Wang, Yang. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (p. 72-74). / Abstracts in English and Chinese. / Chapter 1 --- Literature Review and Model Description --- p.1 / Chapter 1.1 --- Portfolio theory under mean-variance framework --- p.2 / Chapter 1.2 --- Portfolio theory under utility-maximizing framework --- p.5 / Chapter 1.3 --- Model Description --- p.11 / Chapter 2 --- Parameterized optimal rebalancing strategy --- p.14 / Chapter 2.1 --- An open-loop policy of the T-horizon model --- p.16 / Chapter 2.2 --- A closed-loop policy of the T-horizon model --- p.24 / Chapter 2.3 --- Illustrative numerical example --- p.36 / Chapter 3 --- Non-parameterized optimal rebalancing model --- p.46 / Chapter 3.1 --- T=2 period problem --- p.47 / Chapter 3.2 --- T=3 period problem --- p.55 / Chapter 4 --- s-S type policy --- p.59 / Chapter 4.1 --- Exponential K-convex function --- p.60 / Chapter 4.2 --- Revised multiperiod portfolio selection model --- p.62 / Chapter 5 --- Conclusion and summary of work --- p.70 / Bibliography --- p.71

This thesis is concerned with the investigation of numerical methods for the solution of the Hamilton-Jacobi-Bellman (HJB) equations arising in European and American option pricing with proportional transaction costs. We first consider the problem of computing reservation purchase and write prices of a European option in the model proposed by Davis, Panas and Zariphopoulou [19]. It has been shown [19] that computing the reservation purchase and write prices of a European option involves solving three different fully nonlinear HJB equations. In this thesis, we propose a penalty approach combined with a finite difference scheme to solve the HJB equations. We first approximate each of the HJB equations by a quasi-linear second order partial differential equation containing two linear penalty terms with penalty parameters. We then develop a numerical scheme based on the finite differencing in both space and time for solving the penalized equation. We prove that there exists a unique viscosity solution to the penalized equation and the viscosity solution to the penalized equation converges to that of the original HJB equation as the penalty parameters tend to infinity. We also prove that the solution of the finite difference scheme converges to the viscosity solution of the penalized equation. Numerical results are given to demonstrate the effectiveness of the proposed method. We extend the penalty approach combined with a finite difference scheme to the HJB equations in the American option pricing model proposed by Davis and Zarphopoulou [20]. Numerical experiments are presented to illustrate the theoretical findings.