claim at time t to be the value tAt of the replicating portfolio at time t. The preceding lemma shows that this definition is independent of the replicating portfolio as long as this is required to be self-financing and admissible. We can express this value using a numeraire pair (N, N) with EN|X/NT | < . Let be a replicating strategy, so that X = T AT . By unit invariance and the self-financing property of , tAN t = 0AN 0 + ( AN )t. Because is admissible, the right side, and hence the left side is an Nmarginale. Thus its value at t [0, T] is determined by its final value at T: by the martingale property tAt = NttAN t = NtEN T AN T | Ft . The left side is the value Vt of the claim at time t, and the variable inside the conditional expectation on the right is equal to T AN T = X/NT . Thus we obtain the pricing formula for the value Vt at t of a derivative with claim X at time T: (4.18) Vt = NtEN X NT | Ft . 54 4: Finite Economies For t = 0 the conditional expectation reduces to an ordinary expectation, in the case that F0 is trivial. We have derived the pricing formula using a numeraire pair (N, N). At first sight the value given by the formula appears to depend on the numeraire pair, but this is not true, because the formula gives merely a representation of the value process of a replicating strategy, which is unique by Lemma 4.17. Thus we may use any numeraire pair such that EN|X/NT | < . On the other hand, beware of thinking of (4.18) as "the price" of the claim X. Even though this is not visible in the formula, the formula is conditional on the existence of a replicating strategy. There are examples of economies with multiple numeraire pairs for which formula (4.18) evaluates to multiple values. Of course, in these examples there cannot exist a replicating strategy for the claim X, in which case we have not said what we mean by "fair price" in the first place. We discuss this further in Section 4.4. 4.3 Completeness The economy is said to be complete if there exists a numeraire pair that allows to price all contingent claims in this way. 4.19 Definition. The economy is complete if there exists a numeraire pair (N, N) such that for every FT -measurable random variable X with EN|X/NT | < there exists an admissible, self-financing strategy with X = T AT . This definition suggests that completeness requires a special numeraire pair, but this is misleading. Given completeness the set of claims for which EN|X/NT | < is independent of the numeraire pair (N, N). Hence the choice of numeraire pair is irrelevant, and we may choose a convenient one. In fact, under completeness the measure F EN(1F /NT ) is the same on the "final" -field FT for every numeraire pair (N, N) with given initial measure N0 (the restriction of N to F0) and initial numeraire value N0. Essentially, this follows because EN(1F /NT ) is the fair price at time 0 of the claim 1F , which is given by the value at 0 of a replicating portfolio and hence is independent of the numeraire pair. A more formal argument is given in the following lemma. 4.20 Lemma. If the economy is complete and (M, M) and (N, N) are arbitrary numeraire pairs with M0 = N0 and M0 = N0, then the process M/N This argument does not seem to need that N is a numeraire. It suffices that is it a unit and that the processes AN are N-martingales. 4.3: Completeness 55 is the density process of M relative to N. In particular, M-1 T dM = N-1 T dN on FT . Proof. Suppose that the economy is complete and let (N, N) be a numeraire pair as in the definition of completeness. For a given event F FT the claim X = NT 1F is FT -measurable and satisfies EN|X/NT | < . Therefore, by completeness there exists an admissible, self-financing strategy such that NT 1F = T AT . The self-financing property and unit invariance show that the process Z defined by Zt = tAN t can also be represented as Z = 0AN 0 + AN and hence is an N-martingale, by the assumed admissibility of . Clearly ZT = 1F . If (M, M) is another numeraire pair, then, again by self-financing and unit invariance, the process Y defined by Yt = tAM t can be written as Y = 0AM 0 + AM and hence is an M-martingale, by the admissibility of . Clearly Y = (N/M)Z and Y0 = Z0. Being martingales, the processes Z and Y have constant means, and hence EN1F = ENZT = ENZ0 = EMZ0 = EMY0 = EMYT = EMZT NT MT = EM1F NT MT , where we use that M0 = N0 in the last equality of the first line. This being true for an arbitrary event F FT shows that N possesses density NT /MT relative to M on FT , or, equivalently, that M possesses density MT /NT relative to N, or that M-1 T dM = N-1 T dN. This is true for an arbitrary numeraire pair (M, M) and the special numeraire pair (N, N). It follows that the measure M-1 T dM is the same for every numeraire pair (M, M). Because dM/dN = MT /NT , the density process of M relative to N is the process EN(MT /NT | Ft). This process coincides with the process M/N if (and only if) the process M/N is an N-martingale. Because M is a numeraire, there exists a self-financing strategy with M = A. By unit invariance and self-financing M/N = AN = 0AN 0 + AN . Because (N, N) is a numeraire pair, the process AN is an N-local martingale and hence AN and M/N are N-local martingales. (In the case that AN has jumps and is not locally bounded, we use that M/N is nonnegative to ensure the local martingale property.) A positive, local martingale is a supermartingale, by Fatou's lemma, and (hence) is a martingale if its mean is constant. The mean of M/N satisfies EN(MT /NT ) = EM1 = 1 = EN(M0/N0), by assumption. Thus M/N is an N-martingale. 4.21 EXERCISE. Suppose that we drop the conditions M0 = N0 and M0 = N0. Show that M/N(N0/M0)(dM0/dN0) is the density process of M relative to N. The last assertion of the lemma implies that in a complete economy EM(X/MT ) = EN(X/NT ) for every contingent claim X and any pair of 56 4: Finite Economies numeraire pairs. Thus for pricing contingent claims the choice of numeraire pair in formula (4.18) is irrelevant, although one choice may lead to easier computations than another. Because a density process is a martingale, the first assertion of the lemma shows that in a complete economy the quotient M/N of two numeraires that are part of numeraire pairs (M, M) and (N, N) is an Nmartingale. Conversely, if M is a numeraire and M/N is an N-martingale, then there exists a martingale measure M corresponding to M, and we can construct it as in the preceding lemma. 4.22 Lemma. If (N, N) is a numeraire pair and M a numeraire with M0 = N0 such that M/N is an N-martingale, then (M, M) is a numeraire pair for M the probability measure on FT with dM = MT /NT dN. Proof. By the martingale property EN(MT /NT ) = EN(M0/N0) = 1, whence M is a probability measure. Because (N, N) is a numeraire pair, the process AN = (M/N)AM is an N-local martingale. By the definition of M and the assumption that M/N is an N-martingale, the process M/N is the density process of M relative to N. Therefore, AM is an M-martingale, by Lemma 2.14. 4.23 Example (Positive asset as numeraire). If it is strictly positive, then a fixed component M = A(i) of the asset process A is a numeraire. If there exists a numeraire pair (N, N) with N a martingale measure rather than a local martingale measure for AN , then AN and hence its component M/N = (A(i) )N is an N-martingale. Lemma 4.22 then guarantees the existence of a measure M such that (M, M) is a numeraire pair. Thus if there exists a numeraire pair with martingale measure, then we can always use a numeraire pair with as numeraire a strictly positive component of the asset price process, if there is one. ** 4.24 EXERCISE. In an incomplete economy the quotient M/N of two numeraires need not be an N-martingale even if both numeraires are part of numeraire pairs (M, M) and (N, N)?? The definition of completeness requires that every claim can be replicated by an admissible strategy. The definition of an admissible strategy requires that the process AN be an N-martingale for every numeraire pair (N, N). This is rather inconvenient, as there are many numeraire pairs. Actually, the requirement of admissibiliy can be relaxed: it suffices that AN is an N-martingale for just a single numeraire pair (N, N). An economy is complete as soon as every sufficiently integrable claim can be replicated by a self-financing strategy such that AN is an N-martingale for some given numeraire pair (N, N). Also, in a complete economy any strategy for which there exists a numeraire pair (N, N) such that AN is an N-martingale is automatically admissible. 4.3: Completeness 57 4.25 Lemma. The economy is complete if (and only if) there exists a numeraire pair (N, N) such that for every nonnegative FT -measurable random variable X with EN(X/NT ) < there exists a self-financing strategy with X = T AT and such that AN is an N-martingale. 4.26 Lemma. If the economy is complete and is a self-financing strategy such that AN is an N-martingale for some numeraire pair (N, N), then AM is an M-martingale for every numeraire pair (M, M). Proofs. Suppose that we can prove that for any other numeraire pair (M, M) the process M/N is a multiple of the density process of M relative to N. Then by Lemma 2.14 for any self-financing strategy the process AM is an M-martingale if and only if the process (M/N)AM = AN is an N-martingale. Equivalently by unit invariance, the process AM is an M-martingale if and only if the process AN is an N-martingale. The assertion of the second lemma then follows. Furthermore, it is then clear that under the conditions of the first lemma any nonnegative claim can be replicated by an admissible strategy. The completeness of the economy follows then by splitting a general claim X with EN(|X|/NT ) < into its positive and negative parts X+ and X- . The proof that M/N is a multiple of the density process of M with respect to N is in part a repetition of the proof of Lemma 4.20, with the difference that we cannot assume that AM is automatically an M-martingale. This difficulty is overcome by a stopping argument. For simplicity of notation assume that M0 = N0, which can always be arranged by scaling. Fix an event F FT and define a stopping time by Tn = inf{t > 0: Nt/Mt > n}. Because the claim X = NT 1F 1Tn>T is FT -measurable and satisfies 0 X NT , under the conditions of the lemmas there exists a self-financing strategy such that AN is an N-martingale and NT 1F 1Tn>T = T AT . The self-financing property and unit invariance show that the process Z defined by Zt = tAN t can also be represented as Z = 0AN 0 + AN and hence is an N-martingale, by assumption. Clearly ZT = 1F 1Tn>T . If (M, M) is another numeraire pair, then, again by self-financing and unit invariance, the process Y = (N/M)Z = AM can be written as Y = 0AM 0 + AM and hence and hence the stopped process Y Tn can be represented as a stochastic integral as well. Because Y = (N/M)Z and Zt = EN(ZT | Ft) is bounded in absolute value by 1, the stopped process Y Tn is bounded by n, by the definition of Tn. We conclude that Y Tn is an M-martingale, for every fixed n. By optional stopping of the martingale Z, the final value of Y Tn is given by, Y Tn T = YTn 1TnT + YT 1Tn>T = NTn MTn EN(ZT | FTn )1TnT + NT MT ZT 1Tn>T = NT MT 1F 1Tn>T . 58 4: Finite Economies Here we use that EN(ZT | FTn ) = EN(1F | FTn )1Tn>T is zero almost surely on the event {Tn T}. Let V0 = dN0/dM0. Since V0 is F0-measurable, the process V0Y Tn is an M-martingale, just like the process Y Tn . Being martingales, the processes Z and V0Y Tn have constant means, and hence EN1F 1Tn>T = ENZT = ENZ0 = EMV0Y Tn 0 = EMV0Y Tn T = EM1F 1Tn>T V0NT MT . Letting n and applying the monotone convergence theorem yields EN1F = EM1F (V0NT /MT ). Repeating this for an arbitrary event F FT , we see that dN/dM = V0NT /MT . Hence the density process of N relative to M is EM V0NT /MT | Ft). Because N is a numeraire, N = A for some self-financing strategy and hence N/M = AM = 0AM 0 + AM by unit invariance. Because N/M is nonnegative it follows that it is M-super martingale, and so is the process V0N/M. But EMV0NT /MT = 1 = EMV0 = EMV0N0/M0 and hence the process V0N/M is an M-martingale. Combining this with the last conclusion of the preceding paragraph shows that the process V0N/M is the density process of N relative to M. 4.27 EXERCISE. Let (N, N) be a numeraire pair and suppose that is a self-financing strategy whose discounted value process AN is bounded below by a constant and which replicates a claim X in that T AT = X. Show that 0A0 N0EN(X/NT ). Thus the starting value of any replicating self-financing strategy with lower bounded value process (not necessarily admissible) is at least the price given by (4.18). Show this. In a complete economy numeraire pairs (N, N) and (M, M) are connected through the change of measure relation M-1 T dM = N-1 T dN, by Lemma 4.20. This immediately implies that there can be at most one martingale measure for every given numeraire. If the discounted asset processes are locally bounded (in particular if the asset processes are continuous), then this uniqueness property characterizes completeness. 4.28 Theorem (Completeness). Assume that the economy permits a numeraire pair (N, N) such that the process AN is locally bounded. Then the following statements are equivalent: (i) The economy is complete. (ii) If (N , N) is also a numeraire pair and N0 = N0, then N = N on FT . (iii) For every nonnegative N-local martingale M there exists an AN integrable predictable process such that M = M0 + AN . Hunt and Kennedy claim that the process Y is bounded by n on [0, T ]. If that is true the stopping is not necessary and the proof can be simplified. 4.3: Completeness 59 (iv) For every nonnegative N-local martingale M there exists a selffinancing strategy such that M = 0AN 0 + AN . Proof. (i) (ii). By Lemma 4.20 N possesses density process N/N = 1 relative to N. (ii) (iii). Assumption (ii) entails that there exists a unique probability measure on FT with given initial measure such that the process AN is an N-local martingale. Therefore, by Theorem 3.2 (if AN is continuous) or Theorem 3.4 (for general locally bounded AN ) the process AN possesses the martingale representing property (iii). (iii) (iv). This follows from Lemma 4.8. (iv) (i). Given a nonnegative claim X with EN(X/NT ) < , we apply (iv) to the martingale M defined by Mt = EN X/NT | Ft Then we obtain a self-financing strategy such that M = 0AN 0 + AN , which implies that the process AN is a martingale. By unit invariance and selffinancing we have that M = AN . Evaluating this identity at time T, we see that X = MT NT = T AT . The economy is complete by Lemma 4.25. A possible disadvantage of the characterization of completeness provided by the preceding theorem is that its conditions (ii) or (iii) may be difficult to verify. Uniqueness of the "martingale measure" N going with a numeraire N means uniqueness of the probability measure N making the process AN into an N-martingale. Because it is often the process A that is directly described in a model for the economy, not the rebased process AN , standard theorems may not directly apply. In particular, consider the situation that the asset processes are modelled through a stochastic differential equation of the form (4.29) dAt = (t, At) dt + (t, At) dWt. Then the continuous martingale part of A possesses the representing property relative to (, F, {FA t }, P) under the reasonable condition that the solution to the equation is weakly unique, by Theorem 3.14. However, (iii) of the preceding theorem requires that we establish that the process AN possesses the representing property relative to the filtered space (, F, {FA t }, N). If the numeraire is a function of the assets A, then we can approach this by first deriving a diffusion equation for the process AN . Because AN is an N-martingale, this diffusion equation should have a zero drift term if written relative to an N-Brownian motion. We can next deduce the desired result with the help of Theorem 3.14. Similarly, the disadvantage may disappear if the numeraire N takes some other concrete form. 4.30 Example (Black-Scholes). The Black-Scholes economy of Example 4.10 permits a numeraire pair (R, R) with the numeraire Rt = ert , and 60 4: Finite Economies the rebased stock price process satisfies d e-rt St = e-rt St d ~Wt, where ~Wt = Wt +(-r)t/ is an R-Brownian motion. The process e-rt St is strictly positive, and Brownian motion possesses the representing property relative to its augmented filtration. It follows from this that (iii) of the preceding theorem holds for A = (R, S) and the filtration equal to the augmented filtration of Brownian motion. Thus the Black-Scholes economy is complete in this setting. In Example 4.10 it was seen that (R, R) is a numeraire pair for the Black-Scholes economy for any given filtration for which the driven Brownian motion is indeed a Brownian motion. However, for a bigger filtration the Black-Scholes economy is not complete. By enlarging the filtration we add claims, which by definition are FT -measurable random variables, or equivalently local martingales M as in (iii) of Theorem 4.28. These may not be replicable or representable in terms of the Brownian motion, as they are not described in terms of this Brownian motion. In practice one might think of an option on Shell stocks that cannot be replicated using only Philips stocks, even if the stocks satisfy the BlackScholes model. So if the filtration is generated by both stocks and we use as asset process only the Shell stocks, then the economy will not be complete. A solution in this case could be to consider the Shell and Philips stocks jointly. However, one can easily imagine other examples of filtrations containing relevant information available in the market beyond the information in the stocks itself. The representing property of the continuous martingale part Ac (relative to (, F, {FA t }, P)) of the asset price processes is similar to the representing property of the rebased process AN as in (iii) of the preceding lemma, but it does not imply it. Because the process Ac possesses the representing property in many situations, it is useful to investigate which additional conditions are necessary to ensure the representing property of AN and hence completeness. The following theorem shows that, given the existence of a numeraire pair, it suffices to add the existence of a numeraire of bounded variation. 4.31 Theorem (Completeness). If the continuous martingale part Ac,P of a continuous economy A possesses the representing property relative to (, F, {Ft}, P) and the economy permits a numeraire pair (N, N), then the following statements are equivalent: (i) The economy is complete. We write Xc,N for the local martingale part of a continuous semi-martingale X relative to a measure N. This is the unique continuous local martingale such that X - Xc,N is an adapted continuous process of locally bunded variation. 4.3: Completeness 61 (ii) There exists a numeraire of bounded variation. (iii) There exists a strategy with Ac,N = 0 and A = N. Proof. Unless indicated differently, we interpret all statements and notation relative to the martingale measure N. In particular, we abbreviate Ac = Ac,N . Being the value process of a strategy, any numeraire is contin- uous. (i) (iii). Because N is a numeraire, there exist a self-financing strategy with N = A = 0A0 + A. This implies that the local martingale part of N is given by Nc = Ac , for Ac the local martingale part of A. By completeness and Theorem 4.28(iv) there exists a self-financing strategy such that Nc = AN = AN . Now set = - N + A N2 . The process is predictable, as it is adapted and continuous. Because A = N, A = A - A N + A N A N = A = N, dA = dA - N dA + A N2 dA = dN - d(NNc ) N + Nc N dN, where for the last equality we use that dA = d(A) = d(NAN ) = d(NNc ). Using the partial integration formula on d(NNc ) and the fact that [N, Nc ] = [N], we can rewrite the right side as dN N dNc N - Nc dN N - d[N] N + Nc N dN = d(N - Nc ) - d[N] N . We conclude that A = N - Nc - N-1 [N] - N0 is a continuous process of bounded variation, and hence the continuous martingale part Ac of the process A is zero. (i)+(iii) (ii). The process M = exp (/N) A is well defined, strictly positive, continuous, and of locally bounded variation (as Ac = 0). Because N = A, we have (/N)A = 1 and hence M = M(/N)A. By the definition of M and Itô's formula, where the second order term does not appear because M is of locally bounded variation, dM = M(/N) dA. Together these equalities show that M = A for the self-financing strategy = M(/N). Hence M is a numeraire of bounded variation. (ii) (i). In view of Girsanov's theorem, the continuous martingale part of A relative to the measure N is given by Ac,N = Ac,P -L-1 - [L, Ac,P ], It is A-integrable, as is A-integrable, is A-integrable (being self-financing and AN integrable), and 1/N and A/N are locally bounded. Hunt and Kennedy claim that (/N) A is the bounded variation part of log N. Is that true?? 62 4: Finite Economies if L is the density process of N relative to P. By assumption the process Ac,P possesses the representing property for P-local martingales. Therefore, by Lemma 3.7 the N-local martingale Ac = Ac,N possesses the representing property for N-local martingales. Given a claim X with EN(|X|/NT ) < define an N-martingale Kt = EN X/NT | Ft . By the representing property there exists an Ac -integrable predictable process such that K = K0 + Ac . If we can show that Ac can be represented as Ac = AN for some predictable process , then it follows that Kt = K0 + AN for = . By Lemma 4.8 there exists a self-financing strategy such that K = 0AN 0 + AN and this satisfies T AT = NT (AN )T = NT KT = X. The economy is complete in view of Lemma 4.25. Finally, we represent Ac as a stochastic integral relative to AN , using a bounded variation numeraire M with M = A = 0A0 + A for some selffinancing strategy . By unit invariance d(M/N) = dAN . By the partial integration formula, dM = d M N N = M N dN + N d M N + M N , N . Because M is continuous and of bounded variation, its continuous martingale part is zero. Similarly, the continuous martingale part of the last term on the right [M/N, N] is zero. Comparing the continuous martingale parts of the two sides of the display, we deduce that M N dNc = -N d M N c = -N d(AN )c = -N dAN , since AN is a continuous N-local martingale, by assumption. Again by the partial integration formula, dA = d(NAN ) = N dAN + AN dN + [N, AN ]. Hence dAc = N dAN + AN dNc . Combination with the preceding display gives the desired representation of dAc as a multiple of dAN . Warning. Existence of a numeraire pair is an assumption of the theorem, which is not implied by the existence of a numeraire of bounded variation. Even under the conditions of the theorem, the numeraire of finite variation whose existence is guaranteed in (iii) is not necessarily part of a numeraire pair. The preceding theorem applies in particular to economies described through a stochastic differential equation of the type (4.29). If the filtration is chosen equal to the augmented natural filtration FA t of the asset prices (assumed right-continuous), then the condition that Ac possesses the representing property is verified if the solution to (4.29) is weakly unique, by Theorem 3.14. This is true for "nice" functions and . 