Virtual Physics - issue No 13 - December 15, 1996

number 13, December 15, 1996

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a forum for virtual meetings of scientists and students involved in a research activity on
CONTEMPORARY PHYSICS

Editors:

Marcel Ausloos, ausloos@gw.unipc.ulg.ac.be, Institut de Physique, Université de Liège, Belgium,

Kenneth Holmlund, Kenneth.Holmlund@TP.UmU.SE, Umeå University, Sweden

Cameron L. Jones, cjones@swin.edu.au, Swinburne University of Technology, Australia

Zbigniew J.Koziol, (Editor-in-Chief) webex@ra.isisnet.com, WebExperts Inc., Canada

Michal Spalinski, Michal.Spalinski@fuw.edu.pl, Institute of Theoretical Physics, Warsaw University, Poland

Krzysztof P. Wroblewski, chris@nmr.biophys.upenn.edu, University of Pennsylvania, U.S.A.

Copyright (C) 1996 by Zbigniew Koziol.

IN THIS ISSUE:

Letter from the Editor
Financial markets as adaptative ecosystems, by Marc Potters, Rama Conta, and Jean-Philippe Bouchauda
3rd International Summer School on High Temperature Superconductivity, July 1997 Eger, Hungary

____________________________________________________________

LETTER FROM THE EDITOR

Dear Readers of Virtual Physics,

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Financial markets as adaptative ecosystems

Marc Potters, potters@cfm.fr ^a, Rama Cont^a,b Jean-Philippe Bouchaud^a,c

^a Science & Finance, 109--111 rue Victor Hugo, 92532 Levallois Cedex, France
^b Service de Physique de l'Etat Condense, Centre d'etudes de Saclay, Orme des Merisiers, 91191 Gif-sur-Yvette Cedex, France
^c Laboratoire de Physique de la Matiere Condense CNRS URA 190, Universite de Nice- Sophia Antipolis B.P. 70, 06108 Nice, France

Received October 14, 1996
A PostScript version is available at LANL server, http://xxx.lanl.gov/abs/cond-mat/9609172

The option markets offer a very interesting example of the adaptation of a population (the traders) to a complex environment, through trial and errors and natural selection (unefficent traders disappear quickly). The problem is the following: an `option' is an insurance contract protecting its owner against the rise (or fall) of financial assets, such as stocks, currencies, etc. The problem of knowing the value of such contracts has become extremely acute when organised option markets opened twenty five years ago, allowing one to buy or sell options much like stocks. Almost simultaneously, Black and Scholes (BS) proposed their famous option pricing theory, based on a simplified model for stock fluctuations, namely the (geometrical) Brownian motion model. The most important parameter of the model is the `volatility'

, which is the standard deviation of the market price's relative fluctuations. Guided by the Black-Scholes theory, but constrained by the fact that `bad' prices lead to arbitrage opportunities, option markets agree on prices which are close, but significantly and systematically different from the BS formula. Surprisingly, a detailed study of the observed market prices clearly shows that, despite the lack of an appropriate model, traders have empirically adapted to incorporate some subtle information on the real statistics of price changes.

More precisely, a `call' option is such that if the price x(T) of a given asset at time T (the `maturity') exceeds a certain level x_c (the `strike' price), the owner of the option receives the difference x(T)-x_c. Conversely, if x(T) < x_c, the contract is lost. To make a long story short [1,2,3], if T is small enough (a few months) so that interest rate and average returns are negligible compared to fluctuations, the `fair' price C of the option today (t=0), knowing that the price of the asset now is x₀ is given by:

[ Equation 1 ]

(1)

where P(x',T|x₀,0) is the conditional probability density that the stock price at time T will be equal to x', knowing its present value is x₀. Eq. (1) means that the option price is such that on average, there is no winning party. Pricing correctly an option is thus tantamount to having a good model for the conditional probability P(x',T|x₀,0).

There is fairly strong evidence that beyond a time scale of the order of ten minutes, the fluctuations of stock values (on the major markets) are uncorrelated [4], but not identically distributed variables [5]. More precisely, one can write:

[ Equation 2 ]

(2)

where the increments

x_k are distributed as:

[ Equation 3 ]

(3)

which means that

x_k is obtained as the product of a random variable, the distribution of which is independent of k, times a scale factor

_k which stochastically depends on k (see below).

