CHAPTER 25
576
where
„„ the nricc given by the Black-Scholes-Meri
The price in d* tend; * Ppri« *, by
formula for a regular callopt.cn with stnx p
. nnte that the probability that the option will be exertased
option is (25.2)
„\k u/the holder of the «>» <**° J*^ ^ The value of the For a gap put option, the payon is a i   or»"<." r
when.- rf, and <f2 are defined as for equation (25.1). Example 25.7
An asset is currently worth $500,000. Over the next year, it is expected to have a volatility of 20%. The risk-free rate is 5%, and no income is expected. Suppose that an'insurance company agrees to buy the asset for $400,000 if its value has fallen below $400,000 at the end of one year. The payout will be 400,000 - ST whenever the value of the asset is less than $400,000. The insurance company has provided a regular put option where the policyholder has the right to sell the asset to the insurance company for $400,000 in one year. This can be valued using equation (14.21), with S0 = 500.000, K = 400.000, r = 0.05, n = 0.2, 7=1. The value is S3.436.
Suppose next that the cost of transferring the asset is $50,000 and this cost is borne by the policyholder. The option is then exercised only if the value of the asset is less than $350,000. In this case, the cost to the insurance company is A'i - ST when Sr < K2. where K2 = 350,000. K\ - 400,000, and ST is the price of the asset in one year. This is a gap put option. The value is given by equation (25.2), with 50 = 500.000. A', = 400,000, K2 = 350,000, r = 0.05, q = 0. a = 0.2. T = 1, It is SI.896. Recognizing the costs to the policyholder of making a claim reduces the cost of the policy to the insurance company by about 45% in this case.
25.4   FORWARD START OPTIONS
Forward start options are options that will start at some time in the future. Sometimes employee stock options, which were discussed in Chapter 15 can be viewed as forward start options. This is because the company commits (implicitly or explicitly) to granting at-the-money options to employees in the future
Cons,der a forward start at-the-money European call option that will start at time J, and mature at ..me T2 Suppose tha, the asset price is 5„ at time zero and 5, at time T,
Of'(
•j-0 val"e the option, we note fr nd 16 that the value of an at
577
1 ^Xti'f88 formuu,s in    • «
»M asset is proportional to the T, is therefore cS,/So, where c
t-to   r--T r—      - . .
and 16 that me value or an at-the-money"^"" °?"10n pricm8 h 3SSCt price. The value of the forward sta „ Sffi.0" ™ ™« * Proportion js the value at time zero or an at-the-mon«     ■  tm* 7 " ...ml valuation, the value- of .u„ c.      r omior» that
IS uk- J" at-tlie-rnonov n«' 1    ""-reiorc t
ncutra. valuation, the value of the tJSS^Z^       ^ W . Usin nsY-
SUrl °Pl'on at time zero is
where £ denotes the expected value in a risW n„. ■ ■
£[S,1 - S0e(r-",T\ the value or the forvwd ^« ^ and s» are known and
ying stock, q - 0 and the value or the forun". ?     " "    " For a "M-dividcnd-ofa regular at-the-money option with ns. S,art°pUon 15 cxaetly the same as the P   " Wth thc sa™ I* M the forward start option.
payi value
25.5  CLIQUET OPTIONS
A cliquet option (which is also called a ratchet or si riv-, „c ■    • v
put options with rules for determining the . nke nnt ? ^ '? ■""* °f Cal1 °r p  . v ...   """8 ">e stnke price. Suppose that the reset dates are
a, times r, 2r , („ - l)r wuh „, being the end of the clique* life. A simple structure would be as follows, The first opnen has a strike price K (which might equal the initio asset price) and lasts between times 0 and r; the second option provides a payoff at time 2r with a strike price equal to the value of the asset at time t; the third option provides a payoff at time 3t with a strike price equal to the value of the asset at time 2r and so on. This is a regular option plus n - 1 forward start options. The latter can be valued as described in Section 25.2.
Some cliquel options are much more complicated than the one described here. For example, sometimes there are upper and lower limits on the total payoff over the whole period; sometimes cliquets terminate at the end of a period if the asset price is in a certain range. When analytic results are not available, Monte Carlo simulation can often be used for valuation.
25.6 COMPOUND OPTIONS
Compound options are options on options. There are four main types of compound options: a call on a call, a put on a call, a call on a put. and a put on a put. Compound options have two strike prices and two exercise dates. Consider, Tor example, a call on a call. On the first exercise date, T,, the holder of the compound option is entitled to pay the first strike price. JC„ and receive a call option. The call option g.ves the holder the right to buy the underlying asset for the second strike price. K,. on the second exercise date, T, The compound option will be exea-ised on the first exercise date only ,f the value of the optton on that date is greater than the first «*PM. .
