Stochastic Volatility with Levy Processes: Calibration and Pricing

In this thesis, stochastic volatility models with Levy processes are treatedin parameter calibration by the Carr-Madan fast Fourier transform (FFT) method and pricingthrough the partial integro-differential equation (PIDE) approach. First, different models where the underlying log stock price or volatilitydriven by either a Brownian motion or a Levy process are examinedon Standard & Poor's (S&P) 500 data. The absolute percentage errors show that the calibration errorsare different between the models. Furthermore, a new method to estimate the standard errors, which can beseen as a generalization of the traditional error estimation method, is proposedand the results show that in all the parameters of a stochastic volatility model,some parameters are well-identified while the others are not. Next, the previous approach to parameter calibration is modified by making the volatility constrained underthe volatilty process of the model and by making theother model parameters fixed. Parameters are calibrated over five consecutivedays on S&P 500 or foreign exchange (FX) options data. The results show that the absolute percentage errors do not get much larger and are still in an acceptable threshold. Moreover, the parameter calibrating procedure is stabilized due to the constraint made on the volatility process. In other words, it is more likelythat the same calibrated parameters are obtained from different initial guesses. Last, for the PIDEs with two or three space dimensions, which arise in stochastic volatility models or in stochastic skew models,it is in general inefficient or infeasible to apply the same numerical technique to different partsof the system. An operator splitting method is proposed to break down the complicated probleminto a diffusion part and a jump part. The two parts are treated with a finite difference and a finite element method, respectively. For the PIDEs in 1-D, 2-D and 3-D cases, the numerical approach by the operator splittingis carried out in a reasonable time. The results show that the operator splitting method is numerically stable and hasthe monotonicity perserving property with fairly good accuracy, when the boundary conditionsat volatility are estimated by Neumann conditions.