published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

Stochastic volatility models have long provided a popular alternative to the Black- Scholes-Merton framework. They provide, in a self-consistent way, an explanation for the presence of implied volatility smiles/skews seen in practice. Incorporating jumps into the stochastic volatility framework gives further freedom to nancial mathematicians to t both the short and long end of the implied volatility surface. We present three stochastic volatility models here - the Heston model, the Bates model and the SVJJ model. The latter two models incorporate jumps in the stock price process and, in the case of the SVJJ model, jumps in the volatility process. We analyse the e ects that the di erent model parameters have on the implied volatility surface as well as the returns distribution. We also present pricing techniques for determining vanilla European option prices under the dynamics of the three models. These include the fast Fourier transform (FFT) framework of Carr and Madan as well as two Monte Carlo pricing methods. Making use of the FFT pricing framework, we present calibration techniques for tting the models to option data. Speci cally, we examine the use of the genetic algorithm, adaptive simulated annealing and a MATLAB optimisation routine for tting the models to option data via a leastsquares calibration routine. We favour the genetic algorithm and make use of it in tting the three models to ALSI and S&P 500 option data. The last section of the dissertation provides hedging techniques for the models via the calculation of option price sensitivities. We nd that a delta, vega and gamma hedging scheme provides the best results for the Heston model. The inclusion of jumps in the stock price and volatility processes, however, worsens the performance of this scheme. MATLAB code for some of the routines implemented is provided in the appendix.

He, Huifen. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 44-48). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Motivation --- p.1 / Chapter 1.2 --- Literature Review --- p.4 / Chapter 1.2.1 --- Supply Contracts under Price Uncertainty --- p.5 / Chapter 1.2.2 --- Dual Sourcing --- p.6 / Chapter 1.2.3 --- Risk Aversion in Inventory Management --- p.6 / Chapter 1.2.4 --- Hedging Operational Risk Using Financial Instruments --- p.7 / Chapter 1.2.5 --- Financial Literature --- p.9 / Chapter 1.3 --- Organization of the Thesis --- p.10 / Chapter 2 --- A Risk-Neutral Model --- p.12 / Chapter 2.1 --- Framework and Assumptions --- p.12 / Chapter 2.2 --- "Price, Forward and Convenience Yield" --- p.14 / Chapter 2.2.1 --- Stochastic Model of Price --- p.14 / Chapter 2.2.2 --- Marginal Convenience Yield --- p.16 / Chapter 2.3 --- Optimality Equations --- p.17 / Chapter 2.4 --- The Structure of the Optimal Policy --- p.21 / Chapter 2.4.1 --- One-period. Optimal Hedge Decision Rule --- p.21 / Chapter 2.4.2 --- One-period Optimal Orderings Decision Rule --- p.23 / Chapter 2.4.3 --- Optimal Policy --- p.24 / Chapter 3 --- A Risk-Averse Model --- p.28 / Chapter 3.1 --- Risk Aversion Modeling and Utility Function --- p.28 / Chapter 3.2 --- Multi-Period Inventory Modelling --- p.31 / Chapter 3.3 --- Exponential Utility Model --- p.33 / Chapter 3.4 --- Optimal Ordering and Hedging Policy for Multi-Period Problem --- p.37 / Chapter 4 --- Conclusion and Future Research --- p.40 / Bibliography --- p.44 / Chapter A --- Appendix --- p.49 / Chapter A.l --- Notation --- p.49 / Chapter A.2 --- K-Concavity --- p.50

