Physica A 313 (2002) 238­251
www.elsevier.com/locate/physa
An introduction to statistical nance
Jean-Philippe Bouchauda;b;
aService de Physique de ľ Etat Condense, Centre ďetudes de Saclay, Orme des Merisiers,
91191 Gif-sur-Yvette Cedex, France
bScience & Finance, Capital Fund Management, 109-111 rue Victor-Hugo, 92532 France
Abstract
We summarize recent research in a rapid growing eld, that of statistical nance, also called
`econophysics'. There are three main themes in this activity: (i) empirical studies and the discovery
of interesting universal features in the statistical texture of nancial time series, (ii) the
use of these empirical results to devise better models of risk and derivative pricing, of direct
interest for the nancial industry, and (iii) the study of `agent-based models' in order to unveil
the basic mechanisms that are responsible for the statistical `anomalies' observed in nancial
time series. We give a brief overview of some of the results in these three directions. c 2002
Elsevier Science B.V. All rights reserved.
1. Introduction
The last 10 years have witnessed very signicant changes in nance, both as an
academic subject and as a professional eld. Finance is becoming an empirical (rather
than axiomatic) science; correspondingly, nancial engineers (quants) have an increasingly
important role to play in the nancial industry. There are various reasons for
this change, but the most important must surely be the availability of data, and the
possibility to access and process this data very quickly. Fifteen years ago, it was not
so easy to nd data, even on the most liquid markets. Now, one has direct access to
high-frequency data (on the scale of seconds) not only for stocks, currencies or interest
rates, but also for more exotic markets such as option markets, energy markets,
weather derivatives, etc. This means that any statistical model, or theoretical idea, can
and must be tested against available data, as physicists are (probably better than other
 Corresponding author. Service de Physique de ľEtat Condense, Centre ďetudes de Saclay, Orme des
Merisiers, 91191 Gif-sur-Yvette Cedex, France.
E-mail address: bouchau@drecam.saclay.cea.fr (J.-P. Bouchaud).
0378-4371/02/$ - see front matter c 2002 Elsevier Science B.V. All rights reserved.
PII: S 0378-4371(02)01039-7
J.-P. Bouchaud / Physica A 313 (2002) 238­251 239
communities) trained to do. When data was scarce, logical consistency and simplicity
were the only guides; not looking at data nowadays is just an excuse for sparing
mathematically beautiful, but often inadapted models. Correspondingly, quants in banks
are asked to `back-tesť trading strategies, option prices or risk estimates against past
data, and understand in detail any discrepancies or artefacts.
From an academic point of view, nancial time series represent an extremely rich
and fascinating source of questions, where a trace of human activity is recorded and
stored in a quantitative way, sometimes over hundreds of years. What are then the
statistical features of a nancial time series? Does it share common signatures with
other signals that physicists have learned to cope with? Once we have a good model
for price changes, what is it useful for? Do we understand the basic mechanisms, in
terms of human psychology, market micro-structure, etc., that are responsible for the
observed statistical peculiarities of price changes, and their universality across markets
and epochs? These are the questions that were addressed during the 2001 Altenberg
lectures, and which the following notes will brie y summarize. A fuller account can
be found in recently published books on the subject [1­3]. Other sources of information
about this eld can be found in Ref. [4], on www.science-finance.fr and
in the recent journals International Journal of Theoretical and Applied Finance and
Quantitative Finance.
2. `Stylized facts' about nancial time series
The study of price changes (actually relative price changes) on many di erent assets
(stocks, stock indices, currencies, etc.) has revealed a number of robust features that
people in the eld like to call `stylized facts' (see e.g. Ref. [5] for a recent review).
Let us list a few of them, which are most relevant for option pricing and risk control,
and that a microscopic (trader-based) model should in ne be able to account
for:
* Relative price changes are in a good approximation uncorrelated beyond a time scale
of the order of tens of minutes (on liquid markets). This means that the square of the
(log) price di erence grows linearly with time, with a prefactor called the volatility.
This volatility is of the order of 3% per square-root day for individual stocks,
1% for stock indices but only 0.03% for short-term interest rates. A more detailed
analysis, however, shows that some small correlations are present on the scale of a
few days.
