Stochastic Volatility with Levy Processes: Calibration and Pricing

Abstract

In this thesis, stochastic volatility models with Levy processes are treated in parameter calibration by the Carr-Madan fast Fourier transform (FFT) method and pricing through the partial integro-differential equation (PIDE) approach. First, different models where the underlying log stock price or volatility driven by either a Brownian motion or a Levy process are examined on Standard & Poor's (S&P) 500 data. The absolute percentage errors show that the calibration errors are different between the models. Furthermore, a new method to estimate the standard errors, which can be seen as a generalization of the traditional error estimation method, is proposed and 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 under the volatilty process of the model and by making the other model parameters fixed. Parameters are calibrated over five consecutive days 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 likely that 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 parts of the system. An operator splitting method is proposed to break down the complicated problem into 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 splitting is carried out in a reasonable time. The results show that the operator splitting method is numerically stable and has the monotonicity perserving property with fairly good accuracy, when the boundary conditions at volatility are estimated by Neumann conditions.