Granger Independent Martingale Processes

TL;DR

This paper introduces GIMP, a new class of multivariate processes for equity derivatives pricing that enforces martingale conditions and no-Granger-causality, facilitating copula-based modeling and time change extensions.

Contribution

The paper proposes GIMP, a novel class of processes combining martingale properties with no-Granger-causality, and demonstrates their closure under time change, enhancing multivariate derivative modeling.

Findings

GIMP processes satisfy martingale and no-Granger-causality conditions.

GIMP class is closed under time changing with stochastic clocks.

Extension to time-changed GIMP processes maintains martingale properties.

Abstract

We introduce a new class of processes for the evaluation of multivariate equity derivatives. The proposed setting is well suited for the application of the standard copula function theory to processes, rather than variables, and easily enables to enforce the martingale pricing requirement. The martingale condition is imposed in a general multidimensional Markov setting to which we only add the restriction of no-Granger-causality of the increments (Granger-independent increments). We call this class of processes GIMP (Granger Independent Martingale Processes). The approach can also be extended to the application of time change, under which the martingale restriction continues to hold. Moreover, we show that the class of GIMP processes is closed under time changing: if a Granger independent process is used as a multivariate stochastic clock for the change of time of a GIMP process, the new process is also GIMP.