Exotic options are derivatives which have features that makes them more complex than vanilla traded products. Thus, finding their fair value is not always an easy task. We look at a particular example of the exotic options - the power option - whose payoffs are nonlinear functions of the underlying asset price. Previous analyses of the power option have only obtained solutions using probability methods for the case of the constant stock volatility and interest rate. Using Lie symmetry analysis we obtain an optimal system of the Lie point symmetries of the power option PDE and demonstrate an algorithmic method for finding solutions to the equation. In addition, we find a new analytical solution to the asymmetric type of the power option. We also focus on the more practical and realistic case of time dependent parameters: volatility and interest rate. Utilizing Lie symmetries, we are able to provide a new exact solution for the terminal pay off case. We also consider the power parameter of the option as a time dependent factor. A new solution is once again obtained for this scenario. / Thesis (Ph.D.)-University of KwaZulu-Natal, Durban, 2013.

The weather derivative concept was created in the United States of America as a result of the deregulation of its energy industry. When other countries learnt of this concept they decided to enter the market as well. Thus a body called the Weather Risk Management Association was established. This body’s main function is to collate information pertaining to weather risk and to help the process of advancement and growth within the market. The weather risk market has grown tremendously and various participants across the world are using weather derivative products to protect the revenue of their respective companies against adverse weather condition. South Africa entered the weather risk market and it’s contracted its first weather contract in February 2000. The objective of this study is to evaluate the South African financial market perceptions on weather derivatives and to establish the feasibility of use. The study also places emphasis on the importance of evaluating the South African economic conditions in order to achieve the goal of the study. Hence the study evaluates the different aspects in terms of the legal, accounting, taxation, weather data, and structuring and pricing implications of a weather derivative transaction. Thus a survey was designed, forwarded, and received back from professionals in the legal, accounting, taxation, weather data, and structuring and pricing fields. This analysis was conducted to evaluate the South African financial market’s perceptions on weather derivative applications. / Prof. C.H. van Schalkwyk

"This project is an investigation and implementation of pricing derivative securities using the forward measure. It will explain the methodology of building a modified discrete Ho-Lee interest rate model to do so, along with the extraction of historical yield and interest rates to calibrate the model. "

by Choi Ming Yee, Fung Lai Shun, So Wai Ching. / Thesis (M.B.A.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 87-91). / ABSTRACT --- p.ii / TABLE OF CONTENTS --- p.iii / LIST OF FIGURES --- p.v / LIST OF TABLES --- p.vi / LIST OF ABBREVIATIONS --- p.vii / Chapter / Chapter I. --- INTRODUCTION --- p.1 / Chapter II. --- PERSPECTIVES OF INTEREST RATE --- p.3 / Interest Rate and Capital Market --- p.3 / Trade-off between Current and Future Consumption --- p.3 / An Economy without Exchange --- p.4 / An Economy with Capital Market --- p.5 / Determinants of Interest Rate --- p.7 / Credit Considerations --- p.8 / Term Structure --- p.9 / Loanable Funds --- p.11 / Interest Rate Risk --- p.11 / Interest Rate Volatility --- p.15 / Chapter III. --- DEVELOPMENT OF INTEREST RATE DERIVATIVES --- p.20 / The Emergence of Derivatives Markets --- p.20 / Interest Rate Derivatives Market --- p.23 / Interest Rate Futures --- p.24 / Interest Options --- p.25 / Interest Rate Swaps --- p.27 / Forward Rote Agreements (FRAs) --- p.29 / Chapter IV. --- MACROECONOMIC DEVELOPMENT IN ASIA PACIFIC REGION --- p.31 / Chapter V. --- MOTIVATION FOR FINANCIAL LIBERALIZATION --- p.33 / Limitations in Old Systems --- p.33 / Interest Rate Ceilings --- p.33 / Exchange Controls --- p.34 / Portfolio Selection and Credit Rationing --- p.35 / Taxes and Reserve Requirement --- p.37 / Advantage of Liberalization --- p.38 / Chapter VI. --- ECONOMIC VOLATILITY --- p.41 / Capital Mobility and International Integration --- p.41 / Monetary Policy --- p.45 / Chapter VII. --- THE DEMAND AND SUPPLY OF INTEREST RATE DERIVATIVES --- p.48 / Can Hedging Add Value to the Company? --- p.49 / Can Hedging Alter the Discount Rate of a Company? --- p.49 / Chapter VIII. --- THE ASIAN MARKET --- p.58 / New Derivatives Exchanges --- p.61 / Chapter IX. --- FORCES DRIVING DERIVATIVES GROWTH --- p.63 / Sustained Shifts in Volatility --- p.64 / The Demand for New Ways to Transfer Interest Rate Risk --- p.66 / The Demand for Liquidity --- p.69 / Chapter X. --- THE FUTURE --- p.81 / APPENDIX --- p.86 / BIBLIOGRAPHY --- p.87

The credit derivatives market has known an incredible development since its advent in the 1990's. Today there is a plethora of credit derivatives going from the simplest ones, credit default swaps (CDS), to more complex ones such as synthetic single-tranche collateralized debt obligations. Valuing this rich panel of products involves modeling credit risk. For this purpose, two main approaches have been explored and proposed since 1976. The first approach is the Structural approach, first proposed by Merton in 1976, following the work of Black-Scholes for pricing stock options. This approach relies in the capital structure of a firm to model its probability of default. The other approach is called the Reduced-form approach or the hazard rate approach. It is pioneered by Duffie, Lando, Jarrow among others. The main thesis in this approach is that default should be modeled as a jump process. The objective of this work is to value Asset-backed Credit default swaps using the hazard rate approach.

