New Methods for Degenerate Stochastic Volatility Models

We develop a qualitative and quantitative analysis on stochastic volatility models. These models represents a wide known class of models among financial mathematics for the evaluation of options and complex derivatives, starting from the fundamental paper of S.L. Heston (1993, The Review of Financial Studies). Moreover, this thesis proposes an interesting researches on both theoretical studies (extending some recent presented results, as those obtained in Costantini et al. (2012) on Finance and Stochastics) on the solution of Dirichlet problem associated and numerical studies, implementing the ADI methods for the approximation of solution and the model calibration by real data taken from real market. Moreover, we propose a weighted average formulation for the Heston stochastic volatility option price to avoid the estimation of the initial volatility. This approach has been developed in the literature for the estimation of the distribution of stock price changes (returns), showing an excellent agreement with real market data. We extend this method to the calibration of option prices considering a large class of probability distributions assumed for the initial volatility parameter. The estimation error is shown to be less than the case of the simple pricing formula. Our results are also validated with a numerical comparison on observed call prices, between the proposed calibration method and the classical approach.