Partial Differential Equation Pricing of Contingent Claims Under Stochastic Correlation

Abstract

In this paper, we study a partial differential equation (PDE) framework for option pricing where the underlying factors exhibit stochastic correlation, with an emphasis on computation. We derive a multi-dimensional time-dependent PDE for the corresponding pricing problem, and present a numerical PDE solution. We prove a stability result, and study numerical issues regarding the boundary conditions used. Moreover, we develop and analyze an asymptotic analytical approximation to the solution, leading to a novel computational asymptotic approach based on quadrature with a perturbed transition density. Numerical results are presented to verify second order convergence of the numerical PDE solution and to demonstrate its agreement with the asymptotic approximation and Monte Carlo simulations. The effect of certain problem parameters to the PDE solution, as well as to the asymptotic approximation solution, is also studied.

Keywords: Stochastic Correlation, Option Pricing, Numerical Solution, Asymptotic Solution, Partial Differential Equation

JEL Classification: C63