In this project,European Call and Put options,and also American Call and Put options have been priced by some finite difference methods using the C++ programming language.The report describes the following:The theory behind the pricing of options,some pricing methods,and how some finite difference pricing methods have been implemented in C++.

Guan, Peiqiu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (p. 67-71). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Literature Review --- p.6 / Chapter 2.1 --- Characteristic Function --- p.6 / Chapter 2.1.1 --- Definition --- p.6 / Chapter 2.1.2 --- Inverse Fourier Transform --- p.8 / Chapter 2.1.3 --- Fast Fourier Transform (FFT) --- p.9 / Chapter 2.2 --- Levy Processes --- p.13 / Chapter 2.2.1 --- Definition --- p.13 / Chapter 2.2.2 --- Levy-Khinchine Formula --- p.15 / Chapter 2.2.3 --- Levy Processes in Finance --- p.17 / Chapter 2.3 --- Exotic Options --- p.17 / Chapter 2.3.1 --- Barrier Options --- p.18 / Chapter 2.3.2 --- Lookback Options --- p.19 / Chapter 2.3.3 --- Asian Options --- p.20 / Chapter 3 --- FFT Network Model --- p.23 / Chapter 3.1 --- Weaknesses of Traditional Tree Approaches --- p.24 / Chapter 3.2 --- FFT Network Model --- p.30 / Chapter 3.3 --- Basic Transition Probability Matrix --- p.31 / Chapter 3.4 --- Basic FFT Network Pricing Algorithm --- p.35 / Chapter 3.4.1 --- Plain Vanilla Options --- p.35 / Chapter 4 --- FFT Network for Exotic Options --- p.38 / Chapter 4.1 --- Barrier Option Pricing --- p.38 / Chapter 4.2 --- Forward Shooting Grid --- p.41 / Chapter 4.3 --- FSG in FFT Network --- p.43 / Chapter 4.4 --- Lookback and Knock-in Options --- p.45 / Chapter 4.4.1 --- American Lookback Option Pricing Algorithm --- p.48 / Chapter 4.4.2 --- Knock-in American Option Pricing Algorithm --- p.50 / Chapter 4.5 --- Asian Option Pricing --- p.51 / Chapter 4.5.1 --- Asian Option Pricing Algorithm --- p.54 / Chapter 5 --- Numerical Implementation --- p.57 / Chapter 5.1 --- Numerical Scheme --- p.57 / Chapter 5.2 --- Numerical Result --- p.60 / Chapter 6 --- Conclusion --- p.65 / Bibliography --- p.67

Chan, Tsz Him. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (p. 94-96). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Monte Carlo method in Finance --- p.6 / Chapter 2.1 --- Principle of MC and pricing theory --- p.6 / Chapter 2.2 --- An illustrative example --- p.9 / Chapter 3 --- Discretization method --- p.15 / Chapter 3.1 --- The Euler scheme and Milstein scheme --- p.16 / Chapter 3.2 --- Convergence of Mean Square Error --- p.19 / Chapter 4 --- Quasi Monte Carlo method --- p.22 / Chapter 4.1 --- Basic idea of QMC --- p.23 / Chapter 4.2 --- Application of QMC in Finance --- p.29 / Chapter 4.3 --- Another illustrative example --- p.34 / Chapter 5 --- Our Methodology --- p.42 / Chapter 5.1 --- Measure decomposition --- p.43 / Chapter 5.2 --- QMC in SDE simulation --- p.51 / Chapter 5.3 --- Towards a workable algorithm --- p.58 / Chapter 6 --- Numerical Result --- p.69 / Chapter 6.1 --- Case I Generalized Wiener Process --- p.69 / Chapter 6.2 --- Case II Geometric Brownian Motion --- p.76 / Chapter 6.3 --- Case III Ornstein-Uhlenbeck Process --- p.83 / Chapter 7 --- Conclusion --- p.91 / Bibliography --- p.96