Let us first consider the case where _k = ₀ independent of k, corresponding to the classical problem of a sum of independent, identically distributed variables. Although P(x) is strongly non Gaussian (see, e.g. [6], the Central Limit Theorem [7] tells us that for N==T/ large, P(x',T|x₀,0) will be close to a Gaussian. Using then Eq. (1) essentially leads back to the BS formula (although in principle BS use a log-normal, rather than a normal distribution, the difference is not relevant for the present discussion). For finite N, however, there are corrections to the Gaussian, and thus corrections to the BS price. A useful way to characterise these corrections is to introduce the cumulants of the elementary increments x. To a very good approximation, the distribution P₀( x) is even [6]; a classical result is then that the leading correction when N is large is proportional to the kurtosis , defined as [7], which vanishes if x is itself Gaussian, and measures the `fatness' of the tails of the distribution. It is then easy to show that the leading correction to the BS price can be reproduced by using the BS formula, but with a modified value for the volatility (which traders call the `implied volatility' , which depends both on the strike price x_c and on the maturity T=N through:

[ Equation 4 ]

(4)

This is called the `smile effect', because the plot of

versus x_c has the shape of a smile (see Fig. 1). That the volatility had to be smiled up was realised long ago by traders - this reflects the well known fact that the elementary increments have rather `fat' tails: markets are much more jerky than Gaussian random walks.

[ Figure 1 ]

Figure 1. Example of a smile curve: Implied volatility (x_c,T) vs\/ distance from strike price (x-x_c) for a given T. The data shown correspond to all 227 transactions of December options on the German Bund future (Long term interest rates trated on the LIFFE) on November 13, 1995. This is a very `liquid' market, meaning that price anomalies are expected to be small, in particular for short maturities T. Both call and put options are included with put options transformed into call options using the put-call parity [2]. Volatilities are expressed as annualised standard deviation of price differences. According to Eq. (4) the data should fall on a parabola. From a fit of the curvature of this parabola, we extract the `implied kurtosis' _imp for a given N= T/. In this particular case we find _imp=276 at N=144 (9 trading day remaining maturity).

As shown in Fig. 1, the smile formula (4) reproduces correctly the observed prices of `Bund' options provided the kurtosis becomes itself N dependent. The shape of the `implied' kurtosis

_imp(N) as a function of N is given in Fig. 2;

_imp(N) is seen to increase steadily. Why is this so?

[ Figure 2 ]

Figure 2. Plot of the implied kurtosis \kappa_imp and of the `effective' kurtosis _eff, as a function of the reduced time scale N=T/, = 30 minutes. For every day from 1993 to 1995, options transactions were analyse as in Fig. 1 to determined the implied kurtosis that day. The average _imp over all days with the same N is plotted here. The effective kurtosis was computed directly from 5 minute tick data of the Bund future for the same period. The growth of the error bars for the latter quantity comes from the fact that less data is available for larger N, and that a factor N comes in the definition of _eff. Finally, a fit with formula (5), corresponding to a simple Ornstein-Uhlenbeck evolution of the scale parameter _k is shown for comparison. This allows one to extract a correlation time for these fluctuations of the order of a week. The intriguing oscillation of _eff for N > 200 might only be due to the large error intrinsic to measuring this quantity.

Let us study directly the kurtosis

_Nof the distribution of the underlying stock, P(x,T|x₀,0), as a function of N== T/

. If the increments

x were independent and identically distributed (i.e.

_k ==

₀), one should observe that

_N =

/N. In Fig. 2, we have also shown

_eff=N

_N as a function of N. One can notice that not only

_eff is not constant (as it should if

x were identically distributed), but actually

_eff matches quantitatively (at least for N <= 200 with the evolution of the implied kurtosis

_imp! In other words, the price over which traders agree capture rather precisely the anomalous evolution of

_eff.

As we shall show now, this non trivial behaviour of

_eff is related to the fact that the scale of the fluctuations

_k is actually itself a time dependent random variable [5]. This could come from the fact that new information induces reactions of arbitrary sign, increasing the scale of fluctuations; conversely, when fluctuations are too large, risk-averse operators leave the market and this decreases the scale of fluctuations. It is thus reasonable to assume that

_k oscillates around a mean value, with random fluctuations, possibly correlated in time. Writing

refers now to an average over the scale fluctuations), one finds that Eq. (4) still holds, but with

replaced by an effective kurtosis

_eff given by:

[ Equation 5 ]

(5)

where

₀ is the kurtosis of P₀(

x). The simplest possibility is that

_k follows an Ornstein-Uhlenbeck process [7], in which case g(k)=g₀

^k, where

< 1 is related to the correlation time of the

variable. The solid line in Fig. 2 shows a rather good fit of

(N) with this formula, with

20, g₀

4 and

= 0.9913, corresponding to a correlation time of 7.2 days. Note that the effect of a non zero kurtosis on the BS prices was previously investigated in [8,9], although the relation between

_eff and

_imp, or their N dependence, was not investigated.

