Dependency modeling and process simulation in finance.

Summary The first part of this thesis is devoted to numerical methods for the simulation of random processes defined by stochastic differential equations (SDE). We start by studying the algorithm of Beskos et al [13] which allows us to simulate exactly the trajectories of a process that is a solution of a SDE in dimension 1. We propose an extension of this algorithm for the exact computation of expectations and we study the application of these ideas to the pricing of Asian options in the Black & Scholes model. We then turn our attention to numerical schemes. In the second chapter, we propose two discretization schemes for a family of stochastic volatility models and study their convergence properties. The first scheme is adapted to the pricing of path-dependent options and the second to vanilla options. We also study the special case where the process driving the volatility is an Ornstein-Uhlenbeck process and we exhibit a discretization scheme that has better convergence properties. Finally, in the third chapter, we discuss the weak trajectory convergence of the Euler scheme. We provide a first answer by controlling the Wasserstein distance between the marginals of the solution process and the Euler scheme, uniformly in time. The second part of the thesis deals with the modeling of dependence in finance and this through two distinct problems: the joint modeling between a stock index and its component stocks and the management of default risk in credit portfolios. In the fourth chapter, we propose an original modeling framework in which the volatilities of the index and its components are linked. We obtain a simplified model when the index size is large, in which the index follows a local volatility model and the individual stocks follow a stochastic volatility model composed of an intrinsic part and a common part driven by the index. We study the calibration of these models and show that it is possible to calibrate to observed market option prices for both the index and the stocks, which is a considerable advantage. Finally, in the last chapter of the thesis, we develop an intensity model that allows us to simultaneously and consistently model all rating transitions that occur in a large credit portfolio. In order to generate higher levels of dependence, we introduce the dynamic frailty model in which an unobservable dynamic variable acts multiplicatively on the intensities of transitions. Our approach is purely historical and we study maximum likelihood estimation of the parameters of our models based on data of past rating transitions.