Essays on the valuation of complex derivatives.

Summary This thesis presents optimal techniques for the valuation and evaluation of hedging parameters of complex structured products using exotic options. In this respect, when no exploitable analytical formula can be obtained, we resort to Monte Carlo techniques whose main qu'alites are their flexibility, the diversity of their domains of use and their often fruitful application to the solution of multi-dimensional problems. The main results presented in this thesis are the analytical valuation in the usual Brownian framework of path-dependent options whose payoff depends on extreme values of the price of the underlying asset recorded in a discrete and non-continuous manner during the life of the option, the efficient simulation by Monte Carlo of continuous extrema in order to value options with many extrema in their payoff. The use of probabilistic conditioning methods within the simulations to reduce the size of the problems and thus substantially decrease the computational time, and the application of optimal variance reduction techniques to the simulation of multi-support option prices and associated sensitivities. We also propose analytical approximations of prices and hedging parameters of products whose payoff uses functions of a large number of iid or inid variables by using certain methods and asymptotic theorems (Edgeworth developments, distribution of extreme values of iid laws,. ). Finally, techniques for valuing exotic options whose underlying price follows a discontinuous path are presented. Thus, we show how to efficiently value path-dependent options with jumps by Monte Carlo and indicate methods for valuing multi-support options whose joint dynamics of the rates of return is of the multivariate jump-diffusion type.