* The distribution of relative price changes Á is strongly nonGaussian: these distributions
can be characterized by Pareto (power-law) tails Á-1with
an exponent
close to 3 for rather liquid markets [6­11]. Emerging markets have even more
extreme tails, with an exponent that can be less than two--in which case the
volatility is innite.
* Another striking feature is the intermittent nature of the uctuations: localized outbursts
of volatility can clearly be identied. This feature, known as volatility clustering
[12,13,2,1], is very reminiscent of similar intermittent uctuations in turbulent
240 J.-P. Bouchaud / Physica A 313 (2002) 238­251
ows [14]. This e ect can be analysed more quantitatively: the temporal correlation
function of the daily volatility can be tted by an inverse power of the lag, with
a rather small exponent in the range 0.1­0.3 [13,15­18]. This slow decay of the
volatility correlation function leads to a multifractal-like behaviour of price changes
[18­24]: the kurtosis of the (log) price di erence only decays as a small power of
time, rather than the inverse of time as would be the case if volatility was constant
or only had short-ranged correlations [15,1,24]. This slow decay of the kurtosis has
important consequences for option pricing [15,25] (see below).
* The volume of activity also shows long-ranged correlations, very similar to those
observed on the volatility. This is not surprising since volatility and volume are
strongly correlated [26,27].
* Past price changes and future volatilities are negatively correlated--this is the
so-called `leverage e ecť, which re ects the fact that markets become more
active after a price drop, and then to calm down when the price rises. This correlation
is most visible on stock indices and is characterized by a time scale of the
order of 10 days [28]. This leverage e ect leads to an anomalous negative skew in
the distribution of price changes as a function of time, and is again important for
option pricing [29].
* Inter-stock correlations also show very interesting features, such as an apparent increase
of inter-stock correlations in volatile periods [30]. This is most relevant for
risk control, since an increase of correlations means that risk diversication is more
di cult. The full correlation matrix between all pairs of stocks can be studied, and
leads to the following message: most of the eigenvalues of this matrix are well accounted
for by a random matrix theory [31,32]. The dominant part of the correlation
can be explained by a simple `one-factor' model, where stock price changes share
a common factor, called the `markeť [33­35].
* Interest rates corresponding to di erent maturities also evolve in an interesting
correlated manner, which recalls the motion of an elastic string subject to noise.
For recent work on this subject, see Refs. [36,37].
The most important message of these empirical studies is that prices behave very
di erently from the simple geometric Brownian motion which is often assumed in
mathematical nance: extreme events are much more probable, and interesting nonlinear
correlations (volatility­volatility and price­volatility) are observed.
3. Implications for option pricing
3.1. General framework
We now turn to the problem of option pricing and hedging when the statistics for
price increments x have the nonGaussian properties discussed above. The distinctive
feature of the continuous time random walk model usually considered in the theory
of option pricing is the possibility of perfect hedging (for a remarkable introduction
see Ref. [38]), that is, a complete elimination of the risk associated to option trading
[38,39]. This property, however, no longer holds for more realistic models [1].
J.-P. Bouchaud / Physica A 313 (2002) 238­251 241
Let us write down the global wealth balance W|T
0 associated with the writing of
a `calľ option on an asset of price x(t), of maturity T and exercise price xs [1]:
W|T
0 = C(x0; xs; T) exp(rT) - max(x(T) - xs; 0)
+
i
(xi; ti) exp(r(T - ti))[xi+1 - xi - rxi ] ; (1)
where C(x0; xsT) is the price of the call, x0=x(t=0), xi=x(ti) is the price of the asset on
which the option is written, and (x; t) the trading strategy, i.e., the number of stocks
per option in the portfolio of the option writer. Finally, r is the (constant) interest rate,
and the elementary time interval. This wealth balance contains three terms:
* The rst term is the price of the contract, paid at time t = 0 and used to buy bonds
that yield the rate r.
* The second term denes the option contract: the prot of the buyer of the option is
equal to xs -x(T) if x(T)  xs (i.e., if the option is exercised) and zero otherwise--
the option is an insurance contract which guarantees to its owner a maximum price
for acquiring a certain stock at time T. Conversely, a `puť option would guarantee
a certain minimum price for the stock held by the owner of the option.
* The third term counts the prot or loss (over the risk-free rate) incurred due to the
trading strategy (see Ref. [1] for details).
