This paper establishes the need for local volatility coupled with domestic and foreign stochastic interest rates to properly manage some exotic hybrid options. It then computes such a local volatility and identifies a bias with respect to the local volatility with deterministic rates. Performing variance-covariance analysis on the logarithm of the underlying price together with the domestic and foreign spot rates, the paper estimates that bias by calculating the variances of the logarithm of the underlying price with and without stochastic rates at fixed points in time and in space. Equating the resulting variances, the authors express the local volatility with stochastic rates in terms of the one with deterministic rates plus a bias obtaining an exact, fast and robust way of calibrating any local volatility with stochastic rates to market prices.

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 exotic options such as digital coupon options in equity market or Power Reverse Dual Currency (PRDC) in foreign market, when priced with the Black and Scholes assumption can widely be wrong compared with models taking the volatility smile as well as stochastic rates into account. As a result, we combine local volatility with stochastic interest rates to model those products, and we provide a fast and robust way to calibrate this model to market prices.

Since the articles of Geman (1989) and El Karoui et al. (1998), both the forward probability measure and the Gaussian models, are well understood. One can therefore obtain closed-form solution when pricing a call option with deterministic volatility for both the underlying price (which can be either the stock price or the foreign exchange rate) and the bond price. Piterbarg (2005) introduced the market skew to the dynamic of the FX rate by considering the diffusion coefficient to be given by a Constant Elasticity of Variance (CEV). To obtain fast calibration, he reduced a three dimensional problem to a one dimensional one by projecting the dynamic of the underlying price under the forward measure to a one dimensional autonomous representation process. Later, Antonov and Misirpashaev (2006) developed an approximation to project a stochastic volatility model onto a displaced diffusion and recovered Piterbarg’s results in the case of a CEV model.

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