Evidence of Markov properties of high frequency exchange rate data

Abstract

We present a stochastic analysis of a data set consisting of 10 6 quotes of the US Dollar–German Mark exchange rate. Evidence is given that the price changes x( τ) upon different delay times τ can be described as a Markov process evolving in τ. Thus, the τ-dependence of the probability density function (pdf) p( x, τ) on the delay time τ can be described by a Fokker–Planck equation, a generalized diffusion equation for p( x, τ). This equation is completely determined by two coefficients D 1( x, τ) and D 2( x, τ) (drift- and diffusion coefficient, respectively). We demonstrate how these coefficients can be estimated directly from the data without using any assumptions or models for the underlying stochastic process. Furthermore, it is shown that the solutions of the resulting Fokker–Planck equation describe the empirical pdfs correctly, including the pronounced tails.