A natural procedure to x the price of the option C(x0; xs; T) and the optimal strategy

(x; t) was proposed in Refs. [40­42] and further discussed in Ref. [1]. It consists
in imposing a fair game condition, i.e.:
W|T
0 [ ] = 0 (2)
and a risk minimization condition:
W|T
0 [ ]2
(x; t) 
= 0 : (3)
Here, we assume that the variance of the wealth variation is a relevant measure of the
risk. However, other measures are possible, such as higher moments of the distribution
of W, or the `value-at-risk', which is directly related to the weight contained in the
negative tails of the distribution of W|T
0 [43,1].
The notation    in Eqs. (2) and (3) means that one averages over the probability
of the di erent trajectories. The explicit solution of Eqs. (2) and (3) for a general
uncorrelated process (i.e., xi xj =0 for i = j, where xi =xi+1 -xi) is relatively easy
to write if the average bias xi and the interest rate r are negligible, 1
which is the
case for short maturities T. In this case, one nds:
C(x0; xs; T) =

xs
dx (x - xs)P(x ; T|x0; 0) ; (4)

(x; t) =

xs
dx x (x;t)(x ;T)
(x - xs)
2(x; t)
P(x ; T|x; t) ; (5)
1 For the general case, see Refs. [1,44­46].
242 J.-P. Bouchaud / Physica A 313 (2002) 238­251
where 2
(x; t) = x2
|x;t is the `local volatility'--which may depend on x; t--and
x (x;t)(x ;T) is the mean instantaneous increment conditioned to the initial condition
(x; t) and a nal condition (x ; T). The minimal residual risk, dened as
R
= W|T
0 [ 
]2
is in general strictly positive (and in practice rather large), except
for Gaussian uctuations in the continuous limit, where the residual risk is strictly
zero! In this limit, the above Eqs. (2) and (3) actually exactly lead to the celebrated
Black­Scholes option pricing formula. In particular, one can indeed check that 
is
in that case related to C through: 
= @C(x0; xs; T)=@x0, which corresponds to the
so-called -hedge found by Black and Scholes.
3.2. Cumulant expansion and volatility smile
In the case where the market uctuations are moderately nonGaussian, one might
expect that a cumulant expansion around the Black­Scholes formula leads to interesting
results. This cumulant expansion has been worked out in general in Ref. [1] (see
also Ref. [25]), both for the price and for the optimal strategy. If one only retains
the leading order correction which is (for symmetric uctuations) proportional to the
kurtosis ÄT , one nds that the price of options C(x0; xs; T) can be written as a Black­
Scholes formula, but with a modied value of the volatility , which becomes price
and maturity dependent [15]:
imp:(xs; T) = 1 +
ÄT
24
(xs - x0)2
2T
- 1 : (6)
The volatility imp: is called the implied volatility by the market operators, who use
the standard Black­Scholes formula to price options, but with a value of the volatility
which they estimate intuitively, and which turns out to depend on the exercise price
in a roughly parabolic manner, as indeed suggested by Eq. (6).
This is the so-called `volatility smile'. Eq. (6), furthermore, shows that the curvature
of the smile is directly related to the kurtosis ÄT of the underlying statistical process
on the scale of the maturity T = N . When the price distribution is skewed, as is the
case for stock indices where the leverage e ect induces a signicant negative skew,
there are corrections to the above smile formula: the smile itself become asymmetric.
We have tested this prediction by directly comparing the `implied kurtosis', obtained
by extracting from real option prices (on the BUND market) the volatility (which
turns out to be highly correlated with a short time lter of the historical volatility), and
the curvature of the implied volatility smile, to the historical value of the kurtosis ÄT
discussed above. We nd a remarkable agreement between the implied and historical
kurtosis [15,1]. This and the fact that they evolve similarly with maturity, shows that the
market as a whole is able to correct (by trial and errors) the inadequacies of the Black­
Scholes formula, to encode in a satisfactory way both the fact that the distribution
has a positive kurtosis, and that this kurtosis decays in an anomalous fashion due to
volatility persistence e ects. However, the real risks associated with option trading are,
at present, not satisfactorily estimated by market participants. In particular, most risk
control softwares dealing with option books are based on a Gaussian description of the
uctuations.
