Copula Based Martingale Processes and Financial Prices Dynamics

Abstract

In this paper we suggest a new technique to construct Markov processes by means of products of copula functions, in the spirit of Darsow et al, (1992). The approach requires to define: i) a sequence of distribution functions of the increments of the process; ii) a sequence of copula functions representing dependence between each increment of the process and the corresponding level of the process before the increment. We show that the approach is well suited to impose restrictions that may ensure the process to be a martingale. More precisely, we single out two classes of Markov processes endowed with the martingale condition: i) processes with independent increments with zero mean distributions; ii) processes with symmetric increments linked to the initial levels by a symmetric copula. As most of the current financial mathematics literature is limited to the independent increments class, we show that copula functions may be used to produce new martingale processes within this class, and many more in the new class of processes with symmetric increments.