The valuation of many types of financial contracts and contingent claim agreements is complicated by the possibility that one party will default on their contractual obligations. This dissertation develops a general model that prices Black-Scholes options subject to intertemporal default risk using stochastic barrier option pricing theory. The explicit closed-form solution is obtained by generalizing the reflection principle to k-space to determine the appropriate transition density function. The European analytical valuation formula has a straightforward economic interpretation and preserves much of the intuitive appeal of the traditional Black-Scholes model. The hedging properties of this model are compared and contrasted with the default-free model. The model is extended to include partial recoveries. In one situation, the option holder is assumed to recover α (a constant) percent of the value of the writer’s assets at the time of default. This version of the partial recovery option leads to an analytical valuation formula for a first passage option - an option with a random payoff at a random time. / Ph. D.

The development and introduction of financial derivatives have great impact on modern finance. Option pricing theory has become a powerful tool to value and to understand these innovations. It is also an indispensable tool to calculate hedge ratios for risk measurement and management. On the one hand, the introduction of new financial derivatives has been blamed for making financial market more volatile and risky as evidenced in the financial markets of the USA and Japan, especially during the expiration of index futures and index options, On the other hand, the applicability of new pricing models to hedging strategies is essential in monitoring and managing option positions. This study tries to give some answers on first: whether the expiration of financial derivatives increases the volatility of the Hong Kong stock market; second: whether we can better hedge by straightly applying more elaborated option valuation models replacing the standard Black-Scholes model, which market participants commonly employed for hedging option positions. Part I of this article addresses the first question while Part II studies the second question. / Yung Hei Ming. / Source: Dissertation Abstracts International, Volume: 62-09, Section: A, page: 3138. / Supervisor: Zhang Hua. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references. / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest dissertations and theses, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Chu Chun-fai, Carlin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 69-73). / Abstracts in English and Chinese. / Chapter Part I: --- INTRODUCTION --- p.1 / Chapter Part II: --- LITERATURE REVIEW --- p.4 / Chapter - --- Financial option-pricing theory / Chapter - --- Real option-pricing theory / Chapter - --- Real option-pricing theory in Management Information System Area / Chapter Part III: --- CASE BACKGROUND --- p.14 / Chapter - --- Case Background / Chapter - --- Availability of online banking services in Hong Kong / Chapter - --- Online banking investment in the Hong Kong Chinese Bank / Chapter Part IV: --- RESEARCH MODEL --- p.19 / Chapter - --- Research model / Chapter - --- Modelling of the optimal timing problem of HKCB / Chapter - --- Justification of geometric Brownian motion assumption for using Black-Scholes formula / Chapter Part V : --- DATA COLLECTION --- p.30 / Chapter Part VI: --- ANALYSIS RESULT --- p.35 / Chapter - --- Analysis result / Chapter - --- Sensitivity analysis on the selected parameters / Chapter - --- Suggested investment timing / Chapter Part VII: --- DISCUSSIONS AND IMPLICATIONS --- p.44 / Chapter - --- Result discussion / Chapter - --- Implications for researchers / Chapter - --- Implications for practitioners / Chapter Part VIII: --- LIMITATIONS AND CONTRIBUTIONS --- p.48 / Chapter - --- Limitation on data collection process / Chapter - --- Limitations on Black-Scholes model / Chapter - --- Contributions / APPENDIX / Appendix A -Limitation of traditional Discounted Cash Flow analysis --- p.51 / Appendix B -Banks services available to the customers --- p.54 / Appendix C -Sample path of a Geometric Brownian Motion --- p.56 / Appendix D -Discounted Cash Flows analysis of immediate entry of online banking investment --- p.57 / Appendix E -Black-Scholes formula and its interpretation for non-traded --- p.61 / Appendix F -Questionnaire for Online banking investment --- p.64 / Appendix G -Availability of online banking services in May 2001 --- p.67 / Appendix H -Sensitivity analysis on the number of initial usage --- p.68 / Appendix I -Reference List --- p.69

University of Macau / Faculty of Science and Technology / Department of Mathematics

Thesis (MComm (Mathematics))--University of Stellenbosch, 2007. / The word numéraire refers to the unit of measurement used to value a portfolio of assets. The change of numéraire technique involves converting from one measurement to another. The foreign exchange markets are natural settings for interpreting this technique (but are by no means the only examples). This dissertation includes elementary facts about the change of numeraire technique. It also discusses the mathematical soundness of the technique in the abstract setting of Delbaen and Schachermayer’s Mathematics of Arbitrage. The technique is then applied to financial pricing problems. The right choice of numéraire could be an elegant approach to solving a pricing problem or could simplify computation and modelling.

University of Macau / Faculty of Science and Technology / Department of Mathematics

University of Macau / Faculty of Science and Technology / Department of Mathematics

University of Macau / Faculty of Science and Technology / Department of Mathematics

Thesis (MSc (Mathematics))--Stellenbosch University, 2008. / The computation of the greeks of an option is an important aspect of financial mathematics. The information gained from knowing the value of a greek of an option can help investors decide whether or not to hold on to or to sell their options to avoid losses or gain a profit. However, there are technical difficulties that arise from having to do this. Among them is the fact that the mathematical formula for the value some options is complex in nature and evaluating their greeks may be cumber- some. On the other hand the greek might have to be numerically estimated if the option does not posses an explicit evaluation formula. This could be a computationally expensive undertaking. Malliavin calculus offers us a solution to these problems. We can find formula that can be used in combination with Monte Carlo simulations to give results quickly and which are not computationally expensive to obtain and hence give us an degree of accuracy higher that non Malliavin calculus techniques. This thesis will develop the Malliavin calculus tools that will enable us to develop the tools which we will then use to compute the greeks of some known options.

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.