Pricing Long-Maturity Equity and FX Derivatives with Stochastic Interest Rates and Stochastic Volatility

Abstract

In this paper we extend the stochastic volatility model of Schöbel and Zhu (1999) by including stochastic interest rates. Furthermore we allow all driving model factors to be instantaneously correlated with each other, i.e. we allow for a correlation between the instantaneous interest rates, the volatilities and the underlying stock returns. By deriving the characteristic function of the log-asset price distribution, we are able to price European stock options in closed-form by Fourier inversion. Furthermore we present a Foreign Exchange generalization and show how the pricing of Forward-starting options like cliquets can be performed. Additionally we discuss the practical implementation of these new models.

Keywords: Stochastic volatility, Stochastic interest rates, Schöbel-Zhu, Hull-White, Foreign Exchange, Forward starting options

JEL Classification: C10