J.-P. Bouchaud / Physica A 313 (2002) 238­251 243
4. Simple models for herding and mimicry
We now turn to simple models for thick tails in the distribution of price increments
in nancial markets. An intuitive explanation is herding: if a large number of agents
acting on markets coordinate their action, the price change is likely to be huge due to
a large imbalance between buy and sell orders [47]. However, this coordination can
result from two rather di erent mechanisms.
* One is the feedback of past price changes onto themselves, which we will discuss
in the following section. Since all agents are in uenced by the very same
price changes, this can induce nontrivial collective behaviour: for example, an accidental
price drop can trigger large sell orders, which lead to further downward
moves.
* The second is direct in uence between the traders, through exchange of information
that leads to `clusters' of agents sharing the same decision to buy, sell, or be inactive
at any given instant of time.
4.1. Herding and percolation
A simple model of how herding a ects the price uctuations was proposed in
Ref. [48]. 2
It assumes that the price increment x depends linearly on the instantaneous
o set between supply and demand [48,49]. More precisely, if each operator in
the market i wants to buy or sell a certain xed quantity of the nancial asset, one
has [48]: 3
x =
1
i
'i ; (7)
where 'i can take the values -1; 0 or +1, depending on whether the operator i is
selling, inactive, or buying, and is a measure of the market depth. Note that the linearity
of this relation, even for small arguments, has been questioned by Zhang [50].
Recent empirical analysis, however, seems to conrm that the relation is indeed linear
for small arguments, but bends down and even saturates for larger
arguments [51].
Suppose now that the operators interact among themselves in an heterogeneous manner:
with a small probability c=N (where N is the total number of operators on the
market), two operators i and j are `connecteď, and with probability 1 - c=N, they
ignore each other. The factor 1=N means that on average, the number of operators
connected to any particular one is equal to c (the resulting graph is precisely the
same as the random trading graph of Section 3.1). Suppose nally that if two operators
are connected, they come to agree on the strategy they should follow, i.e.,
'i = 'j.
2 See the numerous references of this paper for other works on herding in economics and nance.
3 This can alternatively be written for the relative price increment x=x, which is more adapted to describe
long time scales. On short time scales, however, an additive model is often preferable. See the discussion
in Refs. [1,28].
244 J.-P. Bouchaud / Physica A 313 (2002) 238­251
It is easy to understand that the population of operator clusters into groups sharing
the same opinion. These clusters are dened such that there exists a connection between
any two operators belonging to this cluster, although the connection can be indirect
and follow a certain `path' between operators. These clusters do not have all the same
size, i.e., do not contain the same number of operators. If the size of cluster C is called
S(C), one can write:
x =
1
C
S(C)'(C) ; (8)
where '(C) is the common opinion of all operators belonging to C. The statistics
of the price increments x therefore reduces to the statistics of the size of clusters, a
classical problem in percolation theory [52]. One nds that as long as c  1 (less than
one `neighbour' on average with whom one can exchange information), then all S(C)'s
are small compared to the total number of traders N. More precisely, the distribution
of cluster sizes takes the following form in the limit where 1 - c = 
1:
P(S) ˙S1
1
S5=2
exp - 2
S; S
N : (9)
When c=1 (percolation threshold), the distribution becomes a pure power-law with an
exponent = 3
2 , and the Central Limit Theorem tells us that in this case, the distribution
of the price increments x is precisely a pure symmetrical Levy distribution of index
= 3
2 [1] (assuming that ' = 1 play identical roles, that is if there is no global
bias pushing the price up or down). If c  1, on the other hand, one nds that the
Levy distribution is truncated exponentially, leading to a larger e ective tail exponent
[48]. If c  1, a nite fraction of the N traders have the same opinion: this leads
to a crash. This simple model has been extended in several directions by Stau er and
collaborators [53]. Very recently, a somewhat related model was studied in Ref. [54]
where each agent probes the opinion of a pool of m randomly selected agents. The
agent then chooses either to conform to the majority opinion or to be contrarian if
the majority is too strong. This interesting model leads to various types of behaviour,
including a chaotic phase.
4.2. Avalanches of opinion changes
The above simple percolation model is interesting but has one major drawback:
one has to assume that the parameter c is smaller than one, but relatively close to
one such that Eq. (9) is valid, and nontrivial statistics follows. One should thus explain
why the value of c spontaneously stabilizes in the neighbourhood of the critical
value c = 1. Furthermore, this model is purely static, and does not specify how the
above clusters evolve with time. As such, it cannot address the problem of volatility
clustering. Several extensions of this simple model have been proposed [53,55], in particular
to increase the value of from = 3=2 to  3 and to account for volatility
clustering.
