Option Pricing: Model Calibration, Approximate Solution, And Numerical Computation

Option pricing is a central theme in the theory and practice of finance. The thesis is concerned with the model calibration and option pricing in a local volatility jump-diffusion model, and approximation of the European option price in a stochas-tic factor model.The thesis is organized as follows.Chapter 1 includes some history of mathematical finance, some preliminaries and the main contents and contributions of the thesis.Chapter 2 is concerned with the model calibration of the local volatility jump-diffusion model. From the viewpoint of optimal control, via the regularization tech-nique, the calibration problem is transformed into a four-parameter least-square problem. We not only obtain the existence and stability of the solution, but also investigate the first order optimality condition. The convexity of the problem and the uniqueness of the solution are also considered.Chapter 3 is concerned with the European option pricing problem in a stochastic factor model. An analytical approximate formula is derived which contains a Black-Sholes price and a weighted sum of Greeks. The resulting error is also estimated.In Chapter 4, the local volatility function is calibrated by the Merton implied volatility function. We do interpolation and extrapolation for the smoothness of the implied volatility surface, which permits analytical formulas for the computation of the derivatives. Moreover, we give a numerical result.Chapter 5 is concerned with numerical computation of option pricing in the local volatility jump-diffusion model. We modify the Crank-Nicolson-Explicit method, which divides the grids into two parts. Furthermore, we compare four different numerical methods in the thesis. Chapter 6 discusses the application of Monte-Carlo method to option pricing in a jump-diffusion model. A detailed procedure is given for the simulation, and an example is reported.