Advanced Equity Derivatives: Volatility and Correlation

Chapter 9Stochastic Correlation

Stochastic correlation models may provide a more realistic approach to the pricing and hedging of certain types of exotic derivatives, such as worst-of and best-of options and correlation swaps and correlation options. In this chapter, we review various types of stochastic correlation models and propose a framework for the pricing of realized correlation derivatives that is consistent with variance swap markets.

9.1 Stochastic Single Correlation

Consider the following general model framework for two assets S⁽¹⁾ and S⁽²⁾:

where μ's are instant drift coefficients, σ's are instant volatility coefficients, and ρ is the instant correlation coefficient between the driving Brownian motions W's. Here all the coefficients may be stochastic, and we focus on ρ.

There are some simple ways to make ρ stochastic and comprised between −1 and 1; for example, take where Z is an independent Brownian motion. The dynamics of dρ_t may then be found by means of the Ito-Doeblin theorem. One issue with this approach is that the parameters may not be very intuitive.

A better approach is to specify diffusion dynamics for ρ and examine the Feller conditions at bounds −1 and 1 (see Section 2.4.2.2). A popular process here is the affine Jacobi process, also known as a Fischer-Wright ...