Voit J. The Statistical Mechanics of Financial Markets

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180 6. Turbulence and Foreign Exchange Markets

At this point, recall the Gaussian distribution which describes the Wiener

stochastic process. All Gaussians can be collapsed onto each other by rescal-

ing with the standard deviation, in much the same way as we rescaled the

empirical distribution functions above. For a Wiener process, we know that

σ ∝

√

T − t, so that the empirical standard deviation of an eventually in-

complete data set is not needed. Above, in (6.10), the empirical standard

deviation was used. From the general similarity of the two procedures, one

may wonder if the similarity between turbulence and stochastic processes is

superﬁcial only. Or: can turbulence be described as a stochastic process in

length scales, instead of (or in addition to) time?

In order to pursue this question, we ﬁrst check the Markov property, cf.

Sect. 4.4.1. Markov processes satisfy the Chapman–Kolmogorov–Smoluchow-

ski equation, (3.10), which we rewrite as

p(δ˜v

,λ

|δ˜v

,λ



∞

−∞

d˜v

p(δ˜v

,λ

|δ˜v

,λ

)p(δ˜v

,λ

|δ˜v

,λ

) . (6.13)

λ has been deﬁned above, and the velocities have been rescaled as

δ˜v = δv





/ . (6.14)

This form of rescaling is suggested by theoretical arguments [134, 136]. Under

the assumption that the eddies are space-ﬁlling, and that the downward en-

ergy ﬂow is homogeneous, on can derive a scaling relation ξ

= n/3 between

the exponents ξ

of the structure functions, and their order n which is rather

well satisﬁed by the data at least for small n, cf. Fig. 6.7 below.

Indeed, the empirical data satisfy the Chapman–Kolmogorov–Smoluchow-

ski equation: one can superpose the conditional probability density function

p(δ˜v

,λ

|δ˜v

,λ

) derived from the experimental data, with that calculated

according to (6.13), using experimental data on the right-hand side of the

equation. The result very well matches the p(δ˜v

,λ

|δ˜v

,λ

) measured di-

rectly. As a consequence, one is allowed to iterate the experimental prob-

ability density function for 

(not shown in Fig. 6.4) with (6.13) and the

experimentally determined conditional probability distributions. The results

of this procedure (shown as crosses in Fig. 6.4) exactly superpose the exper-

imental data on scales <

, shown as circles.

When the Chapman–Kolmogorov–Smoluchowski equation is fulﬁlled, one

may search for a description of the scale-evolution of the probability density

functions in terms of a Fokker–Planck equation. Quite generally, one can

convert the convolution equation (6.13) into a diﬀerential form by a Kramers–

Moyal expansion [37],

∂p(δ˜v

,λ

|δ˜v

,λ

)

∂λ

∞



n=1



−

∂

∂(δ˜v

)

(n)

(δ˜v

,λ

) p(δ˜v

,λ

|δ˜v

,λ

)



(6.15)

The Kramers–Moyal coeﬃcients are deﬁned as

6.2 Turbulent Flows 181

(n)

(δ˜v

,λ

lim

→λ

− λ



∞

−∞

d(δ˜v

)(δ˜v

− δ˜v

)

p(δ˜v

,λ

|δ˜v

,λ

) .

(6.16)

If all D

(n)

with n>2 vanish, one obtains a Fokker–Planck partial diﬀerential

equation

∂p(δ˜v

,λ

|δ˜v

,λ

)

∂λ



−

∂

∂(δ˜v

)

(1)

(δ˜v

,λ

) (6.17)

∂

∂(δ˜v

)

(2)

(δ˜v

,λ

)



p(δ˜v

,λ

|δ˜v

,λ

) ,

and the stochastic process is completely characterized by the drift and dif-

fusion “constants” D

(1)

and D

(2)

. It turns out that, within experimental

accuracy, this is the case, and

(1)

(δ˜v, λ) ∼−δ˜v, (6.18)

(2)

(δ˜v, λ) ∼ a(λ)+b(λ)(δ˜v)

. (6.19)

Great care must be taken when estimating these quantities from actual data.