4.4: Incompleteness 63 4.32 Example (Black-Scholes). The Black-Scholes economy given in Example 4.10 is described through a stochastic differential equation that satisfies the Itô conditions. Hence the solution to the equation is weakly unique and possesses the representing property for the filtration generated by the assets. It permits a numeraire pair relative to this filtration. The process Rt = ert is a numeraire of bounded variation. Thus the Black-Scholes economy is complete. 4.4 Incompleteness Real markets are often thought to be not complete. Any replicable claim in an incomplete market can still be priced by the no-arbitrage principle of the preceding sections, and possesses fair price (4.18). However, by definition in an incomplete market some claims are not replicable and a different approach is necessary. The main message of this section is that the no-arbitrage principle allows an interval of possible prices, and each of these prices can be written as in formula (4.18) for some martingale measure N. By Theorem 4.28 an incomplete market allows multiple martingale measures N with a given numeraire N. Every of these measures defines a possible "fair" price. As before let A be a vector-valued semimartingale and let AN be its value expressed in terms of a given numeraire N. Let N be the set of all probability measures N such (N, N) is a numeraire pair. Given a claim X, let (4.33) V t = ess sup NN EN X NT | Ft . Under integrability conditions (nonnegativeness of the claim X suffices) there exist versions of these essential suprema that define a cadlag process V that is an N-supermartingale for every measure N N. (See Lemma 4.36.) The process V in all of the following will be understood to be this version, and it will silently be understood that it is finite. 4.34 EXERCISE (Essential supremum). For any set {X: A} of random variables, there exists a random variable X (possibly with value ) such that (i) X X almost surely for every . (ii) if Y X almost surely for every , then Y X almost surely. This random variable X, which is unique up to null sets, is called the essential supremum of the collection {X: A} and is denoted ess sup X. [Hint: For any countable B A set XB = supB X. Let Bn be a sequence such that E arctg XBn supB E arctg XB and set X = supn XBn .] 64 4: Finite Economies We shall show that N0V 0 is the minimum initial investment needed to superreplicate the claim with certainty: it is the minimal capital needed to start a self-financing strategy that is certain to yield the value of the claim X at time T. In other words, N0V 0 is the minimal price that is risk-free for the seller of the option: after selling the option the seller could buy the portfolio and implement a self-financing strategy that is certain to yield a capital equivalent to the claim at expiry time. Conversely, the amount N0V 0, for V the corresponding essential infimum, is the biggest price that is risk-free for the buyer. Any price not in the interval [N0V 0, N0V 0] allows risk-free profit (for buyer or seller), whereas any price inside the interval is arbitrage-free, but requires risk taking on the part of buyer or seller. The class N of all local martingale measures is convex, and the map N EN(X/NT ) is linear. It follows that the range of this map is convex, and hence an interval in the real line. Thus, if F0 is trival, then any price in the interval (N0V 0, N0V 0) can be written in the form EN(X/NT ) for some N N. 4.35 EXERCISE. Suppose that A is a local martingale on the filtered space (X, F, {Ft}), both when equipped with P and when equipped with Q. Then A is also a local martingale under the measure P + (1 - )Q for any [0, 1]. [Hint: consider the density process of the convex combination and use Lemma 2.14.] The final value of V satisfies V T = X/NT almost surely. The process V is the smallest supermartingale with this property. 4.36 Lemma. If X is a nonnegative FT -measurable random variable such that EN(X/NT ) < for all N N, then (i) the process V is an N-supermartingale for every N N and permits a cadlag version. (ii) any process V with VT X/NT which is an N-supermartingale for every N N satisfies Vt V t almost surely, for every t. 4.37 Corollary. Suppose X 0. Any self-financing strategy with (A)T X and which is admissible or has nonnegative value process A satisfies tAt NtV t almost surely. Proofs. For (ii) of the lemma we note that an N-supermartingale V satisfies Vt EN(VT | Ft) almost surely, which is bounded below by EN(X/NT | Ft) if VT X/NT . This being true for any N N gives that Vt V t almost surely by the definition of the essential supremum. The value process in terms of a numeraire V = AN = (AN )0 +AN of a self-financing strategy as in the corollary is either an N-martingale (if is admissible) or at least an N-supermartingale (if A is nonnegative). It satisfies VT X/NT if (A)T X. Therefore (A)t = Nt(AN )t NtV t almost surely by (ii). 4.4: Incompleteness 65 For the proof of (i) of the lemma fix an arbitrary measure Q in N for reference, and write expectations EQ relative to Q as E. For simplicity of notation write X for X/NT and A for AN . Fix some t [0, T]. If N has density process L relative to Q, then EN(X| Ft) = E(LT X| Ft) Lt = E(KT X| Ft), for K the process K = 1[0,t) + (L/Lt)1[t,T ]. The process K is a positive martingale with mean 1 and hence defines a density process of a probability measure ~N relative to Q. This measure ~N is contained in N (i.e. A is an ~Nlocal martingale), because the process KA = 1[0,t)A+(LA/Lt)1[t,T ] is a Qlocal martingale, both the processes A and LA being Q-local martingales. It follows that the essential supremum in the definition of V t can be restricted to local martingale measures N with density process identically equal to 1 on [0, t]. Write Kt for this collection of density processes. Because an essential supremum can always be written as a supremum over a countable subset, it follows that V t = supn E(K (n) T X| Ft) for a sequence of density processes K(n) Kt. In the present case we can choose the countable subset in such a way that the sequence E(K (n) T X| Ft) is nondecreasing. Indeed, given density processes K(1) and K(2) in Kt, we can define K = K(1) 1C + K(2) 1Cc , C = E(K (1) T X| Ft) > E(K (2) T X| Ft) . The process K is a Q-martingale, as K(1) , K(2) are Q-martingales which coincide on [0, t] and C Ft. Furthermore, the process KA is a Q-local martingale by the same reasoning. The process K is positive, has mean one and is identically one on [0, t]. Therefore, K belongs to Kt. By construction E(KT X| Ft) = 1CE(K (1) T X| Ft) + 1Cc E(K (2) T X| Ft) = E(K (1) T X| Ft) E(K (2) T X| Ft). Thus if E(K (n) T X| Ft) is not increasing for the original sequence K(n) , we can transform it in an increasing sequence by taking successive linear com- binations. It follows that there exists a sequence K(n) in Kt with 0 E(K (n) T X| Ft) V t. By the monotone convergence theorem, for s < t, E(K (n) T X| Fs) = E E(K (n) T X| Ft)| Fs) E(V t| Fs), a.s.. Because each K(n) is a density process of a measure in N and K (n) s = 1, the left side is EN(X| Fs), for some N N, for every n. Therefore it is bounded above by V s, whence V s E(V t| Fs) almost surely. This concludes the proof that the process V is a supermartingale. 66 4: Finite Economies By general theory the supermartingale V permits a cadlag version if its mean t EV t is right-continuous. The monotone convergence theorem applied again to the increasing sequence 0 E(K (n) T X| Ft) V t, but this time unconditionally, gives that EV t = supn E(K (n) T X). Because V t E(KT X| Ft) almost surely for every K Kt, it follows that EV t = supKKt E(KT X). For K Kt and s > t, the process TsK = 1[0,s) + (K/Ks)1[s,T ] is contained in Ks. (Previously K was obtained from L by the operator Tt, and hence the claim follows by preceding arguments.) We have that (TsK)T = KT /Ks KT /Kt = KT as s t, by right-continuity of K. By Fatou's lemma EV t = sup KKt E(KT X) sup KKt lim inf st E((TsK)T X) lim inf st EV s. The supermartingale property of V immediately yields the reverse inequality, whence t EV t is right-continuous. 4.38 EXERCISE. If M and N are (local) martingales, then (i) M1[0,t) + N(Mt/Nt)1[t,) is a (local) martingale. (ii) if M and N agree on [0, t] for some t and C Ft, then M1C + N1Cc is a (local) martingale. The lemma shows that any admissible strategy that yields with certainty at least the value of the claim X at expiry time will cost at least N0V 0 to start at time 0. The next result shows that there exists a strategy at this price that superreplicates the claim. 4.39 Theorem. Assume that X 0 and that the process AN is locally bounded. Then there exists a self-financing strategy such that V = AN C for a cadlag, adapted, nondecreasing process C with C0 = 0. Proof. Because the process V is an N-super martingale for every N N, Theorem 3.5 shows that there exist an AN -integrable predictable process and a cadlag, adapted, nondecreasing process C such that V = V 0 + AN - C. By Lemma 4.8, there exist a self-financing strategy with (AN )0 = V0 and AN = AN . This satisfies V = AN - C and hence fulfills the requirements. The strategy in the lemma is not necessarily admissible in that the process AN is an N-martingale. However, the strategy is "one-sided" admissible in that its value process AN = V + C V is nonnegative if X 0. We interpret the process C in Theorem 4.39 as a cumulative consumption process: at each time instant t the current value of the portfolio is used 4.4: Incompleteness 67 to form a new portfolio and to extract an amount dCt, yielding the portfolio value (in terms of the numeraire) at time t V t = (AN )0 + ( AN )t - Ct. The extracted amount dCt is nonnegative, because the process C is nondecreasing. The final value of the portfolio is (A)T = NT (AN )T = NT (V T + CT ) NT V T = X. Thus notwithstanding the possible consumption during the term of the contract, the portfolio ends up superreplicating the claim. The initial cost 0A0 = N0V 0 of the portfolio should be an acceptable price to the seller of the claim X. An initial cost higher than N0V 0 would provide the seller an opportunity for riskless gain (arbitrage), as would a price of exactly N0V 0 if CT > 0. Of course, the price N0V 0 may not be to the liking of the buyer of the claim. Set V t = ess inf NN EN X NT | Ft . By similar arguments we can argue that N0V 0 is the highest price for the buyer to be sure that he can hedge away the complete risk of buying the claim X. If the prices N0V 0 and N0V 0 do not agree, seller and/or buyer must take some risk, and the price of the claim will be some number between the two extremes. To argue the riskless price N0V 0 for the seller we have assumed that the claim X is nonnegative. The analogous argument from the point of view of the buyer is the seller's argument applied to minus the claim. Application of the preceding would require the assumption that the claim -X is nonpositive, which is incompatible. It is therefore necessary to relax the assumptions. By adding and subtracting it can be seen that Theorem 4.39 is true for any claim X that is lower-bounded by a claim that can be replicated by an admissible strategy. This shows that both the buyer's and the seller's argument are valid for any claim X for which there exist replicable claims X and X with X X X. 4.40 Corollary. Suppose that X X for a claim X for which there exists a self-financing strategy H with X = (HA)T such that H AN is an Nmartingale for every N N. Then the assertion of Theorem 4.39 remains true. Proof. We apply Theorem 4.39 to the nonnegative claim X - X. The supermartingale attached to this claim through formula (4.33) is V (X X) = V (X) - HAN for V (X) the supermartingale attached to X, exactly as given in (4.33). The corollary follows by adding and subtracting, with the desired strategy equal to H + (X - X) for (X - X) the strategy provided for the claim X - X by Theorem 4.39. 68 4: Finite Economies 4.41 Example (Black-Scholes with stochastic volatility). Consider an economy consisting of two asset processes, a deterministic fixed income process Rt = ert and a process S satisfying the coupled SDEs dSt = St( dt + t dW (1) t ), dt = t dW (2) t , for a two-dimensional Brownian motion W = (W(1) , W(2) ). This economy differs from the Black-Scholes model in that the volatility t is a random process rather than a constant. It is an example of a stochastic volatility model. Economies with stochastic volatility are not necessarily incomplete, but this example is. We shall show that the economy (R, S) permits a numeraire pair, but is not complete relative to the augmented filtration generated by S. Note that the volatility process is used to describe the economy, but is not part of the economy (R, S) itself. Two applications of Itô's formula show that the processes S and can be written in the form St = exp t - 1 2 t 0 2 s ds + t 0 s dW(1) s , t = exp(W (2) t - 1 2 t). In particular these processes are strictly positive. Let Ft be the augmented natural filtration of the driving Brownian motion W. The preceding display shows that (S, ) is adapted to this filtration (which is also an implicit assumption in saying these processes solve the SDE). The second equation of the display shows directly that W(2) is adapted to the filtration generated by . The positivity of S and allows to write W(1) as a stochastic integral of (S, ). Thus W is adapted to the filtration generated by (S, ), whence the filtration Ft is the augmented natural filtration of (S, ). From the fact that the process Y = S-1 S has quadratic variation [Y ]t = t 0 2 s ds it follows that the integral of the square volatility process is adapted to the filtration generated by S, which implies that itself is also adapted. We conclude that the augmented filtrations generated by W, by S, or by (S, ) all coincide with the single filtration Ft. We choose Rt = ert as a numeraire. The rebased process S/R can be written in the form St Rt = exp -1 2 t 0 2 s ds + t 0 s d ~W(1) s , for ~W (1) t = W (1) t - t 0 (r - )/s ds. The process L = E (r - )/ W(1) is a P-local martingale, because W(1) is a P-local martingale, The process L a supermartingale, as it is nonnegative, and its mean function can be seen to be constant by direct calculation (condition on !). Thus the process L 4.5: Utility-based Pricing 69 is P-martingale and hence defines a density process. By Girsanov's theorem the process ( ~W(1) , W(2) ) is an R-Brownian motion under the measure R with density process L relative to P. The same calculation as for L then shows that the process R/S is an R-martingale. Thus (R, R) is a numeraire pair for the economy (R, S). However, the measure R is not unique as a martingale measure. Consider the measure ~P with density process L E(Y W(2) ) relative to P for a suitable, sufficiently integrable predictable process Y . The process ~W with coordinates ~W(1) (as before) and ~W (2) t = W (2) t - t 0 Ys ds is a ~P-Brownian motion. That the process S/R is a ~P-martingale follows as before. Thus (~P, R) is a also a numeraire pair. The measure ~P coincides with R only if its density process E(Y W(2) ) with respect to R is equal to 1, i.e. Y = 0. The economy is incomplete by Theorem 4.28. In contrast, the economy (R, S, ) in which the volatility process itself is tradable is complete. (Compare Theorem 5.7.) Thus a claim X, a measurable function of the process (St: 0 t T), may be replicable using the assets (R, S, ), but not using only (R, S). This is a little surprising, because the volatility process is itself a measurable function of the process S, so that it is observed if (R, S) are observed. The explanation is that even though an integral is a function of S, it cannot necessarily be written as an integral S with respect to S. Thus the fair price of a claim X can be written as EN(X/NT ) for some local martingale measure N N (if F0 is trivial). There are at least two approaches for selecting the martingale measure that determines the price. The first is to say that "the market determines the martingale measure". The prices of commonly traded options can be observed on the option market, and the martingale measure can be inferred by calibrating the observed prices to the prices given by (4.18). The second approach digs deeper in the theory of economic behaviour and is based on utility arguments. * 4.5 Utility-based Pricing Fix some nondecreasing function u: R R, which we shall refer to as a utility function. The expected utility (at time 0) of receiving an amount Y at time T is defined as the expected value EPu(Y/NT ) computed under the real world probability measure P. The future payment Y is discounted by a given numeraire N and Y is assumed to be an FT -measurable random variable. Given an initial wealth x at time 0 we aim at maximizing expected utility, where our possible strategies are to invest in the assets and/or to buy or sell an option with a given claim X. If we do not trade in the option, but invest the total initial wealth x 70 4: Finite Economies in the assets, then the maximal expected utility is given by U(x) = sup :0A0=x EPu T AN T = sup x EPu x/N0 + AN T . The first supremum is computed over all strategies with initial investment x belonging to the collection of self-financing strategies with nonnegative value process A. The second supremum may be computed over the same set of strategies, but in view of Lemma 4.8 it remains the same if computed over the collection x of all AN -integrable, predictable processes with x/N0 + AN 0. The limitation to strategies with a nonnegative value process is meant to ensure that the expected utility does not reduce to the trivial value u(). Alternatively, we could buy one option and invest the remainder of our initial capital in the assets. If we buy the option at price y and start with an initial capital x, then the maximal expected utility is Ub(x; y, X) = sup :0A0=x-y = EPu T AN T + X NT . From the buyer's point of view an acceptable price y of the option at time 0 should satisfy Ub(x; y, X) U(x). The maximal value y for which the buyer is interested to buy the option is pb(x; X) = sup{y: Ub(x; y, X) U(x)} and is called the buyer's price. The seller of the option has to deliver the amount X at time T, but collects the price y of the option at time 0. Hence from the seller's point of view the expected utility of an initial capital x is Us(x; y, X) = sup :0A0=x+y EPu T AN T - X NT . For the seller a price y is acceptable only if Us(x; y, X) U(x). The seller's price of the option is the acceptable minimal price, given by ps(x; X) = inf{y: Us(x; y, X) U(x)}. In general buyer's and seller's prices are not the same (??). Of course, there is also no reason why the utility functions of buyer and seller would agree, although they are the same in our notation. It follows easily from the definitions that pb(x; X) = -ps(x; -X). This is what one would intuitively expect: buying is the same as selling the negative at the negated price. In general, the prices depend on the initial capital x, as indicated in the notation, but also on the real world probability measure P. The prices are consistent with the no-arbitrage prices found before. A qualitative restriction such as "admissibility" in the sense of Definition 4.15 may not achieve this. Suppose that is a self-financing strategy with 0A0 = x such that AN is an N-martingale. Then for any R the process is AN -integrable, whence by Lemma 4.8 there exists a self-financing strategy with 0A0 = x and AN = () AN . In particular AN is an N-martingale. It appears EPu(T AN T ) = EPu(x/N0 + AN T ) can be maximized to an extreme value by choosing , at least if u is unbounded. 