Local Volatility Pricing Models for Long-dated FX Derivatives

We study the local volatility function in the Foreign Exchange market where both domestic and foreign interest rates are stochastic. This model is suitable to price long-dated FX derivatives. We derive the local volatility function and obtain several results that can be used for the calibration of this local volatility on the FX option's market. Then, we study two different extensions which allow the volatility of the spot FX rate to have stochastic behavior. First, we introduce a stochastic structure on the local volatility surface and show that local volatilities are risk-adjusted expectations of future instantaneous volatilities. The second extension is obtained by multiplying the FX spot local volatility with a stochastic volatility. Thanks to the Gyöngy's mimicking property, we obtain a calibration method for the local volatility associated to this model.Local volatility; Stochastic volatility; Foreign Exchange; Stochastic interest rates; Calibration