In particular, for a discrete, ﬁnite sampling interval contributions from the

drift terms D

(1)

may contaminate the esimators of D

(2)

and lead to incorrect

estimates of the parameters in (6.19) [137]. This may have aﬀected actual

estimates [136, 137], but our general conclusions are robust. A related ob-

servation has been made from the perspective of integration of stochastic

diﬀerential equations by Timmer [138].

In this perspective, turbulence is described as a stochastic process over a

hierarchy of length scales. The drift term contains the systematic downward

ﬂow of energy postulated by the cascade model. The diﬀusion term describes

the ﬂuctuations around the otherwise deterministic cascade [136], and shows

that there is a strong random component in this energy cascade. This is

connected with the indeterminacy of the number and size of the smaller

eddies produced from one big eddy, as one drifts down the cascade.

6.2.3 Relation to Non-extensive Statistical Mechanics

When the evolution of the probability density of a stochastic process is de-

scribed by a Fokker–Planck equation, an equivalent stochastic diﬀerential

equation for the stochastic variable can be found, and it takes the form of

a Langevin equation, (5.86) [37]. A general form of a non-linear Langevin

equation is [93]

= −γF(x)+ση(t) . (6.20)

It is not necessary to consider explicitly a non-linearity in the diﬀusion term,

as it can be reduced to a constant by a transformation of variables [37].

F (x)=−∂U(x)/∂x is the non-linear force. If U (x)=C|x|

2α

, the non-linear

182 6. Turbulence and Foreign Exchange Markets

Langevin equation generates one of the power-law probability distributions

of non-extensive statistical mechanics, cf. Sect. 5.5.7. The parameters are

identiﬁed as

q =1+

2α

αn +1

β =

2α

1+2α − q

. (6.21)

As in Sect. 5.5.7,

β =1/k

T is the inverse temperature and n the number of

degrees of freedom of the χ

-distribution used to describe the slow temporal

ﬂuctuations of the parameters of the Langevin equation [93].

Assume now that a test particle in a turbulent ﬂow moves for a while

in a region with a (ﬂuctuating) energy-dissipation rate 

on scale r. τ is

a typical time scale during which energy is dissipated, typically the time

of sojourn in the region with 

.Thenβ = 

τΛ is a ﬂuctuating quantity,

and as a model we may assume that it is χ

-distributed. Λ is a constant

necessary to adjust the dimensions of β. At the smallest scale, the dissipation

scale η, β = u

Λ,whereu

is a ﬂuctuating velocity. In the simplest model,

the three components of u

would ﬂuctuate independently and would be

drawn from Gaussian distributions with mean zero. This would suggest that

n = 3, and with α ≈ 1 (weak non-linearity of the forcing potential) would

give q ≈ 3/2. These values are in very good agreement with experimental

data on velocity diﬀerences of a test particle over very small time scales

in turbulent Taylor–Couette ﬂow with a Reynolds number R = 200 [93,

139]. The probability distributions observed there are rather similar to those

depicted in Figs. 6.4 and 6.5. At the dissipation scale, the ﬂuctuations of



r=η

can be viewed in terms of the ordinary diﬀusion of a particle of mass M

which is subject to noise of a temperature T =1/k

β. The changes of the

distributions as the spatial scale of the experiments is varied are embodied

in diﬀerent values of n, q,and

β. Tsallis statistics allows us to relate one

to another. This discussion suggests that Tsallis statistics is applicable to

systems with ﬂuctuating energy-dissipation rates.