4.6: Early Payments 71 4.42 Theorem. Assume that AN is locally bounded and that X 0. Then for any strictly increasing utility function u such that U is strictly monotone both the buyer's price pb(x; X) and the seller's price ps(x; X) are bounded above by N0V 0. Proof. By Theorem 4.39 there exists a self-financing strategy with V = AN - C for an increasing, adapted process C with C = 0. The value process A of this strategy is nonnegative and satisfies 0A0 = N0V 0 and T AN T X/NT . Given any , the process + is contained in as well and ( + )T AN T - X/NT T AN T . If 0A0 = x, then (+)0A0 = x+N0V 0. Thus for every strategy in the definition of U the strategy + is a strategy in the supremum defining Us(x; N0V 0; X) that has at least the same (expected) utility for the seller. We conclude that Us(x; N0V 0; X) U(x), so that ps(x; X) N0V 0. If 0A0 = x - y then ( + )0A0 = x - y + N0V 0 and ( + )T AN T T AN T + X/NT . We conclude that U(x - y + N0V 0) Ub(x; y; X), so that U(x) > Ub(x; y; X) for y > N0V 0 by the assumed strict monotonicity of U. This implies that pb(x; X) y, whence pb(x; X) N0V0. We can prove by an analogous argument that the prices pb(x; X) and ps(x; X) are lower bounded by N0V 0. In fact, this follows from the upper bound given in the preceding lemma applied to the claim -X and the identities pb(x; X) = -ps(x; -X), ps(x; X) = -pb(x; -X), together with the fact that -V 0 relates to -X as V 0 relates to X. Of course, we cannot assume that both X and -X are nonnegative, and hence the preceding lemma would need to be extended to more general claims. This seems to require consideration of strategies with value processes that can be negative (but are bounded below by a suitable process). 4.6 Early Payments The pricing formula (4.18) gives the value of a contract that consists of a single payment at an "expiry time" T. Many contracts include payments made during the term of the contract. We can extend the pricing formula to such contracts by replacing a payment of Y at time S < T by an equivalent amount paid at time T. Given a numeraire N the "equivalent" payment is Y NT /NS, where the factor NT /NS is typically larger than 1 and accounts for the time value of money. We also arrive at this amount if we think of the payment Y as being invested in Y/NS units of the numeraire at time S, so that it grows to the value (Y/NS)NT at time T. If this reasoning is correct, then the just price at time t for a claim consisting of a payment Y at time S is given by (4.18) applied to X = Y NT /NS, 72 4: Finite Economies i.e. NtEN Y/NS| Ft). Repeating this argument, we see that the value at time t of a contract with payments Y1, . . . , Yn at times T1 < Tn < T is given by, for t < T1, (4.43) Nt n i=1 EN Yi NTi | Ft . Thus each of the payments is discounted with the discount factor that is current at the time of payment. This reasoning is believable, but perhaps not as convincing as the arguments based on replication used to "prove" the pricing formula (4.18). In Chapter 7 we find the same formula using replication arguments, within a more general framework allowing a continuous flow of payments over an interval. * 4.7 Pricing Kernels The pricing formula (4.18) employs a change of measure to a martingale measure to express the value process of a claim. Because the martingale measure is equivalent to the original or "true" measure P, an expectation relative to it can also be viewed as a weighted expectation under the original measure. This is formalized by the concept of a pricing kernel. In this section a "numeraire pair" is understood to be a "martingale numeraire pair" and the assets are assumed continuous. 4.44 Definition. A pricing kernel Z is a strictly positive, cadlag semimartingale such that ZA is a P-martingale. If (N, N) is a numeraire pair, then Z = L/N, for L the density process of N relative to P, is a pricing kernel. This follows, because the process AN is an N-martingale if and only if the process LAN = (L/N)A is a P-martingale, by Lemma 2.14. In terms of this kernel the pricing formula (4.18) can be rewritten as Vt = Nt EP LT (X/NT )| Ft EP(LT | Ft) = 1 Zt EP ZT X| Ft . The first equality is a consequence of the formula for re-expressing a conditional expectation using a different underlying measure. This process Z is continuous if, for instance, the asset price is a weakly unique solution of the SDE. Then Ac possesses the representation property, so that the martingale L is a stochastic integral relative to Ac . Continuity appears not to be guaranteed without some sort of condition on A, even though Theorem 7.48 of Hunt and Kennedy appears to make this claim. 4.7: Pricing Kernels 73 Other properties of numeraire pairs can also be translated in pricing kernels. In particular, an economy is complete if there exists a unique pricing kernel. 4.45 Theorem. Suppose that one of the asset price processes is strictly positive. Then: (i) There exists a pricing kernel if and only if there exists a numeraire pair. (ii) A strategy is admissible if and only if (ZA) is a P-martingale for every pricing kernel Z. (iii) The economy is complete if and only if the pricing kernel is unique on [0, T] up to a multiplicative constant (and evanescence). Proof. (i). We have already argued that Z = L/N is a pricing kernel if (N, N) is a numeraire pair and L is the density process of N relative to P. Conversely, a strictly positive component N of the asset price process is always a numeraire. For a pricing kernel Z the process L = NZ, a coordinate of the process AZ, is a P-martingale. If necessary we can multiply N by a constant to ensure that its (constant) mean is equal to 1. Then we can define a probability measure N by dN = (NZ)T dP, and L is the density process of N relative to P. Because LAN = (NZ)AN = ZA is a P-martingale, it follows that AN is an N-martingale, whence (N, N) is a numeraire pair. (ii). For a given numeraire pair (N, N) there exists a pricing kernel Z with NZ = L equal to the density process L of N relative to P. Conversely, for every pricing kernel Z there exists a constant c and a numeraire pair (N, N) such that again N(cZ) = L. We can always scale the pricing kernel such that c = 1. The inverse 1/Z of a pricing kernel Z is a unit. For any self-financing strategy unit invariance gives that AN is an N-martingale if and only if AN is an N-martingale if and only if LAN = A1/Z is a P-martingale if and only if A1/Z = (ZA) is a P-martingale. (iii). Completeness is equivalent to uniqueness of the martingale measure N corresponding to a numeraire N. If Z1 and Z2 are pricing kernels, then the construction of (i) gives two numeraire pairs (N1, N) and (N2, N) with NZi = Li, for Li the density process of Ni relative to P. If N1 = N2, then it follows that L1 = L2 and hence Z1 = Z2. Conversely, if (N1, N) and (N2, N) are numeraire pairs, then Zi = Li/N are two pricing kernels (i = 1, 2). If the pricing kernel is unique, then L1 = L2 and hence N1 = N2. In the following theorem we assume that one of the coordinates of the asset price process is strictly positive, so that it can serve as a numeraire. Theorem 7.48 of Hunt and Kennedy makes the same claims without this assumption. The numeraire is constructed by switching from one nonzero asset to another. Should they not at least assume that at any time point there exists a nonzero asset? 74 4: Finite Economies Under the condition of the preceding theorem there exists only one pricing kernel in a complete economy and this has relation Z = cL/N, for a constant c, to every given numeraire pair (N, N), for L the density process of N relative to P. 5 Extended Black-Scholes Models In this chapter we consider the "standard finance model", an economy consisting of one "risk-free" asset and finitely many other assets. This model is an extension of the Black-Scholes model of Chapter 1 to more than two assets and with greater flexibility in the parameters. Existence of a numeraire pair and completeness of the economy is shown to be equivalent to existence of the "market price of risk" process. The assets are denoted by (R, S), where R is a special "risk-free" asset and S = (S(1) , . . . , S(n) ) is vector-valued. The stochastic process (R, S) is assumed to satisfy the system of stochastic differential equations, for given scalar-, n-vector- and (n × d)-matrix-valued predictable processes r, = ((i) ) and = ((i,j) ), (5.1) dRt = Rtrt dt, dS (i) t = S (i) t (i) t dt + S (i) t d j=1 (i,j) t dW (j) t , i = 1, . . . , n. We shall abbreviate the second equation to dSt = St (t dt+t dWt), where it is understood that the product of St with the vector inside brackets on the right is taken coordinatewise. The significance of factoring the drift and diffusion coefficients as the product of the asset and another process is that this allows to write the solution to the equations in the form Rt = e t 0 rs ds , St = e t 0 s ds+ t 0 s dWs- 1 2 t 0 2 s ds . This follows by the uniqueness of the exponential process as the solution The exponential function is applied coordinatewise and t 0 2 s ds is understood to be a vector-valued process with ith coordinate equal to the sum d j=1 t 0 ((i,j) s )2 ds. 76 5: Extended Black-Scholes Models of the Doléans differential equation. As a consequence, the asset processes are strictly positive. We assume that the process W = (W(1) , . . . , W(d) ) is a d-dimensional Brownian motion on a given filtered space (, F, {Ft}, P), and interpret the predictability of the processes r, and relative to the filtration Ft. The process (R, S) is also assumed adapted to the filtration Ft, but it may itself generate a smaller filtration. Thus we can study arbitrage and completeness relative to three filtrations: the "original" filtration Ft, and the augmented natural filtrations generated by (R, S), and W. Unless mentioned otherwise conditions and assertions are understood to be relative to the original filtration Ft. The "risk-free" asset R may well be a stochastic process, and it may depend on W through the function r: "risk-free" does not mean the same as "deterministic". The difference between the two types of assets is better described by the fact that the risk-free assets are of bounded variation, whereas the risky assets are directly driven by a diffusion term. Nevertheless, it is common to speak of the "risk-free" asset. For brevity we shall also refer to the asset S as the "stocks", even though the model applies equally well to other financial processes, including bonds. The risk-free asset R is a numeraire for the economy consisting of the asset price process A = (R, S), but without further conditions there does not exist a corresponding martingale measure. We shall show that the existence of a numeraire pair requires the existence of a predictable, vector-valued process = ((1) , . . . , (d) ) such that, for Lebesgue almost every t, (5.2) tt = rt1 - t, a.s.. (Here rt1 is the n-dimensional process (rt, . . . , rt).) Furthermore, the existence of such a process , called the market price of risk, together with some integrability conditions is sufficient for the existence of a martingale measure accompanying the numeraire R, and renders the economy complete relative to the natural filtration generated by (R, S). 5.1 Arbitrage The existence of the "market price of risk" process requires that the vector rt1 - t is contained in the range space of the (n × d)-matrix t, for almost every t. This is immediate if the rank of t is equal to the number n of stocks in the economy, as the range of t is all of Rn in that case. If the rank of t is smaller than the number of stocks, then existence of the market price of risk process requires a relationship between the three parameters , r and . This situation is certain to arise if the number of components 5.1: Arbitrage 77 of the driving Brownian motion is smaller than the number of risky assets, i.e. d < n. Hence we can interpret the condition of existence of a process as in the preceding display as implying that the "random inputs" W(i) to the market should be at least as numerous as the (independent) risky assets". Writing a portfolio for the asset A = (R, S) in the form (, ), we can express the value process Vt = tRt +tSt of a self-financing strategy (, ) as Vt - V0 = ( R)t + ( S)t = t 0 sRsrs ds + t 0 s dSs = t 0 Vsrs ds + t 0 s (dSs - Ssrs ds).(5.3) By the partial integration formula and the fact that dRt = rtRt dt, d V R t = -Vt 1 R2 t dRt + 1 Rt dVt = 1 Rt t (dSt - Strt dt), by the preceding display. Hence the discounted value process takes the form, in view of (5.1), (5.4) Vt Rt = V0 + t 0 Ss Rs s s dWs - (rs1 - s) ds . This formula does not make explicit reference to the amount invested in the numeraire, which has been eliminated. A "partial strategy" defines a gain process Vt - V0 through the second line of (5.3), and given we can define a process from the equation Vt = tRt + tSt. By retracing the calculations the resulting strategy (, ) can be seen to be self-financing and to possess value process Vt. Nonexistence of a market price of risk process implies that the vector rt1 - t is not contained in the range of t, for a positive set of times t. Then there exists a vector t such that (S)t is orthogonal to this range (i.e. (S)tt = 0) such that the inner product (S)t(rt1 - t) is strictly negative. We can arrange it so that the latter inner product is never positive and hence, by the preceding display, the corresponding discounted gain process will be zero at time 0 and strictly positive at time T. This suggests that nonexistence of a market price of risk process creates a potential for arbitrage. Because we can scale appropriately, this is not caused by a lack of integrability, i.e. inadmissibility. The only possibility is that there does not exist a numeraire pair. Lemma 5.5 below makes this reasoning rigorous. In the multivariate case the process S in the display is understood to be (1×n)-vectorvalued with coordinates S(i) (i) ; it is multiplied as a vector versus the (n × d)-matrix t giving a (1 × d) vector which is next multiplied with the (d × 1) vector W ; it is also multiplied with the (n × 1) vector r1 - . 78 5: Extended Black-Scholes Models On the other hand, if the market price of risk process exists, then the gains process in the preceding display can be written as a stochastic integral relative to the process ~W, for ~Wt = Wt - t 0 s ds. By Girsanov's theorem, the process ~W will be a Brownian motion after an appropriate change of measure, and hence the discounted gains process will be a local martingale. Given sufficient integrability it will be a martingale under the new underlying measure, excluding the possibility of arbitrage. The new measure will be the martingale measure corresponding to the numeraire R. 5.5 Lemma. If there exists a numeraire pair, then there exists a predictable process with values in Rd such that tt = rt1 - t for Lebesgue almost all t, almost surely. Proof. Let t be the orthogonal projection of the vector rt1 - t onto the orthocomplement of the range of t. Define a process by setting t = -(t/St)t for a given scalar, positive process t, where the quotient is interpreted coordinatewise. Then (S)tt = 0 and -(S)t(rt1 - t) = t 2 t > 0 for every t such that rt1 - t is not contained in the range of t. It can be shown that t = f(rt1 - t, t) for a measurable map f: Rn × Rnd Rn and hence the process is predictable. (Cf. Karatzas and Shreve I.6.9, p26.) The strategy can be made suitably integrable, by choice of . The process defines a discounted gains process V through (5.4), given by Vt/Rt = t 0 (1/Rs) s 2 s ds 0 with strict inequality for t = T unless rt1 - t is contained in the range of t for almost all t. The corresponding strategy (, ) is self-financing and possesses value process Vt = tRt + tSt. If there existed a numeraire paire (N, N), then by self-financing and unit invariance the process V N would be a local martingale and hence a supermartingale, because it is bounded below by 0. Because V0 = 0 and VT 0 we must have Vt = 0 with probability 1 and T 0 (1/Rs) s 2 s ds = 0 almost surely. This implies that t = 0, whence rt1 - t is in the range of t almost surely, for almost all t. 5.6 Lemma. If there exists a predictable process with values in Rd such that tt = rt1 - t for Lebesgue almost all t and E exp 1 2 T 0 s 2 ds < , then there exists a local martingale measure R corresponding to R. If moreover E exp 1 2 T 0 s + s 2 ds < , then there exists a martingale measure R as well. Proof. Because Novikov's condition is satisfied by assumption, the exponential process E( W) is a martingale, and hence the measure R defined 5.2: Completeness 79 by dR = E( W)T dP is a probability measure. By Girsanov's theorem the process ~W defined by ~Wt = Wt - t 0 s ds is a Brownian motion process under R. (See Exercise 2.21 for the vector-valued case.) The solutions R and S to (5.1) can be written in exponential form, and consequently the discounted stock process can be represented as St Rt = e t 0 (s-rs1) ds+ t 0 s dWs- 1 2 [W ]t . In view of the property (5.2) of the market price of risk , the right side is equal to, e - t 0 ss ds+ t 0 s dWs- 1 2 [W ]t = e( ~W )t- 1 2 [W ]t = E( ~W)t. It follows that the discounted stock process S/R is an R-local martingale, and hence (R, R) is a numeraire pair. Because S/R is nonnegative it is also an R-supermartingale and hence is an R-martingale if its mean is constant. The mean process can be written as ERE( ~W)t = EPE( ~W)tE( W)t = EPE ( + ) W t . The assumptions imply that Novikov's condition is satisfied for the exponential process in the right side, and hence the process E ( + ) W is a P-martingale, whence its mean function is constant. We conclude that S/R is an R-martingale. 5.2 Completeness In this section we assume the existence of the market price of risk process and the integrability conditions of Lemma 5.6, so that there exists a martingale measure to the numeraire R. We consider the completeness of the economy both relative to the augmented filtration generated by the asset processes, and relative to the augmented filtration generated by the driving Brownian motion. We assume that the first filtration is right-continuous (as is the second filtration automatically). 5.7 Theorem. Suppose that the conditions of Lemma 5.6 hold. If the stochastic differential equation (5.1) is given by processes rt = r(t, Rt, St), t = (t, Rt, St) and t = (t, Rt, St), for measurable functions r, and , and possesses a weakly unique solution, then there exists a numeraire pair and the economy is complete relative to the augmented natural filtration generated by the process (R, S). Proof. By Lemma 5.6 there exists a numeraire pair. Because the process R is a numeraire of bounded variation, the completeness of the economy 80 5: Extended Black-Scholes Models follows from Theorem 4.31 upon noting that the martingale part (0, S W) of the asset process possesses the representing property by Theorem 3.14. 5.8 EXERCISE. Show that a complete "extended Black-Scholes" economy can have at most one risk-free asset: if the ith row of in (5.1) is identically zero, then S(i) = S (i) 0 R. 5.9 EXERCISE. Consider the economy consisting of a risk-free asset Rt = ert and two additional assets S(1) and S(2) such that dS (i) t = S (i) t ( (i) t dt + (i) t dWt) for a one-dimensional Brownian motion W. For what parameter values is this economy complete? 5.10 EXERCISE. Suppose that in Theorem 5.7 we replace the condition that the stochastic differential equation (5.1) possesses a weakly unique solution by the assumption that the equation dSt = St(rt1 dt + t d ~Wt) possesses a weakly unique solution, for ~Wt = Wt - t 0 s ds. Is the theorem still valid? The natural filtration generated by the asset processes, as used in the preceding theorem, is probably the most natural filtration for use in connection with completeness. An alternative, used by many authors, is the augmented filtration FW t generated by the driving Brownian motion. Because this may be bigger than the natural filtration of the asset price processes, completeness relative to FW t may be more restrictive. The following theorem requires the existence of a market price of risk process, and also that the number of risky assets is not smaller than the number of driving Brownian motions. In contrast (by Theorem 5.7) relative to the filtration generated by the assets the market is typically complete as soon as the market price of risk process exists, which roughly requires that the number of risky assets is no larger than the number of driving Brownian motions. The difference between the two set-ups is seen easily by adding an additional "driving" Brownian motion to the model, but letting it not drive anything, by setting the corresponding column of equal to zero. It will be impossible to replicate claims that are a function of this extra Brownian motion using a portfolio consisting only of the asset price processes. If this were possible, then the extra Brownian could be written as a stochastic integral relative to the other Brownian motions, which is not possible. 5.11 Theorem. Suppose that the conditions of Lemma 5.6 hold. If the number of stocks is equal to the dimension of the Brownian motion W and the process takes its values in the invertible matrices, then there exists a numeraire pair and the economy is complete relative to the augmented natural filtration FW t generated by W. Proof. By Lemma 5.6 there exists a numeraire pair. Because the process R is a numeraire of bounded variation, the completeness of the economy 5.3: Partial Differential Equations 81 follows from Theorem 4.31 upon noting that the martingale part (0, S W) of the asset process possesses the representing property by Lemma 3.9. 