Common Decomposition of Correlated Brownian Motions and its Financial Applications

TL;DR

This paper introduces a new theoretical framework for decomposing correlated Brownian motions into independent components, with applications in financial derivative pricing and correlation modeling.

Contribution

It develops a common decomposition theory for correlated Brownian motions using a change of time, and proposes a new simulation method for correlated processes.

Findings

New decomposition method improves simulation accuracy

Application to pricing two-factor derivatives

Analysis of correlation substitution effects

Abstract

In this paper, we develop a theory of common decomposition for two correlated Brownian motions, in which, by using change of time method, the correlated Brownian motions are represented by a triplet of processes, $(X,Y,T)$, where $X$ and $Y$ are independent Brownian motions. We show the equivalent conditions for the triplet being independent. We discuss the connection and difference of the common decomposition with the local correlation model. Indicated by the discussion, we propose a new method for constructing correlated Brownian motions which performs very well in simulation. For applications, we use these very general results for pricing two-factor financial derivatives whose payoffs rely very much on the correlations of underlyings. And in addition, with the help of numerical method, we also make a discussion of the pricing deviation when substituting a constant correlation model for a general one.