When the usual geometric ^^^^^^ compound options can be valued »nal>'ci,\,n 1, at llme zero of a European call normal distribution.2 With our usual notat.on, the value at time
I--- ,„,•„•'- Inumal of FmncM Economics. 7 (1979V. 63-81;
" See R Geske. "The Valua.ion of Compound Options J M. Rubinstein, "Double Trouble." &, D^mK-r 1991/Januar,
Skenovano pomoci CamScanner
424
19
appendix riSK-NEUTRAL DISTRIBUTIONS
DETERMINING IMPLIED RISK W FROM VOLATILITY SMILES
si ven bv r-u
c_e-^[ (ST-K)g{ST)dST
h      ■ ,h, interest rate (assumed constant). ST is the asset price at time T, and density function of S, DtfTerenttaUng once with res
to A' (rives
respcy
JST=K
dK
Differentiating again with respect to K gives
This shows that the probability density function g is given by
T d2c
9(K) = er
8K-
(19A.1)
This result, which is from Breeden and Litzenberger (1978), allows risk-neutral probability distributions to be estimated from volatility smiles.9 Suppose that c\, c2, and c, are the prices of T-year European call options with strike prices of K —5, K, and K + {, respectively. Assuming <5 is small, an estimate of g{K), obtained by approximating the partial derivative in equation (19A.1), is
rT c, 4- c3 - 2c, e £
For another way of understanding this formula, suppose you set up a butterfly spread with strike prices K-&.K, and K + S, and maturity 7. This means that you buy a call with strike price fC — S, buy a call with strike price K + 8, and sell two calls with strike price K. The value of your position is c, + c3 - 2c2. The value of the position can also be calculated b> integratmg the payoff over the risk-neutral probability distribution, g(ST\ and discounting at the risk-free rate. The payoff is shown in Figure 19A 1. Since i is small. Z TvoT™ *X = g<K> in the whole of the range I - S < 5r < if + «, where The ^alue of the payoff (when i is small) is therefore A«K&. It follows that
^W = C,+i-3-2c,
See D. T. Breeden and R. H. Liizenbcreer "Pr Jounal of Business, 51 (1978), 621-51    ~ ' of Sla|e-Contingcnt Claims Implicit in Opli°n
Prices-
425
Figure 19A.I    I'ayoir rrom butterfly spread payoff
25
which leads directly to
g{K) = /t°-l±Slz2£l
(19A.2)
Example 19A.1
Suppose that the price of a non-dividend-paying stock is S10, the risk-free interest rate is 3%, and the implied volatilities of 3-month European options with strike prices or S6, $7, $8, $9, SIO, SU, Sl2, Sl3, SI4 are 30%, 29%. 28%. 27%. 26% 25%, 24%, 23%, 22%, respectively. One way of apphing the above results is as follows. Assume that g(ST) is constant between ST = 6 and ST = 7, constant between ST = 7 and ST = 8. and so on. Define:
g(ST) = gl   for   6 < ST < 7 g(ST) = g2   for 7^Sr<8 g(Sr)=g3   for  8 < Sr < 9 g(ST) = gA  for 9 C 5r < 10 g{ST) = g5  for l(KSr<U g(ST) = g6  for llCSr<12 g(ST) = <ji   for 12<Sr<13 g{ST) = 08   for   13 < sr < 14 The value of 0, can be calculated by interpolating to get the implied volatility lor a 3-month option with a strike price of S6.5 as 29.5%. Tins means that opuons with strike prices of $6, S6.5. and S7 have unphed volauhues of 30%, 29.5 and 29%, respectively. From DerivaGem their prices are Mnd S3.055,
respectively. Using equation (19A.2), with K = 6.5 and & - 0.x pves ro.03x0^5(4 (us 4- 3.055 - 2 x 3.549) = QM51
0.5-
Similar calculations show that
= 0.0444, g
= 0.1545.   94 = 0.2781
g$
, =0.2813,   96 = 0-1659, 07
= 0.0573. 98=0.0113
594
cfa lit- ur • ■ x'
,l,e value of the exotic option on some b~.' !" :v,'rhedg^byshortin8
be more diffic
to hedging an exoM I nalcl.cs lions whose v»
his portfolio-
f>.
The exotic option is
FURTHER READING
■
iK The State of the Art. London: Thomson ] M Kamal and J. Zou, "More dianYou Ever Wanted
•ilftvand Variance: Options via Swaps," Risk, May 2007,7fi
Clewlow, L.. and C. Stric
Carr.P.a 3ewIow,
Press. 1997. „     , nnd j Zou,   more man . ™ ever wanted to v
Dcnietcrfi. K.. E. ^ * 4 <SUmmer-. '999)- 9~32'
;,boUi Volatility Swaps. . ,    . 0ptions Replication.  Journal of Derfath
Derman, E.. D. Er,cner and I. Kan,,
4 (Summer 1995): 78-95.
bathes, 2
4 (Summer 1995): 78 9 . options," Journal of Financial Economic,, 7 (1979): 63-Sl
Geske, ^"^eValuationofCompom.   P ^ ^ ^ ^ ,
Goldman. B, «/^N 4 n^lbcr 1979); 1111-27. °* Ma^'be r T "'of In Option to Exchange One Asset for Another," ^
Finance, 33 (March 1     . Options: The Sum of Lognormals and the Reciproca,
"Gamma^S*^ 33, 3 (September W)!