5.3 Partial Differential Equations Under the conditions of Theorem 5.7, the process ~W defined by ~Wt = Wt - t 0 s ds is a Brownian motion under the martingale measure R corresponding to the numeraire R. Because option prices can be written as expectations under R, it is useful to rewrite the system of stochastic differential equations (5.1) in terms of the process ~W. If we also assume that the processess r and take the forms rt = r(t, Rt, St) and t = (t, Rt, St), then the equations take the form (5.12) dRt = Rt r(t, Rt, St) dt, dSt = St r(t, Rt, St)1 dt + St (t, Rt, St) d ~Wt. As usual we assume that (R, S) is adapted to the augmented natural filtration F ~W t of ~W. Then, under regularity conditions on r and , the process (R, S) will be Markovian relative to this filtration. If we assume in addition that is invertible, then ~W can be expressed in (R, S) by inverting the second equation, and hence the filtrations Ft and F ~W t generated by (R, S) and ~W are the same. The process (R, S) is then Markovian relative to its own filtration Ft. In that case a conditional expectation of the type ER(X| Ft) of a random variable X that is a measurable function of (Rs, Ss)st can be written as F(t, Rt, St) for a measurable function F. This observation can be used to characterize the value processes of certain options through a partial differential equation. The value process of a claim that is a function X = g(ST ) of the final value ST of the stocks takes the form Vt = RtER g(ST ) RT | Ft = ER e - T t r(s,Rs,Ss) ds g(ST )| Ft . If the process (R, S) is Markovian as in the preceding paragraph, then we can write Vt = F(t, Rt, St) for a measurable function F. We assume that this function possesses continuous partial derivatives up to the second order. For simplicity of notation we also assume that S is one-dimensional. Then, by Itô's formula, dVt = Ft dt + Fr dRt + Fs dSt + 1 2 Fss d[S]t. Here Ft, Fr, Fs are the first order partial derivatives of F relative to its three arguments, Fss is the second order partial derivative relative to its 82 5: Extended Black-Scholes Models third argument, and for brevity we have left off the argument (t, Rt, St) of these functions. (Beware of the different meaning of the subscript t in expressions such as Ft, Rt or St!) A second application of Itô's formula and substitution of the diffusion equation for (R, S) yields d Vt Rt = 1 Rt -Fr + Ft + FrRtr + FsStr + 1 2 FssS2 t 2 dt + 1 Rt FsSt d ~Wt. The process Vt/Rt is the discounted value process of an admissible, selffinancing strategy (replicating the claim X = g(ST )) and hence is an Rmartingale. Because the process ~W is a Brownian motion, this can only be true if the drift term on the right side of the preceding display is zero, i.e. - (Fr)(t, r, s) + Ft(t, r, s) + (rFr)(t, r, s)r + (rFs)(t, r, s)s + 1 2 (2 Fss)(t, r, s)s2 = 0. (There is an unfortunate double use of the symbol r in this formula, both for the function r and for one of the three arguments!) This is the BlackScholes partial differential equation. It was obtained by Black and Scholes in 1973 within the context of the simple Black-Scholes model of Chapter 1, strangely enough by a reasoning that appears not quite correct from today's standpoint. The preceding derivation is closely related to derivation of the Feynman-Kac formula, which, however, goes in the opposite direction. This formula expresses the value of a given function that satisfies a certain partial differential equation as the expectation of a certain function of a Brownian motion. (Cf. e.g. Karatzas and Shreve, Chapter 4.4.) The Black-Scholes partial differential equation is useful for the numerical computation of option prices. Even though the equation is rarely explicitly solvable, a variety of numerical methods permit to approximate the solution F. The equation depends only on the functions r and defining the stochastic differential equation (5.12). Hence it is the same for every option with a claim of the type X = g(ST ), the form of the claim only coming in to determine the boundary condition. Because X = g(ST ) = F(T, RT , ST ), this takes the form F(T, r, s) = g(s). For instance, for a European call option on the stock S, this becomes F(T, r, s) = (s - K)+ . The preceding approach is possible only if the value process V is a smooth function Vt = F(t, Rt, St) of time and the underlying assets. The regularity of the value process depends on the functions r and , determining the evolution of the assets. In the simple Black-Scholes model where these functions are constants the smoothness can be verified. In that case, the conditional distribution of the variable ST = Ste( ~WT - ~Wt)- 1 2 2 (T -t) 5.3: Partial Differential Equations 83 given Ft is log normal, and we can write the value process Vt = RtER(X/RT | Ft) for the claim X = g(ST ) in the form Vt = F(t, St), for the function, with Z a standard normal variable, F(t, s) = e-r(T -t) Eg se T -t Z- 1 2 2 (T -t) . 5.13 EXERCISE. Verify that the function F is infinitely smooth in both t and s. 6 American Options An American option, as opposed to a European option, is a contract derived from a claim process X, which we take to be a cadlag semimartingale. The contract entitles the holder to a cash payment X at a time chosen by the holder of the contract, restricted to be a stopping time with values in an [0, T], which is fixed in the contract. A standard example is an American option on a stock S, which has claim process X = (S - K)+ . Naturally, the holder of the contract tries to maximize the payment X by a clever choice of the stopping time . The value of the contract turns out to be the solution to an optimal stopping problem. However, as for European options, the problem must be formulated relative to a martingale measure, not the measure of the true world. This surprising fact is again the result of a no-arbitrage argument. We shall first express the value of an American option in the value of a "replicating portfolio" through a no-arbitrage argument, and next relate this value to an optimal stopping problem. We work in the finite economy model described in Chapter 4. 6.1 Replicating Strategies Suppose that there exists a self-financing strategy with A X throughout [0, T] and A = X for some stopping time with values in [0, T]. Then the fair price of the American contract X at time 0 is 0A0. We corroborate this by an economic argument. Because A X and we can sell our portfolio at any given stopping time, just as we are allowed to cash X at any stopping time, it is obvious that we should prefer the portfolio over the contract X. This shows that the American contract X is not worth more than 0A0 at time 0. Conversely, if the value of the option were strictly less than 0A0, then 6.1: Replicating Strategies 85 we could create a positive cash flow at time 0 by buying the American contract X and selling the portfolio 0. Until the time we could reshuffle the portfolio -0 according to the strategy -, which can be achieved free of money input. At time we could cash the American claim X , and buy the portfolio . Because A = X by assumption, this would leave us with no money and no obligations, except for the money cashed at time 0. Assuming that it is impossible to create certain profit, we conclude that the initial assumption is wrong and hence the just price of the American claim at time 0 is 0A0. We can repeat this argument to find the value at other times t [0, T]. If there exists a self-financing portfolio such that A X on [t, T] and A = X for some stopping time taking its values in [t, T], then the just price of the American claim X is equal to tAt. Here the replicating portfolio is permitted to (and will typically) depend on t. The preceding no-arbitrage argument is not a proof of a mathematical theorem, but it appears to be convincing. One serious attack on the validity of this reasoning is that the strategies that it is based on may not be unique. This challenge to the argument is solved if we insist that the replicating portfolio be admissible, in view of the following lemma, which shows that the value derived from a replicating portfolio is unique. 6.1 Lemma (Unique value). If and are self-financing admissible strategies with A X and A X on [t, T] and such that A = X and A = X for stopping times and that take their values in [t, T], then tAt = tAt. Proof. The existence of admissible strategies implies the existence of a numeraire pair (N, N). By the definition of admissibility, self-financing and unit invariance, the processes AN and AN are N-martingales. By the optional stopping theorem applied to the identity (AN ) = XN , we see that (AN ) = EN XN | F . By assumption the right side is bounded above by EN (AN ) | F = (AN ) , by the optional stopping theorem. Thus (AN ) (AN ) , and by symmetry also equality. This shows that the N-martingales AN and AN are equal at . By optional stopping they are the same on the interval [0, ], which includes the point t. We shall call a replicating strategy from t for the claim process X an admissible, self-financing strategy such that A X on [t, T] and A = X for some stopping time taking its values in [t, T]. Our next aim is to give a more concrete expression of the value of the American claim through the solution of an optimal stopping problem. We first recall some generalities on optimal stopping. Let T be the set of all stopping times with values in [0, T]. 86 6: American Options 6.2 Optimal Stopping Suppose that X is a nonnegative, cadlag adapted process, indexed by [0, T], on a given filtered space (, F, {Ft}, P) such that (6.2) E sup 0tT Xt < . The optimal stopping problem is concerned with finding the stopping time that maximizes EX over all stopping times with values in [0, T], and finding the value of the maximum. If the time set were a finite set 0 < t1 < t2 < < tn < T, rather than a continuum, then the solution of this problem could be found easily through backwards programming. This consists of recursively optimizing over all stopping times with values in {tn}, {tn-1, tn}, . . .. The first step is to decide for any possible realization of Xtn whether it is better to stop at tn (and collect Xtn ) or to continue (and collect XT ). Given the best strategy from time tn onwards, encoded in a stopping time n with values in {tn, T}, the second step is to decide at time tn-1 whether to stop (and collect Xtn-1 ) or to continue, in which case we would follow the optimal strategy n. This scheme continues backwards in time to time t1. Section 6.4 describes the method in more detail. The essential object to formalize the algorithm is the "Snell envelope", whose value at k gives the optimal (conditional) expectation when the stopping times are restricted to {tk, . . . , tn}. The continuous time solution is similar, but technically much more complicated. The Snell envelope is the smallest cadlag supermartingale Z with Z X, where "smallest" can be understood in that Z Z up to evanescence for every other supermartingale Z with Z X. There exists a version of this supermartingale such that, for every t, (6.3) Zt = ess sup E(X | Ft): t, T , where the supremum is taken over all stopping times with values in [t, T]. (Equation (6.3) is even true for t a stopping time.) We can view the variable in the display as the optimal expected payoff X that can be achieved by stopping the payoff process in [t, T], seen from the time t point of view. In particular, if F0 is the trivial -field, then Z0 is the maximal value of EX over all stopping times . If X is continuous, then there exists an optimal stopping time, given by the first time that the Snell envelope Z and the process X coincide. More generally, we have Zt = E(Xt | Ft) for t the stopping time t = inf{s t: Zs = Xs}. Thus t is the optimal stopping time restricted to [t, T] evaluated from the perspective at time t, conditionally on the past. By (right) continuity Is it necessary to assume that F0 is trivial? 6.3: Pricing and Completeness 87 Zt = Xt and Zs > Xs on [t, t). Thus an optimal time to stop in [t, T] is the first time in this interval that the Snell envelope and the process X coincide. The Snell envelope can be shown to be of class D and hence possesses a Doob-Meyer decomposition Z = M - for a uniformly integrable martingale M and a nondecreasing, cadlag, adapted process with 0 = 0 and ET < . It will be important to know that t = t. We collect some relevant properties in the following proposition. 6.4 Proposition. Let X be a continuous adapted, nonnegative process with E sup0tT Xt < . Then there exists a cadlag supermartingale Z of class D satisfying (6.3) and Zt = E(Xt | Ft) for t = inf{s t: Zs = Xs}. The process Z can be decomposed as Z = M - for a uniformly integrable martingale M and a continuous, nondecreasing process satisfying t = t. For each t the process Z restricted to [t, T] is the smallest supermartingale with Z X. Proof. See e.g. Karatzas and Shreve, Appendix D. 6.3 Pricing and Completeness We apply the optimal stopping theory to the rebased process XN for a given numeraire pair (N, N), and X the claim process of an American option. We assume that the claim process X is nonnegative. The following theorem shows that the fair price process of the option is given by the Snell envelope of the process XN on the filtered probability space (, F, {Ft}, N). At least this is true if the "fair price" can be defined through a replicating strategy, as previously. (Otherwise, we do not have a notion of "fair price".) If the market is complete (in the sense of Chapter 4), then this is true for every claim such that EN sup0tT (Xt/Nt) < . 6.5 Theorem. If is a replicating strategy for t for the continuous, nonnegative claim process X, then, for any numeraire pair (N, N) such that EN supt(Xt/Nt) < , tAt = Nt ess sup EN X N | Ft : t, T . 6.6 Theorem (Completeness). If the market is complete with numeraire pair (N, N), then for every continuous, nonnegative claim process X such that EN supt(Xt/Nt) < , there exists for every t [0, T] a self-financing, admissible strategy such that A X on [t, T] and A = X for some stopping time with values in [t, T]. 88 6: American Options Proofs. Let Z be the Snell envelope of the process XN = X/N on the filtered probability space (, F, {Ft}, N), and let Z = M - be its DoobMeyer decomposition. Let be a replicating strategy as in the first theorem. Because AN is an N-martingale with AN XN on [t, T], and the restriction of Z to [t, T] is the minimal N-super martingale with Z XN on [t, T], it follows that AN Z on this interval. On the other hand, because is a replicating strategy, there exists a stopping time with values in [t, T], such that AN = XN Z , whence by the optimal stopping theorem, tAN t = EN AN | Ft EN(Z | Ft) Zt, by the supermartingale property of Z and the optional stopping theorem. By combining we see that tAN t = Zt, which is the assertion of the first theorem. By completeness there exists a replicating strategy for the claim NT (MT - t), i.e. an admissible, self-financing strategy with NT (MT t) = T AT , or, equivalently, MT - t = T AN T . The left- and right sides of this equation are the values at T of N-martingales on [t, T]. Taking conditional expectations, we conclude that M - t = AN on the interval [t, T]. This shows that AN = Z + - t Z XN on [t, T], because is nondecreasing, whence A X on [t, T]. Furthermore, for t = inf{s t: Zs = XN s } we have that (AN )t = Zt + t - t = Zt = XN t , by Proposition 6.4. This shows that (A)t = Xt . The American option with claim process X is worth more than the European option with claim XT , as the latter can be viewed as the American contract with the restriction that early stopping is not allowed. The difference in value is Nt ess sup EN X N | Ft : t, T - EN XT NT | Ft . This is strictly positive in general. A case of interest where the values are the same is when the process X/N is an N-submartingale. Then, by optional stopping, EN X /N | Ft) EN XT /NT | Ft) almost surely for every stopping time t, and hence early stopping yields no advantage. (On the other hand, it need not be detrimental, because an optimal stopping time need not be unique, and hence early stopping may be preferable for other reasons.) 6.7 Example (American call option). The claim process of an American call option on an asset S is given by X = (S - K)+ , for K a constant fixed in the contract, referred to as the strike price. The claim value XT = (ST - K)+ at expiry time is the value of a European call option on S. For American call options stopping is typically not helpful and the value of an American call option is the same as the value of a European call option. This is the case if there exists a numeraire pair (N, N) with N a martingale measure and such that 1/N is an N-supermartingale. In 6.4: Optimal Stopping in Discrete Time 89 particular, this is true if N is nondecreasing, as is the case for instance for the standard numeraire in the Black-Scholes model. To see this it suffices to show that the rebased claim process X/N is an N-submartingale. By Jensen's inequality, for s < t, EN (St - K)+ Nt | Fs EN St - K Nt | Fs + = Ss Ns - KEN 1 Nt | Fs + , because S/N is an N-martingale. If 1/N is a supermartingale, then this can be further bounded below by (Ss - K)+ /Ns. 6.8 EXERCISE. Suppose that the economy is complete. Show that if (N, N) is a numeraire pair such that 1/N is an N-supermartingale, then 1/M is a M-supermartingale for every numeraire pair (M, M). 6.9 EXERCISE. Express the difference in value between the American and European options as Nt E(T | Ft) - t for the nondecreasing process in the Doob-Meyer decomposition Z = M - of the Snell envelope Z of XN on (, F, {Ft}, N). If the prices of European and American options do not agree, then the computation of the value of an American option may not be easy, and explicit formulas are rarely (or never) available. Typically the optimal stopping problem can be rewritten as a variational problem, but this then needs to be solved numerically. An alternative is to discretize time and compute the Snell envelope of te discretized claim process, possibly by stochastic simulation. 6.4 Optimal Stopping in Discrete Time For an intuitive understanding (and possibly also for numerical implementation) it is helpful to consider the optimal stopping problem in discrete time. Given a time set 0 < t1 < t2 < < tN = T and integrable random variables X1, . . . , XN adapted to a filtration F1 F2 FN , we wish to compute the supremum supT EX of EX over the set T of all stopping times with values in {t1, . . . , tN }, and also the suprema supT :k E(X | Fk) for stopping after tk. The optimal stopping time is computed backwards in time, starting at time tN . If we do not stop at times t1 < < tN-1, then we must stop at time tN , resulting in a payment XN . We encode this by defining a random variable and stopping time by ZN = XN , N = N. 90 6: American Options Next consider an optimal strategy if we stop either at time N -1 or at time N. If we stop at time N - 1, then we receive payment XN-1, whereas if we do not stop we receive payment ZN , whose expected value from time N -1 perspective is E(ZN | FN-1). To maximize the expected payment from the time N - 1 perspective we thus decide to stop if XN-1 E(ZN | FN-1) and to continue otherwise. This gives expected payment from time N - 1 perspective equal to XN-1 E(ZN | FN-1). We encode this optimal strategy in the random variable and stopping time given by ZN-1 = XN-1 E(ZN | FN-1), N-1 = N - 1 if ZN-1 = XN-1 N if ZN-1 > XN-1 . Next we proceed to time N -2. If we do not stop at this time, then the best strategy is to play the optimal strategy N-1 from time N - 1 onwards, which has expected pay-off E(ZN-1| FN-2) from the time N-2 perspective. Thus we decide to stop if XN-2 E(ZN-1| FN-2), giving the optimal expected payment XN-2 E(ZN-1| FN-2) from the time N -2 perspective. We repeat this argument down to time 1. We record the strategy at time k in the random variable and stopping time given by Zk = Xk E(Zk+1| Fk), k = k if Zk = Xk k+1 if Zk > Xk . The gain of stopping at time k rather than continuing to time k + 1 is equal to Xk - E(Zk+1| Fk) if this variable is positive, and there is not gain otherwise. We record this in the jump of a stochastic process , as k+1 = Xk - E(Zk+1| Fk) + . The nondecreasing process is then equal to k = k j=2 j. Thus we have defined a discrete time stochastic process Z = (Z1, . . . , ZN ), a sequence of stopping times k, and a nondecreasing process . The variable Zk gives the value of the optimal stopping problem from the perspective at time tk, and the time k is the optimal stopping time for stopping after time tk. These stopping times have a nice interpretation as the first time after tk that the processes Z and X coincide. 6.10 Theorem. (i) Z is the smallest supermartingale with Z X. (ii) Zk = E(Xk | Fk) = supT :tkk E(X | Fk) for every k. (iii) k = min{tj: j k, Xj = Zj} for every k. (iv) Z = M - for a martingale M. (v) k = k for every k. Proof. Item (i) is immediate from the definition of Z. 6.4: Optimal Stopping in Discrete Time 91 Item (ii) is proved by backward induction on k. The statement is clearly true for k = N. For k < N the definitions give E(Xk | Fk) = E Xk1XkE(Zk+1|Fk) + Xk+1 1Xk k implies that Xk = Zk and k = k+1. By the induction hypothesis k+1 = min{tj: j k + 1, Xj = Zj}, which is the same as min{tj: j k, Xj = Zj} if Xk = Zk. The process M in (iv) clearly must be defined as Z + , and hence it suffices to show that E(Zk + k| Fk-1) = 0 for every k. But by the definitions Zk + k is equal to Zk - Zk-1 + 0 = Zk - E(Zk| Fk-1) + 0 if Xk-1 < E(Zk| Fk-1), and it is equal to Zk - Zk-1 + Xk-1 - E(Zk| Fk-1) = Zk - E(Zk| Fk-1) in the other case. Hence it is equal to Zk - E(Zk| Fk-1) in both cases and clearly a martingale increment. The definition of k shows that k = 0 if k-1 = k. Clearly j = k for any j with k < j k. Therefore 0 = k j=k+1 j = k - k, proving (v). The supermartingale property of Z expresses the fact that stopping later yields lower expected gain. The decomposition Z = M - makes the (expected) decrease of the supermartingale Z visible through the decreasing sample paths of the process -. Property (v) could be rephrased as saying that Z follows the martingale M on any interval (k, k] (which may well be empty of course): from time k onwards we stop only at k; the gains j of stopping earlier are zero. The process is predictable, and the decomposition Z = M - is the Doob decomposition of Z, the discrete time equivalent of the Doob-Meyer decomposition. 