Cheng, Ping. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (p. 91-96). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Background --- p.1 / Chapter 1.1.1 --- Accumulator in a Nutshell --- p.1 / Chapter 1.1.2 --- Criticism over Accumulators --- p.3 / Chapter 1.1.3 --- Significance of Research over Accumulators --- p.4 / Chapter 1.1.4 --- Contribution of this Research --- p.5 / Chapter 1.2 --- Literature Review --- p.6 / Chapter 1.2.1 --- Literature on Option Pricing Theory --- p.6 / Chapter 1.2.2 --- Literature on Occupation Time Derivatives and Accumulators --- p.9 / Chapter 1.2.3 --- Accumulators as Occupation Time Deriva- tives --- p.14 / Chapter 1.3 --- Structure of this Thesis --- p.16 / Chapter 2 --- Theoretical Foundation --- p.17 / Chapter 2.1 --- Black Scholes Framework --- p.18 / Chapter 2.1.1 --- The Model --- p.18 / Chapter 2.1.2 --- Girsanov's Theorem --- p.20 / Chapter 2.1.3 --- Simulation --- p.21 / Chapter 2.2 --- Heston Framework --- p.22 / Chapter 2.2.1 --- Motivation to Extend to Heston Model --- p.22 / Chapter 2.2.2 --- The Model --- p.23 / Chapter 2.2.3 --- The Monte Carlo Method --- p.25 / Chapter 3 --- Pricing under Black-Scholes Framework --- p.30 / Chapter 3.1 --- Structure One (Suspension Feature) --- p.30 / Chapter 3.1.1 --- Introduction --- p.30 / Chapter 3.1.2 --- Model --- p.32 / Chapter 3.1.3 --- Sensitivity Tests --- p.35 / Chapter 3.1.4 --- Simulation Results --- p.38 / Chapter 3.2 --- Structure Two (Knock-out Feature) --- p.40 / Chapter 3.2.1 --- Introduction --- p.40 / Chapter 3.2.2 --- Model --- p.41 / Chapter 3.2.3 --- Sensitivity Tests --- p.45 / Chapter 3.2.4 --- Simulation Results --- p.48 / Chapter 3.3 --- Structure Three (Knock-out & Double Commit- ment Feature) --- p.50 / Chapter 3.3.1 --- Introduction --- p.50 / Chapter 3.3.2 --- Model --- p.51 / Chapter 3.3.3 --- Sensitivity Tests --- p.56 / Chapter 3.3.4 --- Simulation Results --- p.58 / Chapter 4 --- Extension: Pricing under Heston Framework --- p.59 / Chapter 4.1 --- Structure One --- p.59 / Chapter 4.1.1 --- Pricing of the Contract --- p.59 / Chapter 4.2 --- Structure Two and Three --- p.61 / Chapter 4.2.1 --- Simulation Results --- p.62 / Chapter 4.3 --- Heston Parameters Estimates --- p.63 / Chapter 5 --- Discussion --- p.66 / Chapter 5.1 --- Volatility of Accumulators --- p.66 / Chapter 5.2 --- Instability in the Model Parameters --- p.69 / Chapter 5.3 --- Premium over Accumulators --- p.71 / Chapter 5.4 --- Return of the Accumulator Products --- p.72 / Chapter 6 --- Future Work & Conclusion --- p.75 / Chapter 6.1 --- Future Work --- p.75 / Chapter 6.2 --- Conclusion --- p.76 / Chapter A --- Other Parameters Estimation --- p.77 / Chapter B --- Sample Contracts --- p.80 / Chapter B.1 --- Equity Accumulator --- p.80 / Chapter B.2 --- Commodity Accumulator --- p.80 / Chapter B.3 --- FX-Linked Accumulation --- p.80 / Bibliography --- p.91

This study attempts to resolve two important issues. First, it investigates the diversification benefit of crude oil for equities. Second, it examines whether or not crude oil derivatives transactions by oil and gas producers can change shareholders' wealth. With these two major goals in mind, I study the risk and return profile of crude oil, the value effect of crude oil derivatives transactions, and the systematic risk exposure effect of crude oil derivatives transactions. In contrast with previous studies, this study applies the Goldman Sachs Commodity Index (GSCI) methodology to measure the risk and return profile of crude oil. The results show that crude oil is negatively correlated with stocks so adding crude oil into a portfolio with equities can provide significant diversification benefits for the portfolio. Given the diversification benefit of crude oil mixed with equities, this study then examines the value effect of crude oil derivatives transactions by oil and gas producers. Differing from traditional corporate risk management literature, this study examines corporate derivatives transactions from the shareholders' portfolio perspective. The results show that crude oil derivatives transactions by oil and gas producers do impact value. If oil and gas producing companies stop shorting crude oil derivatives contracts, company stock prices increase significantly. In contrast, if oil and gas producing companies start shorting crude oil derivatives contracts, stock prices drop marginally significantly. Thus, hedging by producers is not necessarily good. This paper, however, finds that changes in policy regarding crude oil derivatives transactions cannot significantly affect the beta of shareholders' portfolios. The value effect, therefore, cannot be attributed to any systematic risk exposure change of shareholders' portfolios. Market completeness, transaction costs, and economies of scale are identified as possible sources of value effect. The following conclusions have been obtained in this study. Crude oil provides significant diversification benefits for equities. In the presence of market imperfections, crude oil derivatives transactions by oil and gas producers may change shareholders' wealth, even though crude oil derivatives transactions by oil and gas producers do not have significant effect on the systematic risk exposures of companies.