TAILIEUCHUNG - A Langevin approach to stock market fluctuations and crashes

Although one of the most sophisticated of early agent-based markets, the SFI market was not the first. There were several early simulations that tried to look the impact of random behaving agents on various market structures as in Cohen, Maier, Schwartz & Whitcomb (1983). Another early market looked the interactions of specific trading strategies (Kim &Markowitz 1989). Some of the more interesting early markets were concerned with the dynamics of foreign exchange as in Frankel & Froot (1988) and De Grauwe, Dewachter & Embrechts (1993). These follow much of the later literature in being concerned with the interaction of potentially destabilizing trend following strategies and their. | Eur. Phys. J. B 6 543-550 1998 The European physical Journal B EDP Sciences Springer-Verlag 1998 A Langevin approach to stock market fluctuations and crashes . Bouchaud1 2 a and R. Cont1 2 1 Service de Physique de l Etat Condense Centre d Etudes de Saclay Orme des Merisiers 91191 Gif-sur-Yvette Cedex France 2 Science Finance 109-111 rue Victor-Hugo 92532 Levallois Cedex France Received 27 January 1998 Revised 13 July 1998 Accepted 24 July 1998 Abstract. We propose a non linear Langevin equation as a model for stock market fluctuations and crashes. This equation is based on an identification of the different processes influencing the demand and supply and their mathematical transcription. We emphasize the importance of feedback effects of price variations onto themselves. Risk aversion in particular leads to an up-down symmetry breaking term which is responsible for crashes where panic is self reinforcing. It is also responsible for the sudden collapse of speculative bubbles. Interestingly these crashes appear as rare activated events and have an exponentially small probability of occurence. The model leads to a specific shape of the falldown of the price during a crash which we compare with the October 1987 data. The normal regime where the stock price exhibits behavior similar to that of a random walk however reveals non trivial correlations on different time scales in particular on the time scale over which operators perceive a change of trend. PACS. Stochastic processes - . n Other areas of general interest to physicists 1 Introduction Stock market fluctuations exhibit several statistical peculiarities which are still awaiting for a satisfactory interpretation. More strikingly many of these statistical properties are common to a wide variety of markets and instruments. The most prominent features are 1-3 5-7 1. On short time scales the variations of stock prices are strongly non-Gaussian. 2. Market volatility . the conditional variance of .

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