7 Payment Processes , In Chapter 4 we found the fair price of a contract that yields a single payoff X at an "expiry time" T. Several financial instruments yield payments at multiple times during an interval [0, T]. In this chapter we extend the pricing formula to general payment processes. We shall obtain this extension in a more general framework, allowing the asset processes A to be general cadlag semimartingales, possibly discontinuous. Throughout we assume that A = (A(1) , . . . , A(n) ) is a vector of semimartingales defined on a given filtered probability space (, F, {Ft}, P) that satisfies the usual conditions. As before, a strategy is a predictable process such that the stochastic integral A is well defined, and a numeraire is a strictly positive semimartingale N such that N = 0A0 + A = A for some (self-financing) strategy . In the present chapter a numeraire is a cadlag process that may have jumps. A payment process is defined to be a predictable semimartingale X. The value Xt at time t model is interpreted as the cumulative payments on a contract up to and including time t, and could be viewed as the sum (or integral) of a series of payments of sizes dXt to the holder of the contract. (If X is not of bounded variation, then the latter intuitive interpretation should not be taken too seriously.) In practice a contract often consists of an agreement of a sequence of payments at finitely may predetermined times, whose values depend on the history of the asset process up to the time of payment. Thus the most interesting paymenst processes are of bounded variation, with finitely many jumps. A replicating strategy for X is a predictable process with 0A0 + A = A + X, T AT = 0. This chapter has not been changed in the November 2005 version. It may not be fully consistent with the preceding material. 7: Payment Processes 93 As before, the strategy is interpreted as the contents of an investment portfolio. At time t the value of this portfolio changes by an amount t dAt due to the movement of the asset process. This change in value is used both to make a payment of size dXt and to finance a change d(A)t in the value of the portfolio. We are allowed to reshuffle the contents of the portfolio, as long as we fulfill these financing requirements. At expiry time T the portfolio is liquidated (i.e. T AT = 0) and a final payment of XT is made to the holder of the contract. The requirement that a payment process be predictable can be interpreted as saying that the size of a payment is known just before the time that the payment is made. If the asset process A is continuous, then the filtration FA t is left-continuous and hence this does not mean much as "just before time t" is the same as "at time t". In any case it will be seen below that any payment process that can be replicated is necessarily predictable. 7.1 Example. For given deterministic times 0 < T1 < T2 < < Tn = T and FA Ti--measurable random variables Yi, the process X = n i=1Yi1[Ti,T ] is a payment process. It corresponds to the series of payments Y1, Y2, . . . , Yn made at the times T1, T2, . . . , Tn. To verify the claim it suffices to show that the process X is predictable, or equivalently that a process of the type Y 1[S,T ] is predictable for every Y FA S-. For Y an indicator function Y = 1F of a set F FA s for some s < S, this follows because 1F 1[S,T ] is the limit as n of the left-continuous processes 1F 1(S-n-1 0, and hence in principle uncountably many assets. In this chapter we extend the pricing theory of Chapter 4 to economies with infinitely many assets. The extension is only modest in that the definitions are chosen such that the set-up essentially reduces to that of finite economies. We write the asset processes as A = (Ai : i I), where I is an arbitrary index set, and each Ai is a continuous semimartingale on a given filtered probability space (, F, {Ft}, P). We refer to the family of processess A as the "economy E". For a subset I I we let EI be the "sub-economy" consisting of the family of asset processes AI = (Ai: i I), defined on the same filtered probability space (, F, {Ft}, P). For a finite subset I I the economy EI is a finite economy of the type considered in Chapter 4. We are interested in pricing derivatives with a finite time horizon T > 0, and hence the asset processes are important on the time interval [0, T] only. A "strategy" in the economy E should be a certain family (i : i I) of predictable processes, with the interpretation that at time t we keep i t assets of type Ai in our portfolio. We greatly simplify the set-up by allowing each agent in the economy to trade in only finitely many assets, and define the collection of all strategies in E as the union of all strategies available in the subeconomies EI with I ranging over the finite subsets of I. Thus each agent may trade in finite, but arbitrarily many assets throughout [0, T]. For a family of predictable processes (i : i I) and a subset I I set I = (i : i I). This chapter has not been changed in the November 2005 version. It may not be fully consistent with the preceding material. 98 8: Infinite Economies 8.1 Definition. A strategy in E is a family (i : i I) of predictable processes such that for some finite subset I I: (i) i = 0 for every i / I. (ii) I = (i : i I) is a strategy in EI . The strategy is said to be self-financing if I is self-financing in EI . The value process of is the value process of I in EI . 8.2 EXERCISE. Verify that the preceding definitions of self-financing and value process are well posed, in that they do not depend on the choice of finite subset I. Also verify that if I is a strategy in EI and we set i = 0 for i / I, then J is a strategy in EJ for every finite set J I. [Hint: I AI = J AJ and I AI = J AJ whenever I and J are subsets of I that differ only by i I such that i = 0.] By the preceding definition (and exercise) the value process of a selffinancing strategy is the process I AI = I 0AI 0 + I AI for every finite subset I I such that i = 0 for every i / I. We shall write the value process also as A = 0A0 + A. We define a numeraire exactly as in Chapter 4 as a strictly positive semimartingale that is the value process of some self-financing strategy. Then, in view of the preceding definition, every numeraire in E is also a numeraire in some finite subeconomy EI . 8.3 Definition. (i) A numeraire is a strictly positive semimartingale that is the value process of some self-financing strategy. (ii) A numeraire pair is a pair (N, N) consisting of a numeraire N and a probability measure N on (, F) that is equivalent to P and such that the process t Ai t/Nt is an N-martingale for every i I. (iii) A pricing process Z is a strictly positive, cadlag semimartingale such that ZAi is a P-martingale for every i I. 8.4 Definition. A strategy is admissible if for every numeraire pair (N, N) the process (A/N) is an N-martingale. By notational convention the "stochastic integral" (A/N) in the definition of admissibility is the integral I (AI /N) for an (arbitrary) finite subset I I such that i = 0 for i / I. This might suggest that admissibility might be the same as admissibility in finite subeconomies EI . This appears to be false, because the definition only requires the process (A/N) to be an N-martingale for numeraire pairs (N, N) in E, and not for every numeraire pair in the finite subexperiment EI . Not every numeraire pair in a finite subexperiment need be "extendible" to a numeraire pair in the infinite experiment E. This subtlety makes the following theorems, which are otherwise straigthforward extensions of results for finite economies, worth the effort. 8: Infinite Economies 99 ** 8.5 EXERCISE. Investigate this point. 8.6 Theorem (No arbitrage). If there exists a numeraire pair (N, N) in E and and are self-financing, admissible strategies with T AT = T AT , then A = A on [0, T]. Proof. There exists a finite subset I I such that i = i = 0 for every i / I. By assumption and are self-financing in E, and this implies that I and I are self-financing in EI . By admissibility (A/N) = I (AI /N) = I 0(AI 0/N) + I (AI /N) is an N-martingale, and hence it is completely determined by its final value T (AT /NT ), and similarly for . Because the final values are the same, we have (A/N) = (A/N), and hence A = A. As for finite economies the preceding theorem justifies to define the "fair price" of a European option with claim X at T to be the value of a self-financing, admissible strategy such that X = T AT , if there exists such a strategy. This value can be expressed as an expectation under a numeraire pair (N, N) or pricing process Z, as tAt = NtEN X NT | Ft = 1 Zt EP XZT | Ft . These formulas are identical to the formulas for finite economies, and are immediate from the martingale properties of the processes (A/N) under N, or ZA under P. We silently assume that the variables X/NT and XZT are integrable under N and P, respectively. We shall show that this formula is available for every sufficiently integrable claim X FT as soon as there exists a finite sub-economy that is complete. The following definitions and results all refer to a fixed time horizon T > 0. 8.7 Definition. The economy E is complete if there exists a numeraire pair (N, N) such that for FT -measurable random variable X with EN|X/NT | < there exists a self-financing, admissible strategy with X = T AT . 8.8 Theorem (Completeness). If there exists a numeraire pair (N, N) in E and there exists a finite subset I I such that EI is complete and contains a strictly positive asset process, then E is complete. Proof. We first show that the completeness of EI implies the completeness of EJ for every finite set J I with J I such that N is a numeraire in EJ . Indeed, under these conditions the pair (N, N) is a numeraire pair in EJ and EJ contains a strictly positive asset process. If Z1 and Z2 are pricing processes in EJ , then they are also pricing processes in EI . By completeness of EI and the fact that EI contains a strictly positive asset process, it follows 100 8: Infinite Economies that Z1 = Z2. Thus the pricing process in EJ is unique, whence the economy EJ is complete. By definition a numeraire is a value process of some strategy, and hence there exists a finite subset J I such that N is a numeraire in EJ . Without loss of generality we can choose J I. Then (N, N) is a numeraire pair in the experiment EJ and by the preceding EJ is complete. Therefore, for every X with EN X/NT < there exists a self-financing, admissible strategy J in EJ such that X = J T AJ T . If we set i = 0 for i / J, then is a self-financing strategy in E. The strategy is also admissible in E, but this requires proof. The admissibility of J in EJ guarantees that J (AJ /NJ ) is an NJ -martingale for every numeraire pair (NJ , NJ ) in EJ , but it must be verified that (A/M) is an M-martingale for every numeraire pair (M, M) in E. For every such pair (M, M) there exists a finite subset H I with H J such that M is a numeraire in EH . By the first paragraph of the proof it follows that EH is complete. By the admissibility of J in EJ , the process H (AH /N) = J (AJ /N) is an N-martingale. Thus strategy H is a self-financing strategy in EH such that H (AH /N) is an N-martingale. By Lemma 4.25 the process H (AH /M) = (A/M) is an M-martingale. The preceding theorem is limited, as it is only usable if the infinite economy is not richer than some finite subeconomy. However, it is good enough for most applications. We finish this chapter with some results on pricing processes. 8.9 Lemma. Suppose that there exists a strictly positive asset process in E. Then there exists a pricing process in E if and only if there exists a numeraire pair in E. Proof. If Z is a pricing process and N is a strictly positive asset process in E, then L = ZN/EP(NZ)T is a nonnegative P-martingale with mean 1. Hence dN = LT dP defines a probability measure with density process L relative to P. Because L(Ai /N) = ZAi /EP(NZ)T is a P-martingale, the process Ai /N is an N-martingale, for every i I. Conversely, if (N, N) is a numeraire pair in E and L is the density process of N relative to P, then Z = L/N is a pricing process in E. 8.10 Lemma. If there exists a numeraire pair in E and there exists a finite subset I I such that EI is complete and contains a strictly positive asset process, then E possesses a unique pricing process. Proof. There exists a pricing process in E by the preceding lemma. If Z1 and Z2 are both pricing processes in E, then they are also pricing processes in EI . Because EI is complete and contains a positive asset process, its pricing process is unique, and hence Z1 = Z2. 8: Infinite Economies 101 ** 8.11 EXERCISE. Investigate whether the existence of a unique pricing process in E implies the completeness of E. 9 Term Structures A zero coupon bond, also known as a pure discount bond, is a contract that guarantees a payment of 1 unit at a given time T in the future. The bond is said to "mature" at the maturity time T. In a real market we can typically buy bonds of many different maturities at any given date. Even though in any finite interval only finitely many bonds may be "active", the "maturity periods" of the different bonds overlap, and the totality of these contracts is most naturally modelled on an infinite time horizon [0, ). We denote by Dt,T the value at t of a zero coupon bond maturing at time T. Obviously, the bond will be worthless after time T and hence we may either set Dt,T equal to zero for t > 0, or think of t Dt,T as a process defined on [0, T] only. Because Dt,T models the value at time t < T of one unit to be received at time T, it is also called a discount rate, expressing the "time value of money". A mathematical model for the discount rates Dt,T is called a term structure model. We always assume that each process (Dt,T : 0 t T) is a nonnegative, cadlag semimartingale defined on a given filtered probability space (, F, {Ft}, P), with DT,T = 1. The requirement that the process Dt,T be adapted models the fact that the discount bonds are for sale at time t, whence their prices are known at that time. However, the true meaning depends crucially on the choice of filtration, which we have not fixed. In this chapter we consider a number of standard models for specifying a complete term structure. In practice one is often interested in special derivatives of the discount rates, and a full model for all discount rates may be unnecessary or even undesired. Ad-hoc models for the entities that are relevant for the particular derivative may suffice, as long as it is clear that these are compatible with some term structure, and consistent with other ad-hoc models. We shall see examples of such partial models in later 9.1: Short and Forward Rates 103 chapters. In most models the discount rates Dt,T depend smoothly on the maturity time T, but are of unbounded variation as a function of the time parameter t. This models the fact that at every given time the interest attainable for the period [t, T] should not vary much with T, but will be highly sensitive to changes on the market, caused for instance by the arrival new information. 9.1 Short and Forward Rates Discount rates have close ties to "interest rates" in a wide sense. If at every moment in time it would be possible to obtain an arbitrage-free, continuously compounded, fixed interest of rate r > 0 on an investment, then the only reasonable term structure model would be Dt,T = e-r(T -t) . This is the value of one unit discounted for the fact that the bond has cash value (equal to one) at time T and not at time t. This example shows that - log Dt,T is a natural transformation of the discount rates. The process Yt,T = - log Dt,T T - t is called the yield, and can be viewed as a fixed interest rate over the period [t, T], contracted at time t. Buying a bond with yield Yt,T at time t is equivalent to putting money in a savings account for the period [t, T] against the fixed (continuously compounded) rate Yt,T . In particular, if Dt,T = e-r(T -t) , then the yield is constant and equal to r. The yield T Yt,T as a function of T is known as the yield curve at time t. Typically, the yield curve is increasing, more distant maturities giving higher returns, but this is not necessarily the case, "inverted yield curves" having been observed also. For a general term structure model, the short rate is defined as the limit, if it exists, (9.1) rt = - lim h0 1 h log Dt,t+h. Equivalently, the short rate is the limit of Yt,T as T t, so that the short rate is an "infinitesimal yield" at time t. The short rate has the interpretation of an interest rate (or "yield") on a deposit during the period [t, t + dt], i.e. a deposit that is "withdrawn immediately". In practice the short rate is often identified with the yield on bonds with a short term, such one month or a week. A short rate model takes the short rate as its point of departure. It consists of a model for the evolution of the short rate process r, often a 104 9: Term Structures stochastic differential equation, and a description allowing to recover the processes Dt,T from the short rate. Such a description typically consists of the assumption that, for some probability measure R on the probability space (, F) on which r is defined, (9.2) Dt,T = ER e - T t rs ds | Ft . This formula together with a specification of the process r could be taken as a definition of a term structure, but then carries little intuition. The formula can be interpreted through the pricing formula (4.18) if we assume that the market contains, besides the discount bonds, an asset R that evolves according to the differential equation (with rt the short rate) dRt = rtRt dt, R0 = 1. Then the process R, which takes the form Rt = exp t 0 rs ds , is a numeraire. If the market (assumed to consist of at least the assets R and the discount bonds) is complete, then there exists a corresponding (local) martingale measure R making (R, R) into a numeraire pair, by Example 4.23, and by (4.18) the price at t of a given claim X at time T is equal to Rt ER X RT | Ft = ER Xe - T t rs ds | Ft . Because a discount bond Dt,T corresponds to a payment of X = 1 at time T, we arrive at the short rate model (9.2). Based on this interpretation the measure R is referred to as the risk neutral measure. In view of the definition (9.1) of the short rate r the asset R satisfying dRt = rtRt dt could be interpreted as the result of investing at each time t the current amount Rt in a zero coupon bond Dt,t+dt at the "current interest rate" rt. The existence in the real world of an asset consisting of rolling over bonds with infinitesimal contract periods is questionable, but it is a standard model assumption. It is similar to the assumption of existence of a "risk-free" asset in the extended Black-Scholes model, except that presently the rate rt is derived from the discount rates by (9.1). Actually the preceding argument for the short rate model (9.2) is valid as soon as there exists a numeraire pair (R, R) corresponding to the process Rt = exp( t 0 rs ds) and the market is complete. Thus the process R must be "tradable" in that it is the value process of a portfolio, but it does not need to be one of the basic assets itself. The argument can be generalized to arrive at conclusion (9.2) under the assumption of existence of a numeraire with differentiable sample paths (and market completeness). If (N, N) is an arbitrary numeraire pair in a complete economy containing the discount rate processes, then the pricing formula (4.18) implies that Dt,T = NtEN 1 NT | Ft . 9.1: Short and Forward Rates 105 If the numeraire N possesses absolutely continuous sample paths, then it can be shown that the short rate exists and that N = N0R, and hence we again arrive at a short rate model. 