Calibration and Monte Carlo pricing of the SABR–Hull–White model for long-maturity equity derivatives

Abstract

We model the joint dynamics of stock prices and interest rates using a hybrid SABR-Hull-White model. The asset price dynamics are modeled by the SABR model of Hagan et al and the interest rate dynamics are modeled by the Hull-White short-rate model. We propose a projection formula, mapping the SABR-Hull-White model parameters onto the parameters of the nearest SABR model. Furthermore, a time-dependent parameter extension of this SABR-Hull-White model is introduced to make the calibration of the model consistent across maturities. The inverse of the projection formula enables a rapid calibration of the model. As the calibration quality is subject to the approximation errors of the projection formula, we subsequently apply a nonparametric numerical calibration technique based on the nonuniformly weighted Monte Carlo technique of Avellaneda et al to improve the calibration. In this step, the Monte Carlo weights are not uniform and are chosen in such a way that the calibration market instruments are perfectly replicated.