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 an extension to obtain a more general volatility model and propose a calibration method for the local volatility associated to this model.Recent years, the long-dated FX option's market has grown considerably. Currently most traded and liquid long-dated FX Hybrid products are Power-Reverse Dual-Currency swaps (PRDC) (see for example [Piterbarg, 2006]) as well as vanilla or exotic long-dated products such as barrier options. While for short-dated options (less than 1 year), assuming constant interest rates does not lead to significant mispricing, for long-dated options the effect of interest rate volatility becomes increasingly pronounced with increasing maturity and can become as important as that of the FX spot volatility. Most of the dealers are using a three-factor pricing model for long-dated FX products (see [Piterbarg, 2006, Sippel andOhkoshi, 2002]) where the FX spot is locally governed by a geometric Brownian motion, while each of the domestic and foreign interest rates follows a Hull-White one factor Gaussian model [Hull and White, 1993]. Using such a model does not allow the volatility smile/skew effect encountered in the FX market to be taken into account, and is therefore not appropriate to price and hedge long-dated FX products.Different methods exist to incorporate smile/skew effects in the three-factor pricing model. In the literature, one can find different approaches which consist either of using a local volatility for the FX spot or a stochastic volatility and/or jump. There are many processes that can be used for the stochastic volatility and their choices will generally depend on their tractability and solvability. All arXiv:1204.0633v1 [q-fin.PR]

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 an extension to obtain a more general volatility model and propose a calibration method for the local volatility associated to this model.

We study Vanna-Volga methods which are used to price first generation exotic options in the Foreign Exchange market. They are based on a rescaling of the correction to the Black-Scholes price through the so-called "probability of survival" and the "expected first exit time". Since the methods rely heavily on the appropriate treatment of market data we also provide a summary of the relevant conventions. We offer a justification of the core technique for the case of vanilla options and show how to adapt it to the pricing of exotic options. Our results are compared to a large collection of indicative market prices and to more sophisticated models. Finally we propose a simple calibration method based on one-touch prices that allows the Vanna-Volga results to be in line with our pool of market data.

We propose an integrated model of the joint dynamics of FX rates and asset prices for the pricing of FX derivatives, including Quanto products; the model is based on a multivariate construction for Lévy processes which proves to be analytically tractable. The approach allows for simultaneous calibration to market volatility surfaces of currency triangles, and also gives access to market consistent information on dependence between the relevant variables. A successful joint calibration to real market data is presented for the particular case of the Variance Gamma process.