No abstract

The dynamics of a stock market with heterogeneous agents is discussed in the framework of a recently proposed spin model for the emergence of bubbles and crashes. We relate the log returns of stock prices to magnetization in the model and find that it is closely related to trading volume as observed in real markets. The cumulative distribution of log returns exhibits scaling with exponents steeper than 2 and scaling is observed in the distribution of transition times between bull and bear markets.

In this paper we present an interacting-agent model of speculative activity explaining bubbles and crashes in stock markets. We describe stock markets through an infinite-range Ising model to formulate the tendency of traders getting influenced by the investment attitude of other traders. Bubbles and crashes are understood and described qualitatively and quantitatively in terms of the classical phase transitions. When the interactions among traders become stronger and reach some critical values, a second-order phase transition and critical behaviour can be observed, and a bull market phase and a bear market phase appear. When the system stays at the bull market phase, speculative bubbles occur in the stock market. For a certain range of the external field that we shall call the investment environment, multistability and hysteresis phenomena are observed. When the investment environment reaches some critical values, the rapid changes in the distribution of investment attitude are caused. The first-order phase transition from a bull market phase to a bear market phase is considered as a stock market crash.Furthermore we estimate the parameters of the model using the actual financial data. As an example of large crashes we analyse Japan crisis (the bubble and the subsequent crash in the Japanese stock market in [1987][1988][1989][1990][1991][1992], and show that the good quality of the fits, as well as the consistency of the values of the parameters are obtained from Japan crisis. The results of the empirical study demonstrate that Japan crisis can be explained quite naturally by the model that bubbles and crashes have their origin in the collective crowd behaviour of many interacting agents.

The fractional Poisson process (FPP) is a counting process with independent and identically distributed inter-event times following the Mittag-Leffler distribution. This process is very useful in several fields of applied and theoretical physics including models for anomalous diffusion. Contrary to the well-known Poisson process, the fractional Poisson process does not have stationary and independent increments. It is not a Lévy process and it is not a Markov process. In this letter, we present formulae for its finite-dimensional distribution functions, fully characterizing the process. These exact analytical results are compared to Monte Carlo simulations.