One particularly interesting model is inspired by the recent work of Dahmen and
Sethna [56,57], that describes the behaviour of random magnets in a time-dependent
J.-P. Bouchaud / Physica A 313 (2002) 238­251 245
magnetic eld. Transposed to the present problem (as rst suggested in Ref. [1]), this
model describes the collective behaviour of a set of traders exchanging information,
but having all di erent a priori opinions. One trader can, however, change his mind
and take the opinion of his neighbours if the coupling is strong, or if the strength of
his a priori opinion is weak. More precisely, the opinion 'i(t) of agent i at time t is
determined as
'i(t) = sign

hi(t) +
N
j=1
Jij'j(t)

 ; (10)
where Jij is a connectivity matrix describing how strongly agent j a ects agent i, and
hi(t) describes the a priori opinion of agent i: hi  0 means a propensity to buy, hi  0
a propensity to sell. We assume that hi is a random variable with a time-dependent
mean h(t) and root mean square . The quantity h(t) represents for example condence
in long-term economy growth (h(t)  0), or fear of recession (h(t)  0), leading to
a nonzero average pessimism or optimism. Suppose that one starts at t = 0 from a
`euphoric' state, where h ; J, such that 'i = 1 for all i's. 4
Now, condence is
decreased progressively. How will sell orders appear?
Interestingly, one nds that for small enough in uence (or strong heterogeneities of
agents' anticipations), i.e., for J
 , the average opinion O(t) = i 'i(t)=N evolves
continuously from O(t = 0) = +1 to O(t  ) = -1. The situation changes when
imitation is stronger since a discontinuity then appears in O(t) around a `crash' time
tc, when a nite fraction of the population simultaneously change opinion. The gap
O(tc
) - O(t+
c ) opens continuously as J crosses a critical value Jc( ) [56]. For J
close to Jc, one nds that the sell orders again organize as avalanches of various sizes,
distributed as a power-law with an exponential cut-o . In the `mean-elď case where
Jij  J=N for all ij, one nds =5=4. Note that in this case, the value of the exponent
is universal, and does not depend, for example, on the shape of the distribution of
the hi's, but only on some global properties of the connectivity matrix Jij.
A slowly oscillating h(t) can therefore lead to a succession of bull and bear markets,
with a strongly nonGaussian, intermittent behaviour, since most of the activity is
concentrated around the crash times tc. Some modications of this model are, however,
needed to account for the empirical value  3 observed on the distribution of price
increments (see the discussion in Ref. [53]).
Note that the same model can be applied to other interesting situations, for example
to describe how applause end in a concert hall (here, ' = 1 describes, respectively,
a clapping and a quiet person, and O(t) is the total clapping activity). Clapping can
end abruptly when imitation is strong, or smoothly when many fans are present in the
audience. A static version of the same model has been proposed to describe rational
group decision making [58].
4 Here J denotes the order of magnitude of j Jij.
246 J.-P. Bouchaud / Physica A 313 (2002) 238­251
5. Models of feedback e ects on price uctuations
5.1. Risk-aversion induced crashes
The above average `stimulus' h(t) may also strongly depend on the past behaviour
of the price itself. For example, past positive trends are, for many investors, incentives
to buy, and vice versa. Actually, for a given trend amplitude, price drops tend to feed
back more strongly on investors' behaviour than price rises. Risk-aversion creates an
asymmetry between positive and negative price changes [59]. This is re ected by option
markets, where the price of out-of-the-money puts (i.e., insurance against crashes) is
anomalously high.
