Valuation of currency options with stochastic volatility.

Summary This thesis studies the valuation of European and path-dependent currency options in a stochastic volatility framework. The use of random volatility is justified by econometric studies in the first chapter. The second chapter puts into perspective different European option pricing models with stochastic volatility and shows, with the help of an empirical study on the dollar-mark options market, that the Heston (1993) model is the most suitable to reproduce the observed volatility surface. A calibration procedure is used to estimate the values of the parameters of the volatility diffusion process. The second part of this thesis is devoted to exotic options and their valuation by taking into account the information content of the volatility surface. The third chapter focuses on different methods of extracting implied density functions from option prices. The fourth chapter studies different models of implied trees. They aim at valuing exotic options from a tree deformed in order to value a continuum of European options at their market price. The study of these models and the implementation of Derman and Kani's (1994) model show that these techniques sometimes present arbitrage opportunities and assume that volatility is a deterministic function of the underlying price and time, which is not empirically verified. To overcome these problems the fifth chapter uses the stochastic volatility framework and a finite difference technique to evaluate American and exotic options. Differences in premiums from the Black-Scholes model are explained. The convergence of the prices of the stochastic volatility model with those of a static replication is demonstrated under certain assumptions. These results suggest a hedging strategy for barrier options.