6.3 Foreign Exchange Markets

6.3.1 Why Foreign Exchange Markets?

Foreign exchange markets are extremely interesting for statistical studies be-

cause of the number and quality of data they produce [102, 140]. The markets

have no business-hour limitations. They are open worldwide, 24 hours a day

including weekends, except perhaps a few worldwide holidays. Trading is es-

sentially continuous, the markets (at least for the most frequently traded

currencies) are extremely liquid, and the trading volumes are huge. Daily

volumes are of the order of US$ 10

, approximately the gross national prod-

uct of Italy. Typical sizes of deals are of the order of US$ 10

–10

,andmost

of the deals are speculative in origin. As a consequence of the liquidity, good

6.3 Foreign Exchange Markets 183

databases contain about 1.5 million data points per year, and data have been

collected for many years.

6.3.2 Empirical Results

Ghashghaie et al. [141] analyzed high-frequency data consisting of about

1.5 ×10

quotes of the US$/DEM exchange rate taken from October 1, 1992

until September 30, 1993. The probability density function for price changes

δS

over a time scale τ is shown in Fig. 6.6. The time scale τ of the returns

increases from top to bottom, and the curves have been displaced vertically,

for clarity. Both the fat tails characteristic of ﬁnancial data, and the similarity

to the distributions in turbulence, e.g., Fig. 6.4, are apparent. Speciﬁcally, one

can notice a crossover from a more tent-shaped probability density function

at short time scales to a more parabolic (Gaussian) one at longer scales. This

would imply that the probability density function is not form-invariant under

rescaling, as was found, at least for not too long time scales, in the analysis

of stock market data [62] discussed in Sect. 5.6.1.

There are more analogies between foreign exchange markets and turbu-

lence. For example, one can investigate the scaling of the moments of the

Fig. 6.6. Probability density function for variations of the US$/DEM exchange

rate for time delays ∆t ≡ τ = 640 s, 5120 s, 40 960 s, and 163 840 s (top to bottom).

The full lines are ﬁts using integrals over Gaussian distributions. The identiﬁcation

of the legends with our text is: ∆x ≡ δS

and ∆t ≡ τ .BycourtesyofW.Breymann.

Reprinted by permission from Nature 381, 767 (1996)

1996 Macmillan Magazines

Ltd.

184 6. Turbulence and Foreign Exchange Markets

Fig. 6.7. Dependence of the scaling exponents ξ

of the n

moment of the proba-

bility densities on its order for foreign exchange markets and turbulent ﬂows. The

dotted line is ξ

= n/3. By courtesy of W. Breymann. Reprinted by permission

from Nature 381, 767 (1996)

1996 Macmillan Magazines Ltd.

distribution function with time scale (referring to the ﬁnancial data)

|δS

∼τ

. (6.22)

Examples of the equivalent scaling behavior in turbulence, involving δv(),

have been discussed in Sect. 6.2.1. Figure 6.7 shows the dependence of the

exponents found in foreign exchange rates and turbulence on the order of

the moment. The turbulence data start on the line ξ

= n/3atsmalln,

discussed above [141], and then bend downward, in rough agreement with

a prediction by Kolmogorov. The ﬁnancial data are slightly oﬀ both the

= n/3 line and the turbulence data, but it should be noted that estimates

of the exponents can vary up to 30% depending on details of the estimation

procedure. However, even with diﬀerent methods, the scaling of the exponents

of the moments with their order systematically has a concave shape [140, 142].

As a consequence of this analysis, one would postulate a strong simi-

larity, perhaps a true mapping, between turbulence and foreign exchange

markets [141]. From the cascade model for turbulence, one would then infer

the existence of some kind of cascade in ﬁnancial markets. Details of this

conjectured correspondence are shown in Table 6.1.

