Multi-Currency Fast Stochastic Local Volatility Model

Abstract

We consider a stochastic local volatility model with domestic and foreign stochastic interest rates and identify a bias with respect to the deterministic local volatility with deterministic rates. Relating the local volatility of our model to that of the forward price, we quantify that bias by equating the variance swap contract under the risk-neutral measure with that under the forward probability measure. Assuming a collapse process for the variance with the same random variable for all time and deterministic zero-coupon bond volatility functions, the bias term simplifies and can easily be computed. We then describe a simple implementation of our model. In the discretised dynamics, the local volatility function being piecewise constant, we apply the bootstrapping technique to calculate its value by solving a quadratic equation at each maturity and strike. As a result, we get a fast and robust way of calibrating a stochastic local volatility model with stochastic rates to market prices.