  In the first part of this thesis, we focus on American options in the Heston
model. We first give an analytical characterization of the value function of an
American option as the unique solution of the associated (degenerate) parabolic
obstacle problem. Our approach is based on variational inequalities in suitable
weighted Sobolev spaces. We also investigate the properties of the American
value function. We prove that, under suitable assumptions on the payoff, the
value function is nondecreasing with respect to the volatility variable. Then,
we focus on an American put option and we extend some results which are well
known in the Black and Scholes world. In particular, we prove the strict
convexity of the value function in the continuation region, some properties of
the free boundary function, the Early Exercise Price formula and a weak form of
the smooth fit principle. In the second part, we deal with the numerical
computation of European and American option prices in jump-diffusion stochastic
volatility models. We first focus on the Bates-Hull-White model. We consider a
backward hybrid algorithm which uses a Markov chain approximation (a "multiple
jumps" tree) in the direction of the volatility and the interest rate and a
(deterministic) finite-difference approach in order to handle the underlying
asset price process. We also provide a simulation scheme to be used for Monte
Carlo evaluations. Numerical results show the reliability and efficiency of the
methods. Finally, we analyse the rate of convergence of the hybrid algorithm
applied to general jump-diffusion models. We study first-order weak convergence
of Markov chains to diffusions under quite general assumptions. Then, we prove
the convergence of the algorithm, by studying the stability and the consistency
of the hybrid scheme, in a sense that allows us to exploit the probabilistic
features of the Markov chain approximation.