Similarly, past periods of high volatility increases the risk of investing in stocks, and
usual portfolio theories then suggest that sell orders should follow. A simple mathematical
transcription of these e ects is to write Eq. (7) in a linearized, continuous time
form: 5
dx
dt
 u =
1
h(t) ; (11)
and write a dynamical equation for h(t) which encodes the above feedback e ects
[59,49]:
dh
dt
= au - bu2
- cu + Á(t) ; (12)
where a describes trends following e ects, b risk aversion e ects (breaking the u  -u
symmetry), c is a mean reverting term which arises from market clearing mechanisms
(the very fact that the price moves clears a certain number of orders), and Á is a noise
term representing random external news [59]. Eliminating h from the above equations
leads to
du
dt
= - u - u2
+
1
Á(t)  -
@V
@u
+
1
Á(t) ; (13)
where =(c-a)= and =b= . Eq. (13) represents the evolution of the position u of a
viscous ctitious particle in a `potentiaľ V(u)= u2
=2+u3
=3. If trend following e ects
are not too strong, is positive and V(u) has a local minimum for u = 0, and a local
maximum for u
= - =, beyond which the potential plummets to -. 6
A `potential
barrier' V
= u2
=6 separates the (meta-)stable region around u = 0 from the unstable
region. The nature of the motion of u in such a potential is the following: starting at
u = 0, the particle has a random harmonic-like motion in the vicinity of u = 0 until
an `activateď event (i.e., driven by the noise term) brings the particle near u
. Once
this barrier is crossed, the ctitious particle reaches - in nite time. In nancial
5 In the following, the herding e ects described by Jij are neglected, or more precisely, only their average
e ect encoded by h is taken into account.
6 If is negative, the minimum appears for a positive value of the return u. This corresponds to a
speculative bubble. See Ref. [59].
J.-P. Bouchaud / Physica A 313 (2002) 238­251 247
terms, the regime where u oscillates around u = 0 and where  can be neglected, is
the `normaľ uctuation regime. This normal regime can, however, be interrupted by
`crashes', where the time derivative of the price becomes very large and negative,
due to the risk aversion term b which destabilizes the price by amplifying the sell
orders. The interesting point is that these two regimes can be clearly separated since
the average time t
needed for such crashes to occur is exponentially large in V
[60],
and can thus appear only very rarely. A very long time scale is therefore naturally
generated in this model.
Note that in this line of thought, a crash occurs because of an improbable succession
of unfavourable events, and not due to a single large event in particular. Furthermore,
there are no `precursors' in this model: before u has reached u
, it is impossible to
decide whether it will do so or whether it will quietly come back in the `normaľ
region u 0. Solving the Fokker­Planck equation associated to the Langevin equation
(13) leads to a stationary state with a power law tail for the distribution of u (i.e., of
the instantaneous price increment) decaying as u-2
for u  -. More generally,
if the risk aversion term took the form -b|u|1+
, the negative tail would decay
as u-1-
.
5.2. Dynamical volatility models
The simplest model that describes volatility feedback e ects is to write an ARCH-like
equation [61], which relates today's activity to a measure of yesterday's activity, for
example:
k = k-1 + K( 0 - k-1) + g| xk-1| ; (14)
where 0 is an average volatility level, K a mean-reverting term, and g describes how
much yesterday's observed price change a ects the behaviour of traders today. Since
| xk-1| is a noisy version of k-1, the above equation is, in the continuous time limit,
a Langevin equation with multiplicative noise:
d
dt
= K( 0 - ) + g Á(t) : (15)
The stationary distribution corresponding to this equation is
Peq( ) =
( - 1)
[ ]
e-( -1)=
1+
; (16)
where the tail exponent is by - 1 ˙ K=g2
: over-reactions to past informations (i.e.,
large values of g) decreases the tail exponent . Therefore, one nds a nonuniversal
exponent in this model, which is bequeathed to the distribution of price increments
if one assumes that the volatility uctuations are the dominant cause of large
changes.
Note that the temporal correlation function of the volatility can be exactly calculated
within this model [62], and is found to be exponentially decaying, at variance
with the slow power-law (or logarithmic) decay observed empirically. Furthermore,
248 J.-P. Bouchaud / Physica A 313 (2002) 238­251
distribution (16) does not concur with the nearly log-normal distribution of the volatility
that seems to hold empirically [16,67].
At this point, the slow decay of the volatility can be ascribed to two rather di erent
mechanisms. One is the existence of traders with many di erent time horizons, as
suggested in Refs. [63,15]. If traders are a ected not only by yesterday's price change
amplitude | xk-1|, but also by price changes on coarser time scales |xk - xk-p|, then
the correlation function is expected to be a sum of exponentials with decay rates given
by p-1
. Interestingly, if the di erent p's are uniformly distributed on a log scale, the
resulting sum of exponentials is to a good approximation decaying as a logarithm, as
advocated in Ref. [18]. More precisely:
C( ) =
1
log(pmax=pmin)
pmax
pmin
d(log p) exp(- =p)
log(pmax= )
log(pmax=pmin)
; (17)
whenever pmin
 
pmax. Now, the human time scales are indeed in a natural pseudogeometric
progression: hour, day, week, month, trimester, year [15].
