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.

by Lee Yiu Cho. / Thesis (M.B.A.)--Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaf 47). / ACKNOWLEDGMENT --- p.iii / ABSTRACT --- p.iv / TABLE OF CONTENTS --- p.v / LIST OF CHARTS --- p.vi / Chapter / Chapter I. --- INTRODUCTION --- p.1 / Chapter II. --- THE HANG SENG INDEX OPTION --- p.3 / Chapter III. --- LITERATURE REVIEW --- p.6 / Chapter IV. --- METHODOLOGY & DATA COLLECTION --- p.9 / Methodology --- p.9 / The Black-Scholes Model --- p.9 / Data Collection --- p.11 / Data Manipulation --- p.13 / Limitation of Data --- p.14 / Chapter V. --- EMPIRICAL RESULTS --- p.16 / General Trading Pattern --- p.16 / Comparison of Actual and Theoretical Premiums --- p.17 / Analysis for 2 Sub-periods --- p.19 / Correlation Between Deviations and Variables --- p.22 / The Degree of in-the-money or out-of-the-money --- p.22 / Actual Premium Level --- p.23 / Transaction Volume --- p.25 / Chapter VI. --- CONCLUSION --- p.26 / CHARTS --- p.29 / BIBLIOGRAPHY --- p.47

The study investigates whether the main capital market theories and pricing models provide a reasonably accurate description of the working and efficiency of capital markets, of the pricing of shares and options and the effect the risk/return relationship has on investor behaviour. The capital market theories and pricing models included in the study are Portfolio Theory, the Efficient Market Hypothesis (EMH), the Capital Asset Pricing Model (CAPM), the Arbitrage Pricing Theory (APT), Options Theory and the BlackScholes (8-S) Option Pricing Model. The main conclusion of the study is that the main capital market theories and pricing models, as reviewed in the study, do provide a reasonably accurate description of reality, but a number of anomalies and controversial issues still need to be resolved. The main recommendation of the study is that research into these theories and models should continue unabated, while the specific recommendations in a South African context are the following: ( 1) the benefits of global diversification for South African investors should continue to be investigated; (2) the level and degree of efficiency of the JSE Securities Exchange SA (JSE) should continue to be monitored, and it should be established whether alternative theories to the EMH provide complementary or better descriptions of the efficiency of the South African market; (3) both the CAPM and the APT should continue to be tested, both individually and jointly, in order to better understand the pricing mechanism of, and risk/return relationship on the JSE; (4) much South African research still needs to be conducted on the efficiency of the relatively new options market and the application of the B-S Option Pricing Model under South African conditions. / Financial Accounting / M. Com. (Accounting)

The study investigates whether the main capital market theories and pricing models provide a reasonably accurate description of the working and efficiency of capital markets, of the pricing of shares and options and the effect the risk/return relationship has on investor behaviour. The capital market theories and pricing models included in the study are Portfolio Theory, the Efficient Market Hypothesis (EMH), the Capital Asset Pricing Model (CAPM), the Arbitrage Pricing Theory (APT), Options Theory and the BlackScholes (8-S) Option Pricing Model. The main conclusion of the study is that the main capital market theories and pricing models, as reviewed in the study, do provide a reasonably accurate description of reality, but a number of anomalies and controversial issues still need to be resolved. The main recommendation of the study is that research into these theories and models should continue unabated, while the specific recommendations in a South African context are the following: ( 1) the benefits of global diversification for South African investors should continue to be investigated; (2) the level and degree of efficiency of the JSE Securities Exchange SA (JSE) should continue to be monitored, and it should be established whether alternative theories to the EMH provide complementary or better descriptions of the efficiency of the South African market; (3) both the CAPM and the APT should continue to be tested, both individually and jointly, in order to better understand the pricing mechanism of, and risk/return relationship on the JSE; (4) much South African research still needs to be conducted on the efficiency of the relatively new options market and the application of the B-S Option Pricing Model under South African conditions. / Financial Accounting / M. Com. (Accounting)