9.3 Lemma. Suppose that Dt,T = -1 t ER(T | Ft) for a strictly positive semimartingale defined on a given filtered probability space (, F, {Ft}, R), with absolutely continuous sample paths with derivative process t ˙t that is continuous in R-mean. Then the short rate (9.1) exists as a limit in probability, and (9.2) is valid with R = 0/. Proof. By assumption we can write T = t + T t ˙s ds for a derivative process ˙ such that s ER| ˙s - ˙t| is continuous, and hence ER T 0 | ˙s| ds < , for every T > 0. Thus we can rewrite the discount rate processes as Dt,T = 1 t ER t + T t ˙s ds| Ft = 1 + 1 t T t ER( ˙s| Ft) ds, by Fubini's theorem. This shows that the sample paths of the process T Dt,T are absolutely continuous on [t, ) with derivative s -1 t ER( ˙s| Ft). Because ˙t = ER( ˙t| Ft), we have ER 1 h t+h t ER( ˙s| Ft) ds - ˙t 1 h t+h t ER| ˙s - ˙t| ds 0. Combining the two preceding displays, we conclude that h-1 (Dt,t+h -1) ˙t/t in R-probability as h 0, and hence h-1 log Dt,t+h ˙t/t in probability, by a first order Taylor expansion of the logarithm (the "deltamethod"). Thus the short rate as in (9.1) exists as a limit in probability and is given by rt = - ˙t/t. The absolute continuity and positivity of imply the absolute continuity of t - log t, with derivative t - ˙t/t, by the chain rule. By the preceding paragraph this is equal to rt = ft,t. Hence t = 0 exp - t 0 rs ds and (9.2) is verified. Warning. We might use the equation Dt,T = NtEN(N-1 T | Ft) to define a term structure, rather than deduce the equation from the pricing formula. Then we first construct a strictly positive semimartingale N on a filtered space (, F, {Ft}, N) and next define an economy to consist of (at least) the processes given in the display, and possibly the process N itself. If we define discount rates through Dt,T = NtEN(N-1 T | Ft) for a given strictly positive semimartingale N on a given filtered probability space (, F, {Ft}, N), then t Dt,T /Nt is an N-martingale, for every T. However, it is not necessarily true that N is a numeraire in the economy consisting of all discount rate processes t Dt,T , for T > 0. For example, let N = 1/E(W) for W a Brownian motion on an arbitrary filtered probability space. Then Dt,T = 106 9: Term Structures E(W)-1 t EN E(W)T | Ft = 1, for every T. This does not allow to self-finance any nontrivial value process. This example also shows that in this type of model the initial filtration Ft can be strictly bigger than the filtration generated by the discount rates. Under the short rate model (9.2) the "discounted discount rate" Dt,R/Rt takes the form Dt,T /Rt = ER(R-1 T | Ft) and hence is an Rmartingale. We conclude that the pair (R, R) is a numeraire pair for the economy consisting of all discount rates provided that R is a numeraire. In that case the term structure model defined through (9.2) automatically fulfills the important requirement of no-arbitrage. Whether the economy consisting of all discount rates (and possibly the process R) is complete depends on the further specification of the term structure, for which relation (9.2) provides only the skeleton. Only if both the process r and an underlying filtered probability space (, F, {Ft}, R) are specified, we can appeal to (9.2) to recover the discount rates. A short rate model in the strict sense consists of (9.2) and a specification of the short rate process r as a one-dimensional diffusion process, with the filtration Ft taken equal to the natural filtration of the driving Brownian motion. We discuss these models further in Section 9.2. An advantage of this type of model is that it leads to relatively simple pricing formulas for many options based on the discount rates. Because r becomes a Markov process, the option prices for claims X that are a function of the evolution of r past some fixed time T, which are conditional expectations of the type ER(X/RT | Ft), will be functions of (t, rt) only, for t T. This permits the use of partial differential equations, including numerical methods, to compute these prices in concrete cases. The simplicity of a short rate model in the strict sense is also its weakness. A short rate model models the complete bond market through the single stochastic process r. The Markovian nature of this process could be interpreted as implying that at every time t the state of the bond market is described by the single number rt. This seems unrealistic. More general term structure models can be build on the forward rate, which is the partial derivative, if it exists, (9.4) ft,T = - T log Dt,T . This is interpreted as an interest rate that can be contracted at time t on investments into a savings account at time T. The short rate can be written ft,t and is the interest that can be contracted at time t on investments that are deposited. immediately. To motivate this inpretation of the forward process, consider an owner of an S-bond at a time t < S. The S-bond guarantees a payment of 1 at time S, but suppose that the owner needs the money only at a time T > S. One strategy would be to exchange the S-bond at time t for Dt,S/Dt,T units of T-bonds, which would give the guaranteed payment of Dt,S/Dt,T 9.1: Short and Forward Rates 107 at time T. An alternative strategy would be to keep the S-bond until its maturity at time S, and invest the payment of one unit received at that time into a risk-free account during the period [S, T]. The forward rate for [S, T] contracted at t is by definition the (imaginary) fixed interest rate R = Rt,S,T such that this investment would give exactly the value Dt,S/Dt,T at time T. In other words, the forward rate Rt,S,T is the number R solving the equation eR(T -S) = Dt,S/Dt,T , or Rt,S,T = log Dt,T - log Dt,S T - S . Thus the forward rate is an (imaginary) constant interest rate over the interval [S, T], fixed in a contract entered at a time t < S to the left of the interval. In the special case that t = S, the forward rate is also called the "spot rate", and is exactly the yield process Yt,T = Rt,t,T encountered previously. More importantly, the instantaneous forward rate ft,S is the limit as T S of the process Rt,S,T . This is also referred to as just the "forward rate", and is the process defined in (9.4). If the function T - log Dt,T is continuously differentiable, then the discount rates can be recovered from the forward rates through (9.5) Dt,T = e - T t ft,S dS . In that case, the short rate (9.1) also exists, and is given by rt = ft,t: = limT t ft,T . The forward rate ft,T has the intuitive interpretation as the "interest rate" over a period [T, T +dT] in the future obtainable at time t < T. Unlike is the case for the short rate, it depends on a time horizon T, and hence at every given time t there exist many forward rates. Even though the notion of an instantaneous rate remains a theoretical construct, the existence of multiple interest rates at each given time appears to be realistic. Thus the forward rates permit an attractive, different starting point for defining a model for the term structure. In principle the forward rates permit to describe the bond economy at time t through the infinite-dimensional "state vector" consisting of the set (ft,T : T > t) of all forward rates set at time t. In practice, it may be realistic to reduce the effective dimension of this state vector. One standard approach is to model the forward rates as diffusions relative to a given finitedimensional Brownian motion. We may still assume that the process Rt = exp t 0 rs ds based on the short rate is a numeraire for the bond economy. If the economy is complete, then, as before, this forces the equality (9.2), where R is the martingale measure corresponding to R. Then the discount rates Dt,T are related to the forward rates both through (9.2) and (9.5), where in the former equation rt = ft,t. This implies restrictions on the specification of a 108 9: Term Structures model for the forwards. For instance, we shall see that if the forwards are modelled through diffusion equations dft,T = t,T dt + t,T dWt, where W is a multivariate Brownian motion under R, then necessarily t,T = -t,T t t t,S dS. Thus if using this type of diffusion model we can only freely specify the diffusion function. 9.2 Short Rate Models In a wide sense every term structure model defined through the equation (9.2) with r the short rate process as in (9.1) is a "short rate model". In a more narrow sense a short rate model is a model which also assumes that the short rate process r is defined through a diffusion model on a "standard Brownian space". This is the type of short rate model we discuss in this section. We assume given a one-dimensional Brownian motion W defined on a filtered probability space (, F, {Ft}, R) , with the filtration Ft equal to the augmented filtration generated by W. The short rate process r is assumed to satisfy a diffusion equation of the form drt = (t, rt) dt + (t, rt) dWt. Under appropriate conditions on the functions and , a solution r to this equation will indeed exist and be adapted to the filtration Ft. If (t, rt) is strictly positive, then the equation can be inverted to express W into r, and hence the filtrations generated by r and W are the same. Under (possibly) additional conditions the process r will be a strong Markov process. We assume that all this is the case, and also that the variable exp - t 0 rs ds is R-integrable for every t (which is trivially true if the process r is nonneg- ative). Given the process r we define discount rate processes through (9.2), i.e. with the process R defined by Rt = exp t 0 rs ds , Dt,T = Rt ER 1 RT | Ft . We assume that the process R is a numeraire. It then follows that (R, R) is a numeraire pair for the economy consisting of all discount bonds. Unlike is the case for the Black-Scholes model, which is the accepted model for pricing options on stocks, no single short rate model has gained 9.2: Short Rate Models 109 universal acclaim. Some particular models are: drt = ( - rt) dt + dWt, Vasiček, drt = ( - rt) dt + rt dWt, Cox-Ingersoll-Ross, drt = rt dt + rt dWt, Dothan, drt = ( - rt) dt + rt dWt, Longstaff, d log rt = ( - log rt) dt + dWt. Here , , are given positive constants, which describe the evolution of the short rate process under the "risk-neutral measure" R. Because these three parameters are the only degrees of freedom in the resulting formulas for all discount rate processes and their derivatives, it would be surprising if any of the five models gave a good fit to the real world. For instance, the initial bond prices D0,T are known from the market at time 0, but are also computable from the short rate as ER(1/RT ), and hence can be expressed in the parameters , , of the short rate model. Similarly, the prices of many derivatives are fixed by the model, but also observable in the market. Because in practice it is impossible to choose the three parameters so that the theoretical bond and derivative prices are consistent with the observed prices, it has been proposed to replace the fixed parameters by functions of the time variable. The versions of the Vasiček and Cox-Ingersoll-Ross model in which , and are functions of t are both known as the HullWhite model. The time-dependent version of the Dothan model is the BlackDerman-Toy model. Finally, the Ho-Lee model is given by drt = t dt + dWt. This model also appears as a forward rate model, as will be seen in the next section. As the Vasiček-Hull-White models, it does not preclude the short rate from becoming negative, which is perhaps an undesirable feature. In these models the time-dependent drift function can typically be chosen so that the resulting theoretical initial yield curve T Y0,T = -T-1 log D0,T can be fitted exactly to a given yield curve observed in the market (at time 0). This is called calibration. The remaining parameters of the model can next be calibrated from observed prices on derivatives, or fitted by statistical analysis of the history of the discount rates (less common in practice). In practice the initial yield curve T Y0,T is only observed for a finite number of maturities T, and the model is calibrated to an interpolated initial yield curve. The question of completeness of the short rate models is usually easy to resolve, because a sub-economy consisting of the numeraire R and a single (active) discount rate Dt,T forms a standard Black-Scholes model, as discussed in Chapter 5, and is typically complete for the Brownian filtration Ft. Because the process Dt,T /Rt = ER(R-1 T | Ft) is a martingale relative to 110 9: Term Structures the Brownian filtration Ft, it can be represented as a stochastic integral, by the representation theorem for Brownian motion. If d(Dt,T /Rt) = Ht dWt for a predictable process H, then, by Itô's formula, dDt,T = d Dt,T Rt Rt = Dt,T rt dt + RtHt dWt. By Theorem 5.11 the economy consisting of the processes t (Rt, Dt,T ) is complete relative to the filtration Ft provided that the process H is nonzero. ** 9.6 EXERCISE. Investigate whether this is automatically the case if is strictly positive. Because by our assumptions the short rate process is Markovian and generates the filtration Ft, the conditional law of (rs: s t) given Ft depends on rt only. It follows that the value process Vt = ER Xe - T t rs ds | Ft of a contract with claim X that is a function of rT can be written as a function Vt = F(t, rt), for t T. If F is smooth, then we can use Itô's formula to compute d Vt Rt = 1 Rt Ft + Fr + 1 2 Frr2 - Fr dt + 1 Rt Fr dWt. Here W is an R-Brownian motion and, if the claim is replicable, the discounted value process V/R is an R-martingale. This is possible only if the drift term of the preceding diffusion equation vanishes, whence we obtain the term structure equation Ft(t, r) + (t, r)Fr(t, r) + 1 2 2 (t, r)Frr(t, r) - rF(t, r) = 0. The corresponding boundary condition is F(T, r) = g(r) if X = g(RT ). In particular, we can write the discount rates in the form Dt,T = F(t, rt) for a function F on [0, T]×R satisfying the term structure equation and satisfying the boundary condition F(T, r) = 1. 9.7 Example (Affine structure). A short rate model is said to possess affine structure if the drift and diffusion functions take the forms (t, r) = (t)r + (t), 2 (t, r) = (t)r + (t). Then the coefficients in the term structure equation depend also linearly on r and a solution ought to take the form F(t, r) = eA(t)-B(t)r . Inserting this and the equations for and 2 into the term structure equation yields an equation of the form C(t) + D(t)r = 0 for certain functions C and D. This 9.2: Short Rate Models 111 equation is satisfied identically in (t, r) if and only if C = D = 0, which takes the concrete form Bt(t) + (t)B(t) - 1 2 (t)B2 (t) = -1, At(t) - (t)B(t) + 1 2 (t)B2 (t) = 0. The first equation is a Riccati differential equation for B, while given B the second equation is an ordinary first order differential equation for A. The boundary condition F(T, r) = g(r) for every r translates into equations eA(T ) = g(0) and eB(T ) = g(0)/g(1). In particular, the discount rates take the form Dt,T = F(t, rt; T) for F given by F(t, r) = eA(t,T )-B(t,T )r for certain processes A(t, T) and B(t, T), which as functions of t, for fixed T, must satisfy the differential equations under the boundary condition A(T, T) = B(T, T) = 0. The yields are a logarithmic transformation of the discount rates and satisfy the attractive "linear" equation (T - t)Yt,T = -A(t, T) + B(t, T)rt. 9.8 Example (Vasiček-Hull-White). For the Vasiček-Hull-White model the equations derived in the Example 9.7 are especially simple and lead to explicit formulas. In this model we have (t, r) = (t) - r for a deterministic function and (t, r) = independent of (t, r). The equation for B reduces to a first order linear differential equation equation and hence can be solved explicitly, after which the function A can be found by one integration. In particular, we can write the discount rates in the explicit form Dt,T = eA(t,T )-B(t,T )rt for the functions A(t, T) = 1 2 2 T t B2 (s, T) ds - T t (s)B(s, T) ds, B(t, T) = -1 (1 - e-(T -t) ). In the special case of the Vasiček model, the function is assumed to be constant, and the integral defining A(t, T) can be evaluated analytically. Rather than using the approach of Example 9.7 we can also derive the distribution of the short rate process and next employ equation (9.2). It can be verified that the solution to the Vasiček-Hull-White diffusion equation is given by rt = e-t r0 + e-t t 0 (s)es ds + e-t t 0 es dWs. This is the sum of a deterministic function and the Ornstein-Uhlenbeck process yt = e-t t 0 es dWs. The latter process satisfies the reduced diffusion equation dyt = -yt dt + dWt and is well studied in probability theory. 112 9: Term Structures Because it is the integral of a deterministic function relative to Brownian, the Ornstein-Uhlenbeck process is a zero-mean Gaussian process. Its covariance function can be computed to be Eysyt = 2 e-(s+t) st 0 e2u du = 2 2 e-(s+t) e2st = 2 2 e-|s-t| . It follows that the short rate process is also a Gaussian process, both unconditionally and conditionally given its past. Consequently, by the affine structure, the discount rates Dt,T are log normally distributed, whereas the yields are normal. We can also verify this from (9.2) using the fact that the integral T t rs ds is again conditionally normally distributed given Ft. 9.9 EXERCISE. Carry out the calculations as indicated in the last sentence of the preceding example, i.e. derive the conditional mean and variance of the variable T t rs ds given rt and employ (9.2) to verify the formula for the discount rates. 9.10 EXERCISE. Is the Vasiček-Hull-White model complete? 9.11 EXERCISE. Show that it is possible to determine a parameter (a function : [0, ) R) in the Hull-White model such that the corresponding initial yield curve T Y0,T is exactly equal to a given function. [The significance is that the model can be exactly calibrated to the observed bond rates on the market at time 0.] Is this still possible if is restricted to be constant? 9.12 Example (Cox-Ingersoll-Ross). Bessel process. The preceding short rate models are called "single-factor" models, because they are driven by a single Brownian motion. A multi-factor short rate model describes the short rate process r as a measurable function r = g(s) of a multi-dimensional diffusion process s. An example is the two-factor Hull-White model, in which the short rate r is the first coordinate of the two-dimensional diffusion process (r, s) satisfying, for a two-dimensional Brownian motion (V, W), drt = (t + st - rt) dt + dVt, dst = -st dt + dVt + dWt. The Markov structure of a multi-factor short rate model permits a characterization of the discount rates through a partial differential equation, much in the same way as for a single-factor model. 9.3: Forward Rate Models 113 9.13 EXERCISE. Derive this equation for the two-dimensional Hull-White model. 9.3 Forward Rate Models The best known term structures that take their point of departure in the forward rates are the Heath-Jarrow-Morton models. In these models the forward rate processes t ft,T are assumed to satisfy stochastic differential equations of the type (9.14) dft,T = t,T dt + t,T dWt. Here t t,T and t t,T are stochastic processes that may depend on the horizon T, but W is a single, multivariate Brownian motion that is common to all forward processes. The differentials dft,T are understood to be relative to the argument t, with T being fixed. The corresponding initial conditions, one for each value of T > 0, consist of the specification of a complete curve T f0,T , known as the initial yield curve. We shall assume that there exist versions of the solutions t ft,T to (9.14) such that the processes (t, T) ft,T are jointly continuous. Then the functions T ft,T are integrable, and we can define the discount rate processes Dt,T by (9.5). Furthermore, the short rate, defined as the limit (9.1), exists, and is given by rt = ft,t. We also assume that the process R defined by Rt = exp t 0 rs ds is a numeraire, with corresponding martingale measure R, so that the discount rates also satisfy equation (9.2). The following theorem shows that this necessitates a relation between the drift and diffusion parameters in (9.14). To make this as transparent as possible, we shall interprete (9.14) as a diffusion equation relative to the risk-neutral measure R as in (9.2), i.e. the driving process W is an R-Brownian motion. In that case the drift parameters t,T are completely determined by the diffusion parameters t,T . 9.15 Theorem. Let W be a multivariate Brownian motion defined on a filtered space (, F, {Ft}, R), and suppose that the process t ft,T is a solution to (9.14), for every T > 0, for given continuous processes (t, T) t,T and (t, T) t,T . Then the process t Dt,T /Rt, with Dt,T defined by (9.5) and Rt = exp( t 0 fs,s ds), is an R-local martingale if and only if, If W is d-dimenionsal, then t t,T is an Rd -valued process, for every fixed T > 0, and t,T dWt is understood as an inner product. 114 9: Term Structures with t,T = T t t,S dS, t,T = t,T t,T , a.e. t. In that case, with rt = ft,t, dDt,T = Dt,T rt dt - t,T dWt . Proof. By definition ru = fu,u = f0,u + u 0 dfs,u, where the differential is relative to s, for fixed u. Hence (9.16) t 0 ru du = t 0 f0,u du + t 0 u 0 dfs,u du. Similarly, we can write - log Dt,T = T t ft,u du in the form (9.17) - log Dt,T = T t f0,u du + T t t 0 dfs,u du. The sum of the two double integrals on the far right sides of (9.16) and (9.17) gives a double integral over the area A = {(s, u): 0 s t, s u T}. We can substitute the diffusion equation (9.14) for the forward rates ft,T and use the stochastic Fubini theorem to write this double integral as A dfs,u du = t 0 T s (s,u du ds + s,u du dWs). The sum of the first terms on the right sides of (9.16) and (9.17) is equal to the integral T 0 f0,u du, which is constant in t. Thus adding the equations (9.16) and (9.17) for t 0 ru du and - log Dt,T and next taking the differential relative to t, we find rt dt - d log Dt,T = T t t,u du dt + T t t,u du dWt. Abbreviating the right side to Mt,T dt + t,T dWt, and using Itô's rule we find d Dt,T Rt = d exp log Dt,T - t 0 rs ds = Dt,T Rt d log Dt,T + 1 2 d[log Dt,T ] - rt dt = Dt,T Rt -Mt,T + 1 2 t,T 2 dt - t,T dWt . By assumption W is an R-Brownian motion. Therefore, the process t Dt,T /Rt is an R-local martingale if and only the drift term on the right side The product t,T t,T is understood to be the inner product of two d-dimensional stochastic processes. 9.3: Forward Rate Models 115 of the preceding display is zero, i.e. if and only if Mt,T = 1 2 t,T 2 . This is equivalent to the equality t,T = t,T t,T for almost every t. The last assertion of the theorem follows by another application of Itô's rule. In the preceding theorem the process W in the diffusion equation (9.14) is a Brownian motion under the measure R. If R is the martingale measure corresponding to the short rate numeraire R, then the processes t Dt,T /Rt must be R-local martingales. The theorem shows that in this case the drift functions t,T are completely determined by the diffusion functions t,T . This is different if the diffusion equation is understood relative to another underlying measure, e.g. the "historical measure". If the probability measure P possesses density process L relative to R, then the process ~W = W - C for Ct = t 0 L-1 s- d[L, W]s is a P-Brownian motion, by Girsanov's theorem. By rewriting the diffusion equation in terms of ~W, we see that the drift term under P takes the form t,T dt = t,T t,T dt + dCt). Because the additional term dCt is independent of T, the drift functions are severely restricted, also if the diffusion equation (9.14) is understood relative to a general underlying measure. If the processes t,T and t,T are smooth in the variable T, then the short rate process rt = ft,t also satisfies a stochastic differential equation. Warning. In the following we denote the partial derivative of a process ht,T relative to T by ˙ht,T . 