An equity–interest rate hybrid model with stochastic volatility and the interest rate smile

Lech A. Grzelak and Cornelis W. Oosterlee

Abstract

ABSTRACT

We define an equity-interest rate hybrid model in which the equity part is driven by Heston stochastic volatility and the interest rate is generated by the displaced diffusion stochastic volatility LIBOR market model. We assume a nonzero correlation between the main processes. A number of approximations lead to an approximating model which falls within the class of affine processes described by Duffie, for which we then provide the corresponding forward characteristic function. By using the appropriate change of measure and freezing the LIBOR rates, the dimension of the corresponding pricing partial differential equation can be greatly reduced. We discuss the accuracy of the approximations and the efficient calibration in detail. Finally, using experiments, we show the effect of the correlations and interest rate smile/skew on typical equity-interest rate hybrid product prices. This approximate hybrid model can be evaluated for a whole strip of strikes for equity plain vanilla options in milliseconds.