In conclusion, we have shown by studying in detail the market price of options that traders have evolved from the simple, but inadequate BS formula to an empirical know-how which encodes two important statistical features of asset fluctuations: `fat tails' (i.e. a rather large kurtosis) and more subtle non stationary effects (i.e. the fact that the scale of fluctuations is itself time dependent). These features, although not explicitly included in the theoretical pricing models used by traders, are nevertheless reflected rather precisely in the price fixed by the market as a whole. Financial markets thus behave as adaptative systems with efficient emerging properties.

References

[1] F. Black, M. Scholes, Journal of Political Economy, 81 (1973) 637.
[2] J.C. Hull, Futures, Options and Other Derivative Securities, Prentice Hall (1994).
[3] J.P. Bouchaud, D. Sornette, Journal de Physique I (France), 4 (1994) 863. J.P. Bouchaud, M. Potters, D. Sornette, Rare events and extreme risks in finance, book in preparation.
[4] A. Arneodo, J.P. Bouchaud, R. Cont, J.F. Muzy, M. Potters, D. Sornette, http://xxx.lanl.gov/abs/cond-mat/9607120, submitted to Nature.
[5] R. Engle, Econometrica 50 (1982) 987; T. Bollerslev, Journal of Econometrics, 31 (1986) 307. C. Gourieroux, Modeles ARCH et Applications financieres, Economica, Paris (1992).
[6] R. Mantegna and H.E. Stanley, Nature, 376, 46-49 (1995).
[7] W. Feller, An Introduction to Probability Theory and its Applications, Wiley (1971).
[7] R. Jarrow, A. Rudd, Journal of Financial Economics, 10 (1982) 347.
[8] C.J. Corrado and T. Su, Journal of Financial Research, XIX (1996) 75.

3rd International Summer School on High Temperature Superconductivity

2nd-3rd weeks of July 1997 Eger, Hungary

The objective of the Summer School is to provide an overview on the basic and up-to-date information on the theories and newest results both in the fundamental research and in the applications of high temperature superconductors. The main frame of the School is a series of tutorial lectures held by famous and well-known worldwide scientists and experts of the topic. During the evenings round-table session are going to be organized. The last days of the School will be devoted to a scientific conference for each participant willing to present the results of her/his scientific work.

The program will contain oral lectures and presentations including the following topics:

Ten years of high-Tc superconductivity Survey of theories on high-Tc superconductivity Physical properties of high-Tc materials Characterization and measurements of the physical properties New trends in preparation techniques Crystal structure of high-Tc materials Interdependence of the physical properties and morphology Manufacturing of high-Tc conductors Small and large-scale applications

On the last days of the Summer School a Scientific Session will be organized. Oral presentations will be selected by the Local Organizing Committee. The duration of the presentation is limited to 15 minutes.

For any more information please contact:

dr. Istvan Vajda, vajda@ntb.bme.hu

Department of Electrical Machines and Drives,

Technical University of Budapest

Egry Jozsef utca 18, H-1111 Budapest, Hungary

Phone: +36-1-463.2961, +36-1-463.3600

WWW: http://docs4.mht.bme.hu/~farkas/

Virtual Physics: a forum for virtual meetings of scientists and students involved in a research activity on CONTEMPORARY PHYSICS

Editors:

Marcel Ausloos, ausloos@gw.unipc.ulg.ac.be, Institut de Physique B5, Université de Liège, Sart Tilman, B-4000 Liège, Belgium, tel. (+32 41) 66 37 52 Kenneth Holmlund, Kenneth.Holmlund@TP.UmU.SE, Department of Theoretical Physics Umeå University, S-907 42 Umeå, Sweden, tel. +46-(0)90-167717 Cameron L. Jones, cjones@swin.edu.au, Centre for Applied Colloid and BioColloid Science Swinburne University of Technology, P.O. Box 218 Hawthorn, Victoria, 3122 Australia, tel. +613 9214 8935, fax +613 9819 0834 Zbigniew J. Koziol (Editor-in-Chief), WebEx@ra.isisnet.com, WebExperts Inc., 2-6032 Compton Ave., Halifax, Nova Scotia, B3H 1E7 Canada, tel. (902) 423 2149 Michal Spalinski, Michal.Spalinski@fuw.edu.pl, Institute of Theoretical Physics, Warsaw University, Hoza 69, 00-681 Warsaw, Poland, tel. (+48 2) 628 3031 Krzysztof P. Wroblewski, chris@nmr.biophys.upenn.edu, School of Medicine University of Pennsylvania, Rm. C-501 Richards Bldg., Philadelphia, PA 19104-6089, U.S.A., tel. (215) 898-6396

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