Yet, some recent numerical simulations of traders allowed to switch between different
strategies (active=inactive, or chartist=fundamentalist) suggest strongly intermittent
behaviour [55,64­66,68], and a slow decay of the volatility correlation function
without the explicit existence of logarithmically distributed time scales. Is there a
simple, universal mechanism that could explain these ubiquitous long-range volatility
correlations?
A possibility, discussed in Ref. [69], is that the volume of activity exhibits long-range
correlations because agents switch between di erent strategies depending on their relative
performance. Imagine for example that each agent has two strategies, one active
strategy (say trading every day), and one inactive, or less active strategy. The `score'
of the inactive strategy (i.e., its cumulative prot) is constant in time, or more precisely
equal to the long-term average growth rate. The score of active strategy, on the other
hand, uctuates up and down, due to the uctuations of the market prices themselves.
Since to a good approximation the market prices are not predictable, this means that
the score of any active strategy will behave like a random walk, with an average equal
to that of the inactive strategy (assuming that transaction costs are small). Therefore,
on some occasions the score of the active strategy will happen to be higher than that
of the inactive strategy and the agent will be active, before the score of the active
strategy crosses that of the inactive strategy. The time during which an agent is active
is thus a random variable with the same statistics as the return time to the origin of
a random walk (the di erence of the scores of the two strategies). Interestingly, the
return times of a random walk are well known to be very broadly distributed: the average
return time is actually innite. Hence, if one computes the correlation of activity
in such a model, one nds long-range correlations due to long periods of times where
many agents are active (or inactive). One can consider a specic model (the `Grand
Canonical Minority Game'; for papers on minority game see e.g. Ref. [70]) where this
scenario can be studied more quantitatively, and have found that indeed long-range
correlations in the volume of activity are observed: see Fig. 1. More precisely, the
J.-P. Bouchaud / Physica A 313 (2002) 238­251 249
0 10000 20000 30000
Time
260
280
300
320
340
360
380
400
420
440
Volume
1 10 100 1000

0.01
0.10
1.00
Va()
Minority Game
1/2
0 5 10 15
1/2 (days1/2)
2.0e+05
4.0e+05
6.0e+05
8.0e+05
1.0e+06
Volumevariogram
Data
Minority Game
0 1000 2000 3000
t (days)
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Volume
(a) (b)
Fig. 1. (a) Volume of activity v(t) (number of active agents) as a function of time for the Grand Canonical
Minority Game (GCMG). Inset: The corresponding activity variogram (v(t + ) - v(t))2 , as a function of
the lag , in a log­log plot to emphasize the predicted

singularity at small 's. (b) Total daily volume of
activity (number of trades) on the S&P 500 future contracts in the years 1985­1998. Inset: Corresponding
variogram (diamonds) as a function of the square root of the lag. Note the clear linear behaviour for small

. The full line is the GCMG t, with both axes rescaled and a constant added to account for the presence
of `white noise'trading.
variogram of the volume of activity, dened as V( ) = (v(t + ) - v(t))2
t, can be
very accurately tted by
V( )|fit = V
1 - exp -
0
: (18)
This simple model even allows one to reproduce quantitatively the volume of activity
correlations observed on the New York stock exchange market: see Fig. 1. This
mechanism is very generic and probably also explains why this e ect arises in more
realistic market models [64,68].
As shown in Ref. [69], this mechanism can thus explain the long-range volatility
correlations observed on all nancial markets. However, this interpretation is quite
di erent from the `cascade' picture proposed in Refs. [19,21,18], where the volatility
results from some sort of multiplicative random process (which actually naturally
leads to the log-normal volatility distribution actually observed empirically). More work
is needed to establish which of these two mechanisms is most relevant in nancial
markets.
250 J.-P. Bouchaud / Physica A 313 (2002) 238­251
Acknowledgements
I would like to thank in particular P. Cizeau, R. Cont, I. Giardina, P. Gopikrishnan,
L. Laloux, T. Lux, A. Matacz, M. Mezard, and M. Potters for many discussions on
these matters.
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