The FX Smile
The IUP Journal of Financial Risk Management, Vol. VII, No. 4, pp. 20-33, December 2010
Posted: 12 Feb 2011
Date Written: December 2010
The study considers a stochastic local volatility model with domestic and foreign stochastic interest rates and identifies a bias with respect to the deterministic local volatility with deterministic rates. Relating the local volatility of the model to that of the forward price, the study quantifies the 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. A simple implementation of the model is also described. In the discretized dynamics, the local volatility function being piecewise constant, the bootstrapping technique is applied to calculate its value by solving a quadratic equation at each maturity and strike. As a result, a fast and robust way of calibrating a stochastic local volatility model with stochastic rates to market prices is obtained.
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