Scaling in Stock Market Data: Stable Laws and Beyond
The concepts of scale invariance and scaling behavior are now increasingly applied outside their traditional domains of application, the physical sciences. Their application to financial markets, initiated by Mandelbrot [1,2] in the 1960s, has experienced
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Scaling in Stock Market Data: Stable Laws and Beyond Rama Cont*'**'***, Marc Potters** and Jean-Philippe Bouchaud*'** *Service de Physique de /'Etat Condense, Centre d'Etudes de Sac/ay, 91191 Gif-sur-Yvette, France **Science & Finance, 109-111 rue Victor Hugo, 92532 Leva/lois, France *** Laboratoire de Physique de Ia Matiere Condensee, URA 190 du CNRS, Universite de Nice, 06108 Nice cedex 2, France
1.
Introduction
The concepts of scale invariance and scaling behavior are now increasingly applied outside their traditional domains of application, the physical sciences. Their application to financial markets, initiated by Mandelbrot [1, 2] in the 1960s, has experienced a regain of interest in the recent years, partly due to the abundance of high-frequency data sets and availability of computers for analyzing their statistical properties. This lecture is intended as an introduction and a brief review of current research in a field which is becoming increasingly popular in the theoretical physics community. We will try to show how the concepts of scale invariance and scaling behavior may be usefully applied in the framework of a statistical approach to the study of financial data, pointing out at the same time the limits of such an approach. B. Dubrulle et al. (eds.), Scale Invariance and Beyond © Springer-Verlag Berlin Heidelberg 1997
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Statistical description of market data
Quantitative research mainly focuses on "liquid" financial markets i.e. organized markets where transactions are frequent and the number of actors is large. Typical examples are foreign exchange markets, organized futures markets and stock index markets and the market for large stocks. Prices are recorded several times a minute in such markets, creating mines of data to exploit. Such markets are complex systems with many degrees of freedom [3], where many internal and external factors interact at each instant in order to fix the transaction price of financial assets. Various factors such as public policy, interest rates and economic conditions doubtlessly influence market behavior. However, the precise nature of their influence is not well known and given the complex nature of the pricing mechanism, simple deterministic models are unable to reproduce the properties observed in financial time series. Furthermore, although the details of the price fixing mechanism- market microstructure - may be different from one market to the other, what is striking is the universality of some simple statistical properties of price fluctuations, prompting a unified approach to the study of different types of markets. As in the case of other types of complex systems with universal characteristics, a stochastic approach has proved to be fruitful in this case. An investor buying a financial asset at time t and selling it at time t + T is primarily interested in the (relative or absolute) variation of the price between t and t + T. The main object of study in this framework is therefore the probability density function (PDF) of the increment
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