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.

Hybrid options provide risk managers a general risk-protection tool but their high dimension makes it difficult to model. It is now well-known that some long-dated exotic options such as digital coupon options in equity market or the Power Reverse Dual Currency (PRDC) in foreign market, when priced with the Black-Scholes assumption (Black and Scholes, 1973) can be widely wrong compared to the models taking the volatility smile as well as stochastic rates into account. The deterministic local volatility models are widely used because they give perfect fit to market prices allowing for simple hedging of exotic options. However, the local volatility is just a useful simplification to describe price processes with non-constant volatility. As described by Gyongy (1986), only the marginal distributions are matched and not the joint distributions, leading to different dynamics and hedge ratios. Moreover, there are empirical evidences that volatility displays characteristics of a stochastic process on its own (Bakshi et al., 2000). That is, option prices are driven by at least one random factor other than asset price. For instance, the study by Chernov et al. (2002) found evidence of substantial volatility feedback and concluded that various diffusion-based models require at least two-factor stochastic volatility model to simultaneously summarize volatility evolution and capture the fat-tailed properties of daily returns.

Therefore, for models to capture market behavior, the introduction of stochastic volatility process independent from the deterministic instantaneous volatility is necessary. Jex et al. (1999) are the first ones to combine in a diffusion model, deterministic local volatility and stochastic volatility. They used a two-dimensional tree to diffuse the transition probabilities in order to recover the local volatility. Lee (2001) used probabilistic and asymptotic methods to get the local volatility of stochastic volatility diffusions independent from the underlying asset prices. Alexander and Nogueira (2004) assumed a parametric functional form for the local volatility and derived a continuous stochastic local volatility model. Similar to interest rates, Ren et al. (2007) considered an extension of the deterministic local volatility by incorporating an independent stochastic component to volatility. However, they did not consider combining stochastic volatility with stochastic rates since their approach is limited by the dimensionality of the PDE solver of the Fokker-Planck equation.

Financial Risk Management Journal, FX Smile, Stochastic Local Volatility Model, Foreign Market, Stochastic Process, Asymptotic Methods, Probabilistic Methods, Fokker-Planck Equation, Domestic Market, Stochastic Local Variance, Bootstrapping Technique.