Combination of transition probability distribution and stable Lorentz distribution in stock markets

Abstract

It is well known that the stock price is a stochastic series of time. The probability distribution of the stock price changes should be stable. However, the empirically observed probability distribution of log-returns is not Gaussian and has power-law tails. We present a combination of Lorentz stable distribution and transition probability distribution description for the stock index changes. The Lorentz stable distribution describes the stochastic changes of stock prices and the transition distribution presents features of stock price transiting from S(t) to S(t+Δt) for small time interval. By making use of the Fokker–Planck equation, we give the explicit expression of the transition probability distribution for the stock price changes. The two stage formalism is in agreement with empirical observations on high frequency time series of the S&P 500 Index.