9.18 Lemma. Let W be a multivariate Brownian motion defined on a filtered space (, F, {Ft}, R), and suppose that the process t ft,T is a solution to (9.14), for every T > 0, for given processes (t, T) t,T and (t, T) t,T whose sample paths are partially differentiable relative to T with continuous partial derivative processes (t, T) ˙t,T and (t, T) ˙t,T . Then drt = df0,t + t,t + t 0 ˙s,t ds + t 0 ˙s,t dWs dt + t,t dWt. Proof. By definition rt = ft,t = f0,t + t 0 dfs,t, where the differential is relative to s, for fixed t. Inserting the diffusion equation (9.14) for the forward rates, and writing s,t = s,s + t s ˙s,u du and similarly for s,t, we 116 9: Term Structures find rt = f0,t + t 0 (s,t ds + s,t dWs) = f0,t + t 0 (s,s ds + s,s dWs) + t 0 t s ( ˙s,u du ds + ˙s,u du dWs) = f0,t + t 0 (s,s ds + s,s dWs) + t 0 u 0 ( ˙s,u ds du + ˙s,u dWs du), by the stochastic Fubini theorem. Taking differentials gives the result. 9.19 EXERCISE. Rewrite the assertion of the lemma as drt = (t,t + ˙ft,t) dt + t,t dWt for ˙ft,T the partial derivative of T ft,T . [Hint: differentiate the identity ft,T = f0,T + t 0 (s,T ds + s,T dWs) with respect to T.] The preceding lemma should not be mistaken to imply that a HeathJarrow-Morton forward rate model is a multi-factor short rate model in disguise. The lemma merely asserts that the short rate process satisfies a stochastic differential equation, but this need not be of the standard diffusion type. Neither the drift function, nor the "diffusion function" need be expressible in (t, rt), as required for a single-factor short rate model, or even in the value of an underlying vector-valued Markov process. On the other hand, in simple examples the equation for r may well reduce to a diffusion equation. 9.20 Example (Hoo-Lee). Consider the Heath-Jarrow-Morton model (9.14) driven by a one-dimensional Brownian motion and with diffusion coefficients t,T = equal to a constant, for every T. Then t,T = (T -t) and hence the diffusion equations for ft,T and Dt,T take the forms dft,T = 2 (T - t) dt + dWt, dDt,T = Dt,T rt dt - (T - t) dWt . The forward curves t ft,T can be viewed as random perturbations of the parabola t 2 (Tt - 1 2 t2 ), where the random deviations from this fixed curve are the same for every T. In economic terms "the only possible movements of the yield curve are parallel shifts", or "all rates along the yield curve fluctuate in the same way". Given an initial forward curve T f0,T , the short rate rt = ft,t can be computed as rt = f0,t + 1 2 2 t2 + Wt. If the initial yield curve is differentiable, then we can write this in differential form and obtain a short rate model with nonrandom, time-dependent drift function and diffusion term Wt. This is known as the Hoo-Lee model. 9.3: Forward Rate Models 117 9.21 Example. The choice t,T = e-(T -t) in the Heath-Jarrow-Morton model yields a model with an exponentially increasing influence of the diffusion term. This leads to t,T = - e-(T -t) - 1 , dft,T = - 2 e-(T -t) e-(T -t) - 1 dt + e-(T -t) dWt, drt = (t - rt) dt + dWt, for t = mt + mt and mt = f0,t - 1 2 2 (1 - e-t )2 /2 . [Or +???] This is again a short rate model with time-dependent drift parameters. 9.22 EXERCISE. Verify the calculations of Example 9.21 and investigate the completeness of this model relative to the filtration generated by W. If the process R is a numeraire and R a corresponding martingale measure, then the processes t Dt,T /Rt must be R-martingales. By Theorem 9.15 this can only be true for a Heath-Jarrow-Morton model (9.14) if t,T = t,T t,T . This condition and some integrability is also sufficient for (R, R) to be a numeraire pair. 9.23 Theorem. Suppose that the conditions of Theorem 9.15 hold and that Rt = exp( t 0 rs ds) is a numeraire. If t,T = t,T t,T and ER exp T 0 1 2 s,T 2 ds < , then (R, R) is a martingale numeraire pair for the economy consisting of all discount rate processes t Dt,T . Proof. By Theorem 9.15 the discounted discount rate processes t Dt,T /Rt are R-local martingales. It suffices to show that they are also R- martingales. By the last assertion of Theorem 9.15 the discount rate processes satisfy a stochastic differential equation. This equation and an application of Itô's formula give that d Dt,T Rt = - Dt,T Rt t,T dWt. (Cf. the proof of Theorem 9.15.) This shows that the processes Dt,T /Rt satisfy the stochastic differential equation dXt = Xt dMt for the local martingale M = -t,T W. This is the Doléans equation and hence the processes t Dt,T /Rt can be shown to be a martingale by verification of Novikov's condition for M. This is the condition as in the theorem. The last assertion of Theorem 9.15 shows that the discount rate processes t Dt,T satisfy a stochastic differential equation. Therefore, the economy formed by the process R together with finitely many discount rate processes t Dt,Ti , for i = 1, . . . , n, is an extended Black-Scholes 118 9: Term Structures model of the type discussed in Chapter 5. In the parameterization of Theorem 9.15 the diffusion model for the asset processes t Dt,Ti is given under the martingale measure R and possesses drift r instead of m as in Chapter 5. If we take this as the initial measure, then a "market price of risk" process as in Theorem 5.7 can be taken equal to zero, and hence exists. It follows that the economy is complete relative to the augmented natural filtration generated by the asset processes t Dt,Ti provided that these are weakly uniquely defined by the stochastic differential equation. If the initial filtration Ft has the property that for every T > 0 there exist finitely many times T1 < T2 < < Tn such that (Ft)0tT is generated by the processes (Rs: 0 s t) and (Dt,Ti : 0 t Ti, i = 1, . . . , n), then the Heath-Jarrow-Morton economy is also complete relative to the filtration Ft. Alternatively, we may apply Theorem 5.11 to address the completeness relative to the augmented natural filtration FW t of the driving Brownian motion W. If W is d-dimensional, then a sufficient condition for completeness is that for each T > 0 there exist times T1 < T2 < < Td such that the (d × d)-matrices with ith row t,Ti are invertible, for every t [0, T]. 10 Vanilla Interest Rate Contracts In this chapter we consider a number of standard contracts, called "vanilla" or "over the counter", because they are commonly traded, as opposed to "exotic" contracts, which are tailored to special demands. We are interested in a description of these contracts, and particularly in their valuation. Throughout the chapter it is silently understood that the bond economy permits numeraire pairs, and if convenient also that it is complete, or embedded into a complete economy. The completeness assumption implies that to every numeraire, for instance a discount rate, exists a corresponding martingale measure. This free choice of numeraire is convenient for obtaining the valuation formulas. All contracts considered in this chapter are derivatives of the zero coupon bonds, described in Chapter 9. A zero coupon bond is a contract that guarantees a cash-flow of 1 unit at a time T in the future. To make the dependence on the maturity explicit, we shall also refer to such a bond as a T-bond. A graphical display of this contract is given in Figure 10.1, the upward arrow indicating a payment to the owner of the contract at time T. The value of a zero coupon bond at a time t < T is by definition the discount rate Dt,T . 1 t T Figure 10.1. Zero coupon bond or "T -bond". The value at time t is Dt,T . The value of a general derivative with payments X1, . . . , Xn at times T1 < < Tn that are functions of the discount rates up to these times is equal to Nt n i=1 EN Xi NTi | Ft . 120 10: Vanilla Interest Rate Contracts Here we are free to choose a numeraire pair (N, N). Rather than the obvious pair (R, R) corresponding to the short rate, we may choose a numeraire pair for which the evaluation of this expression is straightforward. Often one even chooses both a special numeraire pair and a special model for the distribution of the relevant discount rate that makes the calculations easy. If everything fails we can always use the numeraire R and determine the value of the derivative numerically by stochastic simulation under the corresponding martingale measure R. 10.1 Deposits A deposit is a contract that guarantees a fixed interest rate L on a given capital over a prespecified period. A graphical display is given in Figure 10.2. The downward arrow indicates a payment of a single unit by the buyer of the contract at time t, for which he in return receives a payment of 1 + L units at the expiry time T of the contract. The number , referred to as an accrual factor or daycount fraction indicates the duration of the deposit. In practice this could be the number of days divided by 360, but for us it will do to think of it as an absolute number. The number L is the rate of return, and is also specified in the contract, usually as a percentage. 1 + L t T 1 Figure 10.2. Deposit. The value at time t is zero if L = Lt[t, T ]. The parameter measures the length of the time interval [t, T ]. The value of the deposit contract at time t is positive if the return rate L is high and negative in the opposite case. The number L such that the value at time t is zero is called LIBOR (from "London Inter Bank Office Rate") and is denoted by Lt[t, T]. Thus a deposit with return rate L set equal to the LIBOR guarantees a payment of 1 + Lt[t, T] at time T after making the initial investment of 1 unit at time t, and no other cashflows. Because buying (1+Lt[t, T]) zero coupon bonds with maturity T returns the same payment, the no-arbitrage principle forces the costs of the two contracts to be the same. The bonds can be acquired at cost Dt,T per bond at time t. Hence (10.1) 1 = (1 + Lt[t, T])Dt,T , Lt[t, T] = 1 - Dt,T Dt,T . 10.2: Forward Rate Agreements 121 It follows that we can think of the LIBOR as a derivative of the discount rates. A different interpretation of the LIBOR results from noting that Lt[t, T] = 1/Dt,T - 1 is the profit made by investing 1 unit at time t in discount bonds with maturity T. The LIBOR relates to the yield through the formula exp (T -t)Yt,T = 1 + Lt[t, T]. Thus a third way of interpreting the LIBOR is to view it as a fixed interest rate for the interval [t, T], which, unlike the yield, is not continuously compounded, but applied once to multiply the capital. 10.2 Forward Rate Agreements The forward rate agreement (FRA) is graphically displayed in Figure 10.3. It incorporates two payments at the expiry time T, one receiving and one buying. The buying payment K is at a fixed rate of return K, whereas the receiving payment is proportional to the LIBOR LS[S, T] for the period [S, T]. Because this LIBOR will only be "set" at some time S in the future, from the current time t perspective this payment is a random variable. In financial jargon it is referred to as a floating payment. The purpose of the forward rate agreement is to exchange the unknown rate of return LS[S, T] for a rate K that is written in the contract at time t. Depending on K this may not be without cost. The value of K such that the FRA has zero value at time t is called the forward LIBOR and is denoted by Lt[S, T]. The FRA enables one to obtain a certain return of Lt[S, T] on a sum of money that we know will come in our possession during a time interval [S, T] in the future. If the sum of money is 1, the following strategy could be used. time t sell an FRA at the forward LIBOR rate K = Lt[S, T] at no cost. time S receive 1 unit; deposit 1 unit at rate LS[S, T] until T. time T pay LS[S, T] - Lt[S, T] on FRA, cash deposit giving 1 + LS[S, T], total value sums up to 1 + Lt[S, T]. Following this scheme, we are certain to receive a return rate of Lt[S, T] on the money received at the future time S, a rate that is fixed at the current time t. Thus an FRA is an instrument to "swap" a random future rate for a fixed rate. For K unequal to the forward LIBOR the FRA possesses a nonzero value process. We can determine the value of an FRA at time t by the general theory of Chapter 4, under the assumption that the economy is complete. The FRA contract guarantees a single payment of LS[S, T] K at time T. Therefore, given a martingale numeraire pair (N, N), the 122 10: Vanilla Interest Rate Contracts LS[S, T] t S T K Figure 10.3. Forward rate agreement (FRA). The value of the contract at time t is zero if K = Lt[S, T ]. The parameter measures the length of the time interval [S, T ]. value of the contract at time t is given by Vt = Nt EN LS[S, T] - K NT | Ft = Nt EN LS[S, T] - K EN 1 NT | FS | Ft = Nt EN LS[S, T] - K DS,T NS | Ft . Here we have used the fact that, by the definition of a martingale numeraire pair (N, N), the process t Dt,T /Nt is an N-martingale, for every fixed T, and 1/NT = DT,T /NT . We now express LS[S, T] into the discount rates through (10.1) and rewrite the right side as Vt = NtEN 1 - DS,T - KDS,T NS | Ft = Dt,S - (1 + K)Dt,T , again by the martingale property of the discounted discount rates. We have carried out the preceding calculation without ever specifying the numeraire pair or indicating a term structure model. This may be explained from the fact that there exists a very simple hedging strategy for an FRA, which also does not require specification of a model. For buying an FRA this takes the following steps. Given an amount of Dt,S - (1 + K)Dt,T at time t: time t buy an S-bond at cost Dt,S, sell (1 + K) T-bonds at cost -(1 + K)Dt,T . time S cash S-bond; deposit 1 unit at rate LS[S, T] until T. time T pay off (1 + K) T-bonds at cost (1 + K), cash deposit giving 1 + LS[S, T], total value sums up to LS[S, T] - K . By definition the forward LIBOR Lt[S, T] is the value of K setting the value of the FRA at time t equal to zero. Solving the equation 0 = Dt,S - (1 + K)Dt,T for K allows to express the LIBOR in the discount rates, as (10.2) Lt[S, T] = Dt,S - Dt,T Dt,T . 10.3: Swaps 123 We can substitute this equation back into the expression for the value of an FRA to obtain that the value at time of an FRA is given by Vt = Dt,T Lt[S, T] - K . 10.3 Swaps A swap or interest rate swap can be considered an FRA with payments that are spread over multiple time points in the future. As can be seen in the graphical display given in Figure 10.4, the swap consists of a floating leg and a fixed leg. For simplicity we assume that there is a single set of payment dates T1 < < Tn, although in practice the payment dates for the two legs may not match perfectly. The rate of return K in the fixed leg is the same for every payment and is fixed in the contract. The return rates on the payments in the floating leg may differ and are equal to the LIBORs setting at the beginning of the periods. Thus a swap allows to exchange random LIBORs that are set in the future for a fixed return rate written in the contract. A swap whose value at the current time t is zero is called a par swap, and the corresponding value of K is the swap rate, denoted by yt[T0, T1, . . . , Tn]. If t < T0, then this is a "forward swap rate", whereas for t = T0 the swap rate is "spot". The spot swap rate is commonly quoted on market screens and hence is an important indicator for the bond market. 1LT0 [T0, T1] 2LT1 [T1, T2] nLTn-1 [Tn-1, Tn] 1 2 n t T0 T1 T2 Tn-1 Tn 1K 2K nK Figure 10.4. An Interest rate swap. The value of the swap at time t is zero for K = yt[T0, T1, . . . , Tn]. The parameter i measures the length of the time interval [Ti-1, Ti]. Because a swap is a repetition of FRAs, its valuation is similar to that of an FRA. The payment at time Ti is equal to i LTi-1 [Ti-1, Ti]-K and hence by (4.43), given a numeraire pair (N, N), the value of the swap at time t is equal to Vt = NtEN n i=1 i LTi-1 [Ti-1, Ti] - K NTi | Ft . Each of the terms of the sum on the right side can be evaluated exactly as 124 10: Vanilla Interest Rate Contracts for an FRA. This yields Vt = n i=1 Dt,Ti-1 - (1 + iK)Dt,Ti = Dt,T0 - Dt,Tn - KPt[T0, T1, . . . , Tn], for Pt[T0, T1, . . . , Tn] the "present value of unity" defined by (10.3) Pt[T0, T1, . . . , Tn] = n i=1 iDt,Ti . The present value of unity is the value of the fixed leg of the swap if the return rate K is unity. It is 10 000 times the present value of a basis point (PVBP), a "basis point" being 0.01 %, which is a more usual quantity quoted on the market. By definition the value of a swap is zero if K is equal to the swap rate yt[T0, T1, . . . , Tn]. This readily gives the formula (10.4) yt[T0, T1, . . . , Tn] = Dt,T0 - Dt,Tn Pt[T0, T1, . . . , Tn] . We can substitute this back into the value formula for a swap to see that this can also be written as Vt = Pt[T0, T1, . . . , Tn] yt[T0, T1, . . . , Tn] - K . As for an FRA there is a simple hedging strategy for a swap, which could have been used for the valuation. Given the amount Dt,T0 - Dt,Tn - KPt[T0, T1, . . . , Tn] at time t: time t buy an T0-bond at cost Dt,T0 , sell Tn-bond at cost -Dt,Tn , sell iK Ti-bonds for i = 1, . . . , n at total cost -KPt[T0, . . . , Tn]. time T0 cash T0-bond, deposit 1 unit at rate LT0 [T0, T1] until T1. time T1 deposit returns 1 + 1LT0 [T0, T1], pay out 1LT0 [T0, T1], pay off 1K T1-bonds at cost 1K, deposit 1 unit at rate LT1 [T1, T2] until T2. time Ti deposit returns 1 + iLTi-1 [Ti-1, Ti], pay out iLTi-1 [Ti-1, Ti], pay off iK Ti-bonds at cost iK, deposit 1 unit at rate LTi [Ti, Ti+1] until Ti+1. time Tn deposit returns 1 + nLTn-1 [Tn-1, Tn], pay out nLTn-1 [Tn-1, Tn], pay off nK Tn-bonds at cost nK, pay off Tn-bond at cost 1. 10.4: Caps and Floors 125 10.4 Caps and Floors Caplets and Floorlets are European call options and put options on the spot LIBOR rates. Their payment schemes are shown in Figures 10.5 and 10.6. Caps and Floors are repetitive caplets and floorlets, much as a swap is a repetition of FRAs. The payment scheme of a cap is shown in Figure 10.7. These contracts allow to profit from a potential rise in the value of the LIBOR, without full exposure to a possible decrease in the interest rate. Thus they are less conservative than FRAs or swaps, which are meant to take away all uncertainty by "swapping" future interest rates for a fixed rate. Because the pay-offs of a caplet and floorlet are functions of the variable LS[S, T] - K , which is the value at time T of a FRA, we can also view these contracts as options on an FRA. As the value of an FRA can be expressed in the discount rates, in the end caps and floors are derivatives of the discount rates. (LS[S, T] - K)+ t S T Figure 10.5. Caplet. (K - LS[S, T])+ t S T Figure 10.6. Floorlet. Unlike is the case for FRAs and swaps, there is no universal hedging strategy for caplets and floorlets. The replicating strategy necessarily depends on the model used for the underlying discount rates. We might valuate these contracts using a model for the full term structure of the economy. However, because the payoffs on caplets and floorlets are functions of the LIBOR, their prices are expectations of the discounted LIBOR under a martingale measure, and hence an easier way to proceed is to model the LIBOR and the numeraire directly under a martingale measure. Here we are free to choose a convenient numeraire. If we use the discount rate Nt = Dt,T as a numeraire, then in view of (10.2) the forward LIBOR process t Lt[S, T] is a linear combination of two assets divided by the numeraire. By the definition of a martingale numeraire pair the forward LIBOR must be a martingale under a martingale measure N corresponding to the numeraire Nt = Dt,T . Because NT = DT,T = 1 the price of a derivative X based on the LIBOR takes the form 126 10: Vanilla Interest Rate Contracts Vt = Dt,T EN(X| Ft). We can evaluate this expectation as soon as we specify the distribution of the LIBOR under the martingale measure. The model must incorporate the martingale property of the LIBOR. The simplest possible model that also ensures positivity of the LIBOR Lt = Lt[S, T] is given by the differential equation dLt = Ltt dWt, where W is an N-Brownian motion relative to the filtration Ft, and t is a deterministic function. This equation is solved by the Doléans exponential Lt = E( W) and hence LS = Lte S t s dWs- 1 2 S t 2 s ds , t < S. Because W possesses independent increments, the conditional distribution of LS given Ft is the same as the distribution of Lt exp(tZ - 1 2 2 t ), for Lt = Lt[S, T] considered constant, Z a standard normal variable, and 2 t = S t 2 s ds. The value of a caplet can now be calculated as Vt = NtEN LS[S, T] - K + NT | Ft = Dt,T E LtetZ- 1 2 2 t - K + = Dt,T Lt log(Lt/K) t + 1 2 t - K log(Lt/K) t - 1 2 t . This is known as Black's formula. An alternative approach to valuing a caplet is to employ one of the term structure models discussed in Chapter 9. If we use the numeraire R with corresponding martingale measure R, then we must evaluate ER e - T t rs ds 1 DS,T - 1 - K + | Ft . In most cases it is not possible to compute the expectation analytically. An exception is the Vasiček-Hull-White model, for which the preceding display reduces to ER e - T t rs ds e-A(t,T )+B(t,T )rS - 1 - K + | Ft . This can be evaluated using the fact that the random vector ( T t rs ds, rS) is bivariate-normally distributed given Ft. The resulting expression depends on the parameters (, , ) of the Vasiček-Hull-White model, and will generally not agree with Black's formula.