Thesis (M.S.) -- Worcester Polytechnic Institute. / Keywords: option; financial mathematics; differential equation; stiff. Includes bibliographical references (p.).

Research conducted in mathematical finance focuses on the quantitative modeling of financial markets. It allows one to solve financial problems by using mathematical methods and provides understanding and prediction of the complicated financial behaviors. In this thesis, efforts are devoted to derive and extend stochastic optimization models in financial economics and establish practical algorithms for representing and solving problems in mathematical finance. An option gives the holder the right, but not the obligation, to buy or sell an underlying asset at a specified strike price on