Market dynamics after large financial crash

arXiv:0807.2083v1 [q-fin.ST] 14 Jul 2008 Market dynamics after large financial crash G. Buchbinder∗and K. Chistilin† Physics Department, Omsk State University, Peace Avenue, 55a, 644077 Omsk, Russia (Dated: October 25, 2018) The model describing market dynamics after a large financial crash is considered in terms of the stochastic differential equation of Ito. Physically, the model presents an overdamped Brownian particle moving in the nonstationary one-dimensional potential U under the influence of the variable noise intensity, depending on the particle position x. Based on the empirical data the approximate estimation of the Kramers-Moyal coefficients D1,2 allow to predicate quite definitely the behavior of the potential introduced by D1 = −∂U/∂x and the volatility ∼√D2. It has been shown that the presented model describes well enough the best known empirical facts relative to the large financial crash of October 1987. PACS numbers: 89.65.Gh, 02.50.Ey, 05.40.-a I. INTRODUCTION The dynamics of the financial markets has been at- tracting attention of the physics community for two decades [1, 2, 3, 4]. During this time a large volume of the empirical researches has been done. Of specific interest is the investigation of the behavior of the market in the time periods of the large financial crises when the statistical properties of the market are drastically distin- guished from its properties in quiet days [3, 5, 6, 7]. The empirical analysis has found the occurrence of a number of peculiarities in market dynamics appearing in these periods. Originally they have been revealed when in- vestigating the large financial crash on 19 October 1987 (Black Monday) at New York Stock Exchange. However these peculiarities are not specific for October 1987 and have been later revealed in the financial crashes of 1997 and 1998 years for different countries and Exchanges [7]. Firstly, it has been established that the large finan- cial crashes are outliers which are not possible within the scope of the typical price distributions. For their realiza- tion the complementary factors absent in quiet days are needed [5, 7]. Secondary, the emergence of the certain periodic pat- terns in the dynamics of the different financial assets are detected both before the crash, and immediately after it. In particular, a log-periodic oscillations in time evolution of the price of an asset appear before the extreme event [7, 8, 9, 10, 11, 12]. The aftershock period is charac- terized by an exponentially decaying sinusoidal behavior of a price [7, 8]. The characteristic behavior of volatil- ity is also revealed after the crash. In the moment of the crash of October 1987 the implied volatility of the S&P500 index makes a shock and then sharply decays as power law decorated with log-periodic oscillations [7, 8]. ∗Electronic address: glb@omsu.ru †Electronic address: Chistilin˙K@mail.ru In the day of the maximum drop the central part of the empirical return distribution moves toward negative re- turns and then begins to oscillate between positive and negative returns [6]. At last, the relaxation dynamics of an aftercrash pe- riod is characterized by what is termed as the Omori law which shows that the rate of of the return shocks larger than some threshold decays as the power law with expo- nent close to 1 for different thresholds [13, 14]. There are a large number of works devoted to the mod- eling of the market dynamics immediately before crash [7, 8, 9, 10, 11, 12, 15, 16] (see also references in the article of Sornette [7]). These works based on the em- pirical observations exploiting the analogy with critical phenomena have been directed to the investigation of a possibility of predictions of the large financial crash on the base of the modeling of the dynamics of different fi- nancial indexes (S&P500, DJ, etc). The ”microscopic” approach has been suggested in the work [17] where the model of the market dynamics taking into account a possibility of occurrence of the crash is developed. In physical terms, the model is reduced to a motion of a Brownian particle in the stationary cubic potential and crashes are considered as rare activated events. In Refs. [18, 19] the generalization of the above model has been suggested for the case of the stochastic volatil- ity which was considered within the scope of the modified Heston model. Physically, the model represents an over- damped Brownian particle moving in the stationary cu- bic potential under the influence of the fluctuating noise intensity. Since the statistic properties of the volatility in the Heston model are uniform in a time [20], both the above models describe the uniform stochastic process. In fi- nancial terms, this means that statistic properties of the market are the same regardless of where market is ei- ther in normal regime or close to a crash. It seems this is poorly consistent with the assertions that crashes are outliers and the statistic properties of the market are 2 drastically change in extreme d

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