The idea of a cascade, perhaps an information cascade, is not completely

speculative. It has been born in an analysis of time-scale-dependent volatility

in FX and commodity markets in the economics literature [143], and has also

been hypothezized for the S&P500 stock market index [144]. As we have

seen in the previous chapter, volatility is a long-time-correlated variable. It

therefore can be predicted, in principle. Obviously, the better the stochastic

6.3 Foreign Exchange Markets 185

Table 6.1. Postulated correspondence between fully developed three-dimensional

turbulence and foreign exchange markets. Adapted from [86]

Hydrodynamic Turbulence Foreign Exchange Markets

Energy Information

Spatial distance Time delay

Intermittency (laminar periods Volatility clustering

interrupted by turbulent bursts)

Energy cascade in space hierarchy Information cascade in time hierarchy

|δv|

∼

|δS|

∼τ

volatility process and its driving mechanisms are understood, the better a

prediction one can hope to generate.

In a heterogeneous market, the diﬀerent types of traders present, e.g.,

long-term investors, day traders, etc., in general act with diﬀerent time hori-

zons. A day trader will observe market volatility on a very short scale. On

the other hand, a long-term investor will not watch the market often enough

to even perceive short-term volatility. The question of how statistics reﬂects

the various types of operators in the marketplace then is reduced to the cor-

relations between the volatilities characterizing the various actors. In FX and

commodity markets, M¨uller et al. [143] have studied the correlation of ﬁnely

deﬁned volatility with coarsely deﬁned volatility.

We deﬁne the ﬁnely and coarsely deﬁned volatilities by absolute values of

return and, to be speciﬁc, use a one-week time scale

ﬁne

(N)=



i=1

|δS

(N,i)| and σ

coarse

(N)=|δS

(N)| . (6.23)

The ﬁne volatility is the sum of daily volatilities while the coarse volatility is

the weekly return directly. N labels weeks and, where necessary, i labels the

business days of the week. A lagged correlation

[σ

coarse

(N + τ) −σ

coarse

(N + τ)]



ﬁne

(N) −σ

ﬁne

(N)







var[σ

coarse

(N)] var[σ

ﬁne

(N)]

(6.24)

measures the correlation of the coarse volatility with the ﬁne volatility τ weeks

earlier. Empirically, it turns out that ρ

− ρ

−τ

< 0forτ>0 quite generally

[143]. This implies that the coarse volatility predicts the ﬁne volatility better

than vice versa. This result is observed both for daily data (assumed in the

equations above) and for high-frequency intraday data. It can be explained

by a hypothesis of heterogeneous markets: coarse volatility matters both for

a long-term investor and for a day trader. It will set the overall scale for the

latter, and a day trader will take diﬀerent positions depending on the level

186 6. Turbulence and Foreign Exchange Markets

of volatility. On the other hand, short-term volatility is only important for

the short-term trader.

More formally, in complete analogy to turbulence, one can search for a

stochastic process across time scales in foreign exchange markets. It may

not surprise the reader that indeed the probability density functions of the

US$/DEM exchange rate satisfy the Chapman–Kolmogorov–Smoluchowski

equation and allow one to reduce it to a Fokker–Planck equation [145]. Dif-

ferences are only found in details, such as the precise functional form of the

rescaling of δS

. The drift and diﬀusion constants are found to be

(1)

(δS,τ)=−0.93 δS , (6.25)

(2)

(δS,τ)=0.016τ +0.11(δS)

. (6.26)

The numerical prefactor of τ in (6.26) is given in units of days. The com-

parison of the scale dependence of the “experimental” probability density

function with the one obtained by solving the Fokker–Planck equation using

the appropriate empirical probability density function for long time scales is

shown in Fig. 6.8.

In fact, the numbers given in (6.25) and (6.26) above are not the orig-

inal results of Friedrich et al., but have been corrected by improved data

analysis and a more robust ﬁtting procedure, based on conditional instead of

unconditional probability distributions and accounting explicitly for possible

observational noise [146]. Firstly, one may calculate the power spectrum of

the time series

Π(ω)=



∞

−∞

dt e

iωt

S(t)S(0) . (6.27)

For approximately two decades in frequency, it decreases as ω

−2

before level-

ing oﬀ to a constant, i.e., white noise, at high frequency. The presence of white

noise suggests that the signal may be composed of two components, the in-

trinsic signal with Π(ω) ∼ ω

−2

, and white observational noise. The presence

of observational noise in similar ﬁnancial data has been shown independently

in an investigation using more traditional approaches of time-series analysis

[147]. However, there observational noise is found neither in the prices nor in

the returns but in the time series of squared returns.

