Dynamical Stochastic Processes of Returns in Financial Markets

We show how the evolution of probability distribution functions of the returns from the tick data of the Korean treasury bond futures (KTB) and the S$\&$P $500$ stock index can be described by means of the Fokker-Planck equation. We derive the Fokker- Planck equation from the estimated Kramers-Moyal coefficients estimated directly from the empirical data. By analyzing the statistics of the returns, we present the quantitative deterministic and random influences on both financial time series, for which we can give a simple physical interpretation. Finally, we remark that the diffusion coefficient should be significantly considered to make a portfolio.