Cross hedging with stochastic correlation

This paper is concerned with the study of quadratic hedging of contingent claims with basis risk. We extend existing results by allowing the correlation between the hedging instrument and the underlying of the contingent claim to be random itself. We assume that the correlation process ρ evolves according to a stochastic differential equation with values between the boundaries −1 and 1. We keep the correlation dynamics general and derive an integrability condition on the correlation process that allows to describe and compute the quadratic hedge by means of a simple hedging formula that can be directly implemented. Furthermore, we show that the conditions on ρ are fulfilled by a large class of dynamics. The theory is exemplified by various explicitly given correlation dynamics.