As a consequence of observational noise, one should work with a smoothed

signal where this observational noise has been averaged out. The width of

the averaging window deﬁnes a minimal time scale of 4 minutes. With this

analysis, the expressions for D

(1)

(δS,τ)andD

(2)

(δS,τ) are obtained. The tail

index µ of the unconditional probability distribution can also be calculated

from the drift and diﬀusion coeﬃcients. A value of µ =4.2 ±0.8 is obtained,

quite in the range of the data analyzed in Sect. 5.6 [146, 148]. It is not clear

to what extent this improved analysis still is aﬀected by possible errors in

(2)

related to the ﬁnite time scales used [137, 138].

6.3 Foreign Exchange Markets 187

Fig. 6.8. Probability densities for variations of the US$/DEM exchange rates (dots)

compared to the solutions of a Fokker–Planck equation (solid lines) with the initial

distribution taken as the one for τ = 40 960 s. Time delays were τ = 5120 s, 10 240 s,

20 480 s, and 40 960 s. Both the data set and the notation are the same as in Fig.

6.6. By courtesy of J. Peinke. Reprinted from Friedrich et al.: Phys. Rev. Lett. 84,

5224 (2000),

2000 by the American Physical Society

Quite some time before this work, possible analogies between hydrody-

namic turbulence and ﬁnancial time series [141] have been questioned because

of diﬀerent (and, in some instances, less well deﬁned) power-law scaling in the

S&P500 index and air-ﬂow data at R = 1500 [149]. One interesting aspect

of this work is that the power spectrum of the S&P500 data is ω

−2

in the

entire frequency range considered, and no crossover to observational noise is

observed. This may be related to the time scale τ = 1 hour of the S&P500

returns analyzed.

With a Fokker–Planck equation for the probability distributions of ﬁnan-

cial data at hand, it would be interesting to search for improvements, e.g.,

in the theory of option pricing, etc., to include the eﬀects of non-Gaussian

statistics. This will be pursued further in Sect. 7.6 below.

188 6. Turbulence and Foreign Exchange Markets

An interesting phenomenological analogy between turbulence and ﬁnan-

cial markets also follows from realizing the similarity of the probability dis-

tributions and their scale dependences to the spectroscopic lineshapes of im-

purity molecules in disordered solids [86, 87]. Ideally, the optical absorption

spectrum of a molecule in a crystal consists of a series of delta functions.

Imperfections is real systems always lead to a broadening. There, a change

of the lineshape from a Lorentzian distribution to a Gaussian is observed

when the density of the disordered units is varied: when the inﬂuence of

the disordered matrix units (which are present in important concentrations)

on a molecule is dominant, the lineshape is a Gaussian, as required by the

central limit theorem. When, on the other hand, the interaction of certain

two-level systems which are quite dilute with the host molecule dominate its

absorption, Lorentzian lineshapes are observed. Models for these line shapes

usually assume additive contribution of the individual perturbing elements

in the neighborhood of the molecules probed.

