Empirical laws of a stock price index and a stochastic model

Recent works by econo-physicists [5, 8, 15, 19] have shown that the probability function of the share returns and the volatility satisfies a power law with an exponent close to 4. On the other hand, we investigated quantitatively the return and the volatility of the daily data of the Nikkei 225 index from 1990 to 2003, and we found that the distributions of the returns and the volatility can be accurately described by the exponential distributions [11]. We then propose a stochastic model of stock markets that can reproduce these empirical laws. In our model the fluctuations of stock prices are caused by interactions among traders. We indicate that the model can reproduce the empirical facts mentioned above. In particular, we show that the interaction strengths among traders are a key variable that can distinguish the emergence of the exponential distribution or the power-law distribution.