409-^7
RitchkeTp "On Pricing Barrier Options," Journal of Derivatives, 3, 2 (Winter 199 5): 19-28. Ruchken P.. L. Sankarasubramanian. and A.M. Vijh. "The Valuation of Path Dependent
Contracts on the Average," Management Science, 39 (1993). 1202-13. Rubinstein. M.. and E. Reiner. "Breaking Down the Barriers," Risk, September (1991): 28-35. Rubinstein. M„ "Double Trouble." fits*. December/January (1991/1992): 53-56. Rubinstein. M., "One for Another." Risk, July/August (1991): 30-32. Rubinstein. M.. "Options for the Undecided," Risk, April (1991): 70-73. Rubinstein. M., "Pay Now, Choose Later," Risk, February (1991): 44-47. Rubinstein. M., "Somewhere Over the Rainbow," 7?«*, November (1991): 63-66. Rubinstein, M.. "Two in One," Risk May (1991): 49.
Rubinstein, M., and E. Reiner. "Unscrambling the Binary Code," Risk, October 1991: 75-83. Stulz, R.M., "Options on the Minimum or Maximum of Two Assets," Journal of Financial
Economics. 10 (1982): 161-85. Turnbull, S.M., and L. M. Wakeman, "A Quick Algorithm for Pricing European Average
Options. Journal of Financial and Quantitative Analysis, 26 (September 1991): 377-89.
Practice Questions (Answers in Solutions Manual)
25.1. Explain the difference heiMfon a r
wt rw   u  l 0rward starl °Ption and a chooser option.
Sa^J*-*. - ta*. .oo^ck call - ■ »»«
25.3
Opt
ions
595
ronSidcr a chooser option where the holder has the rid,. ,„ i.
and a European put at any time during ,    8      choosc belwecn a European ■y lime during a 2-year period. The maturity dales and
C8
Sl0kC PISrHl IS "he        aTrC "T re8ardlC&5 °f When ^ is made, is
„ ever optimal to make the choice before the end of the 2-year period? Explain your answer.
25.4
Suppose that c, and p, arc the prices or a European average price call and a European average price put with strike price K and maturity T,     and p2 are the prices^ a European average strike call and European average strike put with maturity T. and c, and Pi are Out prices of a regular European call and a regular European put with strike price K and maturity T. Show that c, + c2 - c3 = p, + p2 - Pj.
25 5. The text derives a decomposition of a particular type of chooser option into a call maturing at time T2 and a put maturing at time T,. Derive an alternative decomposition into a call maturing at time T, and a put maturing at time 7%.
25.6. Section 25.8 gives two formulas for a down-and-out call. The first applies to the situation where the barrier, H, is less than or equal to the strike price, K. The second applies to the situation where H ^ K. Show that the two formulas are the same when H = K.
25.1. Explain why a down-and-out put is worth zero when the barrier is greater than the strike price.
25.8. Suppose that the strike price of an American call option on a non-dividend-paying stock grows at rate g. Show that if g is less than the risk-free rate. r. it is never optimal to exercise the call early.
25.9. How can the value of a forward start put option on a non-dividend-paying stock be calculated if it is agreed that the strike price will be 10% greater than the slock price at the time the option starts?
If a stock price follows geometric Brownian motion, what process does A(t) follow where A(0 is the arithmetic average stock price between time zero and time r?
Explain why delta hedging is easier for Asian options than for regular options.
25.12. Calculate the price of a 1-year European option to give up 100 ounces of silver in exchange for 1 ounce of gold. The current prices of gold and silver are S380 and S4, respectively; the risk-free interest rate is 10% per annum; the volatility of each commodity price is 20%; and the correlation between the two prices is 0.7. Ignore storage costs.
25.13. Is a European down-and-out option on an asset worth the same as a European down-and-out option on the asset's futures price for a futures contract maturing at the same time as the option?
25.14. Answer the following questions about compound options:
(a) What put-call parity relationship exists between the price of a European call on a call and a European put on a call? Show that the formulas given in the text satisfy the relationship.
(b) What put-call parity relationship exists between the price of a European call on a put and a European put on a put? Show that the formulas given in the text satisfy the relationship.
25.15. Does a floating lookback call become more valuable or less valuable as we increase the frequency with which we observe the asset price in calculating the minimum.
25.10'.
25.11.