In a ﬁnancial market, the traders would take the role of the dye mole-

cules in glasses. The environment inﬂuencing their behavior is information

which becomes available at various moments of time. The time passed since

the arrival of a piece of information plays the role of spatial distance in

the molecule-in-a-glass problem. The inﬂuence function which, in the spec-

troscopy problem, is taken by the dipole–dipole interaction, becomes a mem-

ory function af(t − t



) in a market. a is the amplitude, t is the time of

a trading decision, and t



is the time of arrival of a piece of information

[86, 87]. The probability distribution of the price changes observed then is

determined by the functional form of f(t − t



). If the frequency of informa-

tion arrival is large with respect to the inverse time scale of the returns under

consideration, the precise form of f (t −t



) does not matter: the central limit

theorem requires that the resulting probability distribution will be Gaussian,

independently of the details of the memory functions. On the other hand,

when the frequency of information arrival is low or the time scale of the re-

turns short enough, the functional form of f(t −t



) matters. For example, for

an exponential memory kernel, the short-time probability distributions have

very ﬂat wings with a pronounced spike at zero return. Such spikes are not

observed in real markets, but they were generated in numerical simulations

of artiﬁcial ﬁnancial markets to be discussed in Sect. 8.3.2. On the other

hand, for a stretched-exponential decay in the memory kernel, a set of time-

scale-dependent probability distribution functions similar to Fig. 6.6 with

a truncation in the wings were obtained. Finally, for an algebraic memory

function, the probability distributions at short times were of the form of the

truncated L´evy distributions discussed in Sect. 5.4.4. The interesting conclu-

sion from this work is that, in terms of fundamental analysis, traders would

account for, resp. the market would reﬂect, information with a memory which

is scale-free (stretched-exponential or power-law memory function) [86, 87].

In turbulence, the role of the dye molecule/trader would be played by the

6.3 Foreign Exchange Markets 189

measurement device (anemometer), and that of the perturber would be taken

by the eddies depicted in Fig. 6.1.

6.3.3 Stochastic Cascade Models

The idea that turbulent ﬂows or foreign exchange markets are described by

a stochastic cascade across spatial or time scales can be formalized. Here,

we restrict ourselves to capital markets [144, 150]. Our discussion in the

preceding section is equivalent to postulating for returns on a scale τ

δS

(t)=σ

(t)ε(t) . (6.28)

Here, ε(t) is a scale-independent random variable, and σ

(t) is a positive

random variable depending on the scale, and identiﬁed with the standard

deviation on that scale τ. There is a hierarchy of scales τ

= T>···>τ

···>τ

. If the cascade is purely multiplicative, the σs are related by

(k)

(t)=a

(k)

(t)σ

(k−1)

(t) ,σ

(k)

(t) ≡ σ

(t), (6.29)

with time-dependent random factors a

(k)

(t). If our discussion of turbulent

signals by a cascade in Sect. 6.2.2, and a similar analysis of foreign exchange

quotes underlying the solid lines in Fig. 6.6, are rephrased in terms of (6.29),

the probability distribution of the a

(k)

(t) is log-normal with a k-dependent

width, cf. (6.12). This gives for the volatility at scale τ

(m)

= σ

(0)

k=1

(k)

(t) . (6.30)

One particularly simple realization would be a geometric progression in the

inverse scales, e.g., τ

= τ

k−1

/2, and to associate two random numbers a

(k)

and a

(k)

with the passage from one level to the next lower [144].

In this model, there is a deﬁnite direction for the net ﬂow of information

from large to small scales. Namely, one can calculate the cross-correlation

coeﬃcient [144]

,τ

(∆t)=

ln σ

(m)

(t)lnσ

(n)

(t + ∆t)

var(ln σ

(m)

)var(ln σ

(n)

)

. (6.31)

One ﬁnds that C

,τ

(∆t) >C

,τ

(−∆t)ifτ

>τ

and ∆t > 0. This

can be interpreted as a ﬂow of information contained in ln σ

(n+∆)

(t)to

ln σ

(n)

(t + ∆t).

More sophisticated updating schemes for the random numbers a

(k)

,re-

lating the volatilities on neighboring time scales, have been devised [150]. At

, one draws the a

(k)

) from a log-normal distribution with k-dependent

width. In later time steps, the factors at the top of the hierarchy, a

(1)

n+1