[True ???] We can use the same approach to calculate the value of a floorlet. It is more interesting to derive this from the put-call parity, which is based on the identity x+ - (-x)+ = x applied with x equal to LS[S, T] - K . If 10.5: Vanilla Swaptions 127 we write the values processes of caplets and floorlets by V cap and V floor , then the identity and the pricing formula (4.18) yield V cap t - V floor t = NtEN LS[S, T] - K NT | Ft = NtEN LS[S, T] - K DS,T NS | Ft . This is true for every martingale numeraire pair (N, N). If the numeraire is again chosen equal to the discount rate Dt,T , then DS,T /NS = 1, and the LIBOR t Lt[S, T] is an N-martingale. Then the right side reduces to Dt,T Lt[S, T] - K . In this argument we have not used any model for the term structure, or the LIBOR. The put-call parity is true independently of the distribution of the assets. Because a cap pays a series of caplets, the value of a cap is the sum of the values of a series of caplets. We omit the details. 10.5 EXERCISE. What is the value of a caplet at a time t with S t < T? 10.6 EXERCISE. What is the value of a cap at time t [Ti, Ti+1)? 1(LT0 [T0, T1] - K)+ 2(LT1 [T1, T2] - K)+ n(LTn-1 [Tn-1, Tn] - K)+ 1 2 n t T0 T1 T2 Tn-1 Tn Figure 10.7. Cap. 10.5 Vanilla Swaptions A swaption is a European option on a swap. In the context of swaptions the "call" and "put" forms of options are referred to as "payer's" and "buyer's" swaptions. Given fixed times T0 < T1 < < Tn, the claim of a payer's swaption consists of a payment at time T0 of the amount PT0 [T0, T1, . . . , Tn] yT0 [T0, T1, . . . , Tn] - K + . This is exactly the positive part of the value at T0 of a swap at time T0. The payment scheme of a swaption is displayed in Figure 10.8. PT0 (yT0 - K)+ 1 2 n t T0 T1 T2 Tn-1 Tn Figure 10.8. Swaption. 128 10: Vanilla Interest Rate Contracts The swap rate yt = yt[T0, . . . , Tn] is the quotient of Dt,T0 - Dt,Tn and the PVBP Pt = Pt[T0, . . . , Tn]. This and the form of the claim makes the PVBP into a convenient numeraire for valuating the swaption. Let N be a martingale measure corresponding to Pt. Then t Dt,T /Pt is an Nmartingale for every discount rate Dt,T , and hence so is the swap rate yt. The value of the swaption is equal to (choose N = P) Vt = NtEN PT0 (yT0 - K)+ NT0 | Ft = PtEN (yT0 - K)+ | Ft . To turn this into a concrete formula it suffices to model the distribution of yT0 under the martingale measure N. Because the swap rate is positive and is an N-martingale, a simple model is the geometric Brownian motion, i.e. yt = e t 0 s dWs- 1 2 2 s ds , for a deterministic function and a process W that is an N-Brownian motion relative to the filtration Ft. The corresponding value of the swaption is given by Vt = Pt[T0, . . . , Tn]E yt[T0, . . . , Tn]etZ- 1 2 2 t - K + . Here 2 t = t 0 2 s ds and the expectation is to be taken relative to the standard normal variable Z, keeping yt[T0, . . . , Tn] fixed. This can be further evaluated by similar calculations as used for caplets, leading to an expression of the same form as Black's formula. 10.7 EXERCISE. The claim of a receiver's swaption takes the form PT0 [T0, T1, . . . , Tn] yT0 [T0, T1, . . . , Tn]-K . Show that the put-call parity (or payer-receiver parity) takes the form V pay t - V rec t = Pt(yt - K), for t < T0. 10.6 Digital Options A digital option is an "all or nothing option", giving a fixed return if a certain event happens and no return in the opposite case. The digital caplet pays one unit at time T if the LIBOR LS[S, T] set at S < T is above a certain prespecified level and nothing otherwise. (Cf. Figure 10.9.) For a digital floorlet these possibilities are exchanged. A convenient numeraire is the discount rate t Dt,T . Under a martingale measure corresponding to this numeraire the LIBOR t Lt[S, T] is a martingale, and hence can be reasonably modelled as a geometric Brownian 10.7: Forwards 129 motion. The approach is identical to the one taken for caplets, and, with the same notation as before, leads to the value of digital caplet given by Vt = Dt,T log Lt[S, T]/K - 1 2 2 t t . 10.8 EXERCISE. Derive a put-call parity between digital caplets and floor- lets. 1LS [S,T ]>K t S T Figure 10.9. Digital caplet. The digital swaption pays one unit at some time T if the swap rate yt[T0, . . . , Tn] is bigger (digital payer's swaption) or smaller (digital receiver's swaption) than some constant, and nothing otherwise. The payment time T may or may not be one of the times Ti. The payment scheme of a digital payer's swaption is displayed in Figure 10.10. It appears that this derivative does not permit pricing with an equally simple model as before. . 1yT0 [T0,...,Tn] i. We can summarize this payment scheme by saying, if the contract is acquired at time t, then the payment at time Tj is equal to Tj t - Tj-1t, j = 1, . . . , n. If the contract is entered at time 0, then the total payment over the interval [0, Ti] is equal to Ti - 0. Thus the futures process can be interpreted as the cumulative payments over time intervals. 0 T1 Ti-1 t Ti Ti+1 Tn Ti - t Ti+1 - Ti Tn - Tn-1 Figure 11.1. Payments required on futures contract acquired at time t. As usual we assume the existence of a martingale numeraire pair (N, N) and market completeness. Then the pricing formula (4.43) shows that the value of the futures contract (to the other party in the contract) at time t 11.2: Continuous Time 133 is equal to (11.1) Vt = NtEN n j=1 Tj t - Tj-1t NTj | Ft . 11.2 Definition. The process is said to be a futures price process in discrete time for the settlement X if is a cadlag semimartingale with T = X and Vt = 0 for every t [0, T]. Under some integrability conditions the futures price process exists and is uniquely determined. To see this we can rewrite the equation Vt = 0 for t [Ti-1, Ti) in the form t = EN(Ti /NTi | Ft) + EN n j=i+1(Tj - Tj-1 )/NTj | Ft EN(1/NTi | Ft) . This can be solved recursively, starting with T = X, next using the formula for the intervals [Tn-1, T), [Tn-2, Tn-1), etc. To obtain a concrete representation of the futures process, we must model the joint distribution of the settlement amount X and the numeraire process N under the martingale measure N. Furthermore, the recursions may be difficult to implement in practice. 11.2 Continuous Time In the continuous time model the payments are made continuously over time. To motivate the definition we can take a limit along a sequence of discrete time futures models with time points 0 = Tn 0 < Tn 1 < < Tn n = T. If maxi |Tn j - Tn j-1| 0, then n j=1 Tj t - Tj-1t NTj P GT - Gt, for the process G defined by (11.3) Gt = t 0 1 Ns ds + 1 N , t . This suggests to replace the process Vt in (11.1) in the continuous time model by the process (11.4) Vt = NtEN GT - Gt| Ft , We use that iHsi (Xsi - Xsi-1 ) converges to t 0 H dX + [H, X]t in probability if the mesh widths of the partition 0 = s0 < < sn = t tends to zero. The "correction term" [H, X] arises because the sums use the final value of the process H in the partitioning interval. 134 11: Futures and to define a process to be a futures process for the claim X if is a cadlag semimartingale with V = 0, exactly as in the preceding definition. This is equivalent to the statement Gt = EN(GT | Ft) for every t, and hence that the process G is an N-martingale. In the following we assume that the numeraire N is continuous. 11.5 Definition. The process is said to be a futures price process in continuous time for the claim X if is a cadlag semimartingale with T = X and such that the process G in (11.3) is an N-martingale on [0, T] for some numeraire pair (N, N). A different way to arrive at this definition is to consider t - 0 to be the cumulative payments on a futures contract over the time period [0, t]. Then, as seen in Chapter 7, for a continuous numeraire N the cumulative discounted payment over the interval [0, t] is equal to Gt, and the value at time t of the contract is equal to Vt, by (7.3). The more general approach of Chapter 7 also shows how to extend the definition to numeraires with jumps. If the numeraire N is continuous and of bounded variation, then the term [1/N, ] in the definition of G vanishes. In that case we can invert the relation G = N-1 to obtain that = 0 +N G. If is a futures process, then G is an N-martingale, and hence is an N-local martingale. If is also an N-martingale, then we obtain the pricing formula, since T = X, t = EN(X| Ft). This observation helps to find sufficient conditions for the existence of a futures process. 11.6 Lemma. If ENX2 < and N is a continuous numeraire of bounded variation that is bounded away from zero, then t = EN(X| Ft) is a futures process with T = X. Proof. If ENX2 < , then t: = EN(X| Ft) is an N-martingale bounded in L2. If N-1 is bounded, then G = N-1 is also an N-martingale bounded in L2, and hence is a futures process. ** 11.7 EXERCISE. Investigate uniqueness of the futures process, for instance under the conditions of the lemma. 11.8 EXERCISE. Calculate the futures price process for the settlement equal to the price ST of a stock in the Black-Scholes model of Chapter 1. The futures prices are often compared to the values Ft of a forward on the same underlying asset. Let N be a numeraire of bounded variation. In 11.2: Continuous Time 135 view of formula (10.9), t - Ft = EN(X| Ft) EN(X/NT | Ft) EN(1/NT | Ft) = covN(X, 1/NT | Ft) EN(1/NT | Ft) . This is called the futures correction. If the settlement X and the numeraire at expiry time are positively dependent, then the futures price is higher than the forward price. The forward is also entered at zero cost at time t, and includes a total payment of X - Ft over the contract period [t, T]. The difference is that this payment is made in one installment at the expiry time, whereas the payments T - t = X - t made under the futures contract are made continuously during the contract period. 11.9 Example (Euro-dollar contract). The Euro-dollar contract has claim X = 100 1 - LS[S, T] . The forward LIBOR Lt[S, T] and the short rate rt are typically positively dependent, and hence so are the forward LIBOR and the process Nt = exp T 0 rs ds . This implies that the claim X and the variable 1/NT = exp - T 0 rs ds are also typically positively correlated. Assuming that N is a numeraire, the futures correction shows that t < Ft. 11.10 Example (Euro-dollar, V-H-W). We can calculate the futures process and the futures correction analytically in the Vasiček-Hull-White model. In this model the short rate process is an Ornstein-Uhlenbeck process with a deterministic drift and hence is a Gaussian process, under the martingale measure R corresponding to the numeraire Rt = exp t 0 rs ds . The discount rates can be expressed in the short rate as Dt,T = At,T e-Bt,T rt for the constants A(t, T) and B(t, T) given in Example 9.8, and hence by (10.1) the LIBOR satisfies LS[S, T] = exp -A(S, T) + B(S, T)rS ) - 1. We conclude that the futures price process for the settlement amount X = LS[S, T] is given by t = 1 e-A(S,T ) ER eB(S,T )rS | Ft - 1 . Given Ft the random variable rS possesses a log normal distribution, and hence the expectation is easy to compute. The futures correction takes the form t - Ft = - 1 e-A(S,T ) covR eB(S,T )rS , e - T 0 rs ds Dt,T /Nt . This is more work to compute, but just as straightforward, the vector rS, T 0 rs ds possessing a bivariate normal distribution under R. We omit the details. 136 11: Futures 11.11 EXERCISE. Calculate the futures correction for the settlement equal to the price ST of a stock in the Black-Scholes model of Chapter 1. 12 Swap Rate Models The swap rate yt[T0, T1, . . . , Tn], given by (10.4), is by its definition an indicator for the "interest rate" on the bond market in a given time interval. A swap rate model is a term structure that models the discount rates , or functions thereof, in terms of the swap rate. This is particularly attractive if we are interested in linear combinations of discount rates with maturities in the time interval spanned by the time points T0 < T1 < < Tn only. Throughout the chapter we fix the time points 0 < T0 < T1 < < Tn and abbreviate the swap rate and present value of a base point (PVBP) corresponding to these time points to yt = yt[T0, T1, . . . , Tn] and Pt = Pt[T0, T1, . . . , Tn]. This gives corresponding stochastic processes y and P. The processes P and yP are linear combinations of discount rate processes and hence are numeraires in the bond market (provided that y is positive). We silently assume that the bond market is complete, so that there exist corresponding martingale measures S and Y, giving the numeraire pairs (P, S) and (yP, Y). 12.1 Linear Swap Rate Model The swap rate y is a difference of bond prices divided by the numeraire PVBP, and hence the swap rate y is a martingale under the corresponding martingale measure S. For any maturity time T the process t Dt,T /Pt is an S-martingale also. The linear swap rate model postulates that these two martingales are affinely proportional in that, for a constant A and a deterministic function T BT , (12.1) Dt,T Pt = A + BT yt. 138 12: Swap Rate Models This expresses the discount rate Dt,T into the discount rates Dt,Ti . The model leads to particularly simple calculations for derivatives that are functions of the quotients Dt,T /Pt, because their law (under the martingale measure) will be determined as soon as we know the distribution of the swap rate. Suitable constants A and BT for the linear swap rate model (12.1) can be derived from the implied relations 1 = i i Dt,Ti Pt = i i A + BTi yt , D0,T = A + BTi y0 P0. If we want the first relation to hold for every value of the swap rate yt, then we are led to set i i A = 1. The initial bond prices D0,T and the initial swap rate P0 are known from the market at time 0, and hence, given A, we can solve the constant BT from the second equation. 12.2 Exponential Swap Rate Model The exponential swap rate model postulates that all discount rates are determined by a single univariate process z through a relationship of the form Dt,T = e-CT zt , for a deterministic function T CT . In view of the definition of the swap rate (10.4) this implies the relationship yt = e-CT0 zt - e-CTn zt i ie-CTi zt . If the constants CT are fixed this expresses zt into the swap rate yt. Substituting this inverse relationship in the exponential form postulated for Dt,T we obtain a model for the discount rates in terms of the swap rates. The constants CT can be derived from the known values of the bond prices and the PVBP at time 0. The martingale property of the processes t Dt,T /Pt under the martingale measure S corresponding to the numeraire Pt[T0, . . . , Tn] gives the relations D0,T P0 = ES Dt,T Pt = ES e-CT zt i ie-CTi zt . We may use these relations for a suitable choice of t together with a model for zt (or indirectly the swap rate yt) under the martingale measure Sr and solve for the constants CT . 12.3: Calibration 139 12.3 Calibration The value at time 0 of a swaption with strike price K as considered in Section 10.5 is given by V0(K) = P0 ES yT0 - K + . Because a vanilla swaption is a commonly traded instrument, this value is known from the market at time 0 for a large number of strike prices K. A formal differentiation of V0(K) relative to K yields K V0(K) = -P0 ES1yT0 >K. The expectation on the right side is the survival function of the swap rate yT0 . In principle, the left side and the PVBP P0 are known from the market at time 0. This enables us to infer the distribution of the swap rate under the martingale measure S from observed market prices. An alternative is to model the swap rate as a geometric Brownian motion, as in Section 10.5. The parameters of this model can then be inferrred from the prices of a few swaptions. 12.4 Convexity Corrections Ad-hoc models used for option pricing, as used in Chapter 10, carry the danger that different, but related derivatives are not priced consistently. This could create the possibility of arbitrage within the model, which is certainly unrealistic, as the prices are based on the assumption of no arbitrage. It is thus important to price related products using a single model set-up. In this section we give an example. Consider pricing two (k = 1, 2) European options that guarantee a single payment at time T0 of sizes, for given times 0 < T0 < T (k) 1 < < T (k) n and a given FT0 -measurable variable Z, X(k) = n i=1 c (k) i DT0,T (k) i Z, k = 1, 2. The two claims possess the same form and include the same variable Z, but the maturities T (k) i of the discount rates on which they are based and the corresponding weights c (k) i may be different. One way to price these claims individually would be to choose as a numeraire the process Q (k) t = n i=1 c (k) i Dt,T (k) i , 140 12: Swap Rate Models and to model the distribution of the variable Z under the corresponding martingale measure. The option price process would then be equal to Q (k) t EN(k) (Z| Ft), where N(k) is the martingale measure. To price both options in this manner requires to model the variable Z under two different martingale measures, and it could be hard to make this modelling consistent between the two options. A different approach is to adopt the linear swap rate model of Section 12.1, choosing suitable times 0 < T0 < T1 < < Tn. In terms of the linear swap rate model we can write the claims as X(k) = A n i=1 c (k) i PT0 Z + n i=1 c (k) i BT (k) i yT0 PT0 Z. Here yt = yt[T0, T1, . . . , Tn] and Pt = Pt[T0, T1, . . . , Tn], for times T0 < T1 < < Tn that are chosen to represent the times T (k) i , but need not coincide with the latter times. The claims are sums of two terms, the first taking the form of a constant multiple of the product of the PVBP PT0 and the variable Z, the second having the same form except that yT0 PT0 takes the place of the PVBP. If the claim were given by only one of the two terms, then a convenient pricing strategy would be to choose the process P, or the process yP, as a numeraire, and to model the distribution of Z under the corresponding martingale measure. In the present situation we can use both numeraires, on the two corresponding terms of the option. A minor extension of the pricing formula (4.18) permits to express the value process of the claim X(k) as V (k) t = Pt A n i=1 c (k) i ES(Z| Ft) + ytPt n i=1 c (k) i BT (k) i EY(Z| Ft). Here S and Y are the martingale measures corresponding to the two numeraires P and yP. Expressing the process Q (k) t in the two numeraire processes according to the linear swap rate model, we can write this in the form V (k) t = Q (k) t w (k) t ES(Z| Ft) + (1 - w (k) t )EY(Z| Ft) , for the "weights" given by w (k) t = Pt/ i i Q (k) t / i c (k) i . Both numerator and denominator of the weights are a weighted average of discount rates, and the weights are unity if the times Ti and T (k) i and the raw weights i and c (k) i agree. We can compare the formula for the value process V (k) with the formula obtained if the two claims were priced individually, using the numeraires Q(k) . As we noted the latter procedure leads to a value process of 12.4: Convexity Corrections 141 the form ~V (k) = Qt (k) EN(k) (Z| Ft), where N(k) is the martingale measure corresponding to the numeraire Q(k) . Without a specification of the conditional expectations of the variable Z under the measures N(k) , S, and Y, a direct comparison of the different pricing formulas is impossible. However, if we would employ simple ad-hoc models for the distributions of Z under these measures, then quite possibly we would use the same model under the measures N(k) and S, as the numeraires Q(k) and P are similar in form: both numeraires are a linear combination of discount rates with maturities spread over a time interval following T0. If we would use the same model, then the conditional expectations EN(k) (Z| Ft) and ES(Z| Ft) would be identical, and the difference V (k) - ~V (k) between the two price processes would be equal to Q (k) t (1 - w (k) t ) EY(Z| Ft) - ES(Z| Ft) . This is known as a convexity correction. To turn the expression for V (k) into a concrete formula, it suffices to specify the distribution of the variable Z under the measures S and Y. Because these are the martingale measures corresponding to the numeraires P and yP these distributions are related, under the assumption of completeness, by their FT0 -density dY dS = yT0 PT0 /(y0P0) PT0 /P0 = yT0 y0 . The density process of Y relative to S is the process y/y0. If we specify the distribution of Z under S, then its distribution under Y is fixed. A simple possibility is to model the swap rate as a geometric Brownian motion under S, i.e. dyt = tyt dWt for a deterministic function and an S-Brownian motion W relative to the filtration Ft. By Girsanov's theorem the process ~W given by ~Wt = Wt - (y/y0)-1 [y/y0, W]t is an Y-Brownian motion. Because y = (y) W, this process takes the form ~Wt = Wt - t 0 s dWs. It follows that yt = y0e t 0 s dWs- 1 2 t 0 2 s ds = y0e t 0 s d ~Ws+ 1 2 t 0 2 s ds . If the function the function is strictly positive, then the augmented filtrations generated by the swap rate y and the driving Brownian motion W are the same. If we also assume that this filtration coincides with the given filtration Ft, then the preceding model for the swap rate is enough to determine the conditional expectations ES(Z| Ft) and EY(Z| Ft). Furthermore, if s is sufficiently regular, then the swap rate will be a Markov process, and these conditional expectations will be measurable functions of the swap rate yt at time t. * 12.2 EXERCISE. If ES(Z| Ft) = Ft(yt) for a measurable function Ft, show that EY(Z| Ft) = Ft yt exp( T0 t 2 s ds) . [True?]