Variance reduction for sensitivities: application to interest rate products.

Summary This thesis studies variance reduction techniques for the problem of approximating functionals of diffusion processes, motivated by applications in computational finance to derivatives pricing and hedging. The main tool is Malliavin's stochastic calculus of variations, which yields simulatable representations of both sensitivities and the optimal strategy for variance reduction. In the first part we present a unified view of the control variates and importance sampling methodologies, and give a practical factorization of the optimal strategies. We introduce a parametric importance sampling algorithm and carry out its study in detail. To solve the corresponding optimization problem, we validate two procedures based respectively on stochastic approximation and minimizing an empirical counterpart. Several numerical examples are given which highlight the method's potential. In a second part we combine integration by parts with a Girsanov transform to obtain several stochastic representations of sensitivities. Going beyond a strictly elliptic framework, we show on a class of HJM models with stochastic volatility how to efficiently construct a covering vector field in the sense of Malliavin-Thalmaier. The last chapter, of a more applied nature, deals with a practical case of pricing and hedging exotic rates options.