Su, Yiwen. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (p. 76-84). / Abstract also in Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Literature Review --- p.9 / Chapter 2.1 --- Hedge Ratio Review --- p.9 / Chapter 2.2 --- Estimating the Hedge Ratio --- p.13 / Chapter 2.2.1 --- Static Hedge Ratio --- p.13 / Chapter 2.2.2 --- "Dynamic Hedge Ratio, Multivariate GARCH Frame-work and DCC Model" --- p.14 / Chapter 3 --- Futures Market Efficiency --- p.19 / Chapter 3.1 --- Market Efficiency and Cointegration Test --- p.20 / Chapter 4 --- Model Specifications and Hedging Strategy --- p.24 / Chapter 4.1 --- Model Specifications --- p.24 / Chapter 4.1.1 --- BGARCH-DCC Model --- p.25 / Chapter 4.1.2 --- Symmetric BGARCH-DCC Model --- p.28 / Chapter 4.1.3 --- Asymmetric BGARCH-DCC Model --- p.31 / Chapter 4.2 --- Hedge Ratio --- p.33 / Chapter 4.2.1 --- MV Hedge Ratio --- p.34 / Chapter 4.2.2 --- Zero-VaR Hedge Ratio --- p.35 / Chapter 4.3 --- Evaluation of Hedge Effectiveness --- p.38 / Chapter 5 --- Data Description and Empirical Results --- p.39 / Chapter 5.1 --- Preliminary Data Analysis --- p.39 / Chapter 5.2 --- Estimation Results --- p.42 / Chapter 5.3 --- Dynamic Hedging Performance --- p.53 / Chapter 6 --- Conclusion --- p.68 / Chapter A --- Equation Derivation --- p.72 / Bibliography --- p.76

Cheung, Kwan Hung Edwin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 39-40). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- The Levy Process --- p.4 / Chapter 2.1 --- Levy-Khintchine representation --- p.5 / Chapter 2.2 --- Variance Gamma process --- p.6 / Chapter 3 --- Minimum-Variance Static Hedge with Index futures --- p.8 / Chapter 3.1 --- Capital Asset Pricing Model with static hedge --- p.10 / Chapter 3.2 --- Continuous CAPM under Levy process --- p.11 / Chapter 4 --- Option pricing under Levy process --- p.15 / Chapter 4.1 --- Option pricing under the fast Fourier transform --- p.16 / Chapter 4.2 --- The modified fast Fourier transform on call option price --- p.19 / Chapter 5 --- Empirical Results --- p.23 / Chapter 5.1 --- Proposed model for empirical studies --- p.25 / Chapter 5.2 --- Calibration Procedure and Estimates of Betas --- p.26 / Chapter 5.3 --- Hedging performance of Betas --- p.32 / Chapter 6 --- Conclusion --- p.37 / Bibliography --- p.39

Starting from the most famous Black-Scholes model for the underlying asset price, there has been a large variety of extensions made in recent decades. One main strand is about the models which allow a jump component in the asset price. The first topic of this thesis is about the study of jump risk premium by an equilibrium approach. Different from others, this work provides a more general result by modeling the underlying asset price as the ordinary exponential of a L?vy process. For any given asset price process, the equity premium, pricing kernel and an equilibrium option pricing formula can be derived. Moreover, some empirical evidence such as the negative variance risk premium, implied volatility smirk, and negative skewness risk premium can be well explained by using the relation between the physical and risk-neutral distributions for the jump component. Another strand of the extensions of the Black-Scholes model is about the models which can incorporate stochastic volatility in the asset price. The second topic of this thesis is about the replication of exponential variance, where the key risks are the ones induced by the stochastic volatility and moreover it can be correlated with the returns of the asset, referred to as leverage effect. A time-changed L?vy process is used to incorporate jumps, stochastic volatility and leverage effect all together. The exponential variance can be robustly replicated by European portfolios, without any specification of a model for the stochastic volatility. Beyond the above asset pricing and hedging, portfolio optimization is also discussed. Based on the Merton (1969, 1971)'s reduced portfolio optimization and the delta hedging problem, a portfolio of an option, the underlying stock and a risk-free bond can be optimized in discrete time and its optimal solution can be shown to be a mixture of the Merton's result and the delta hedging strategy. The main approach is the elasticity approach, which has initially been proposed in continuous time. In addition to the above optimization problem in discrete time, the same topic but in a continuous-time regime-switching market is also presented. The use of regime-switching makes our market incomplete, and makes it difficult to use some approaches which are applicable in complete market. To overcome this challenge, two methods are provided. The first method is that we simply do not price the regime-switching risk when obtaining the risk-neutral probability. Then by the idea of elasticity, the utility maximization problem can be formulated as a stochastic control problem with only a single control variable, and explicit solutions can be obtained. The second method is to introduce a functional operator to general value functions of stochastic control problem in such a way that the optimal value function in our setting can be given by the limit of a sequence of value functions defined by iterating the operator. Hence the original problem can be deduced to an auxiliary optimization problem, which can be solved as if we were in a single-regime market, which is complete. / published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

Firms can use financial derivatives to hedge risks and thereby decrease the probability of bankruptcy and increase total expected tax shields. Firms also can adjust their operational policies in response to fluctuations in prices, a strategy that is often referred to as "operational hedging". In this paper, I investigate the relationship between the optimal financial and operational hedging strategies for a firm, which are endogenously determined together with its capital structure. This allows me to examine how operational hedging affects debt capacity and total expected tax shields and to make quantitative predictions about the relationship between debt issues and hedging policies. I also model the effects of asymmetric information about firms' investment opportunities on their financing and hedging decisions. First, I examine the case in which both debt and hedging contracts are observable. Then, I study the case in which firms' hedging activities are not completely transparent. The models are tested using a data set compiled from the annual reports of North American gold mining companies. Supporting evidence is found for the key predictions of the model under asymmetric information.

Firms can use financial derivatives to hedge risks and thereby decrease the probability of bankruptcy and increase total expected tax shields. Firms also can adjust their operational policies in response to fluctuations in prices, a strategy that is often referred to as "operational hedging". In this paper, I investigate the relationship between the optimal financial and operational hedging strategies for a firm, which are endogenously determined together with its capital structure. This allows me to examine how operational hedging affects debt capacity and total expected tax shields and to make quantitative predictions about the relationship between debt issues and hedging policies. I also model the effects of asymmetric information about firms' investment opportunities on their financing and hedging decisions. First, I examine the case in which both debt and hedging contracts are observable. Then, I study the case in which firms' hedging activities are not completely transparent. The models are tested using a data set compiled from the annual reports of North American gold mining companies. Supporting evidence is found for the key predictions of the model under asymmetric information. / Business, Sauder School of / Graduate

by Wong Man Kit, Andy, Yu Miu Ki. / Thesis (M.B.A.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 53-54). / ABSTRACT --- p.ii / ACKNOWLEDGEMENT --- p.iii / TABLE OF CONTENTS --- p.iv / Chapter / Chapter I. --- INTRODUCTION --- p.1 / Credit Risk --- p.2 / Operational risk --- p.3 / Liquidity risk --- p.3 / Legal risk --- p.3 / Market Risk --- p.3 / Model risk --- p.4 / Chapter II. --- LITERATURE REVIEW --- p.5 / Value at Risk (VaR) --- p.5 / Minimum Variance --- p.7 / Dollar equivalence --- p.8 / Statistical Hedging --- p.8 / Risk and Return in an Imperfect Hedge --- p.8 / Expected return and standard deviation in a hedged position --- p.9 / Risk and Return in an actual hedge --- p.11 / Optimal Hedge Ratio --- p.13 / Deriving Optimal Hedge Ratio h* --- p.15 / Computing the minimum risk hedge ratio by regression --- p.16 / Basis Risk --- p.18 / Sources of Basis Risk --- p.19 / Variation in the equilibrium price relationship between cash and futures --- p.19 / "Random ""noise"" in the price process" --- p.19 / Mismatch between cash position and the underlying for the future --- p.20 / Hedging Effectiveness --- p.21 / Chapter III. --- DATA AND METHODOLOGY --- p.25 / Data --- p.25 / Data Collection --- p.25 / Data Selection --- p.25 / Data Manipulation --- p.26 / Methodology --- p.27 / Part I: The Selection of the Portfolios --- p.27 / Part II: The Determination of the Hedge Ratio --- p.28 / Part III: Hedged vs. Unhedged --- p.29 / Part IV: Data Analysis & Comparison --- p.31 / Chapter IV. --- FINDINGS --- p.35 / High volatility of Hong Kong market --- p.35 / Manipulation of institutional investors --- p.36 / Hong Kong financial market are less mature --- p.36 / Less efficient information flow --- p.37 / Less Sophisticated Investors --- p.38 / Results and Discussion --- p.39 / Empirical Results --- p.40 / Explanation for the differences --- p.42 / Limitations --- p.47 / Learning Period --- p.47 / Cross Hedging --- p.47 / Mismatch between the futures and the underlying index --- p.48 / Missing Stock Data in the S&P500 --- p.49 / Chapter V. --- CONCLUSION --- p.50 / Tradeoff between risk and return --- p.50 / Hedge Effectiveness --- p.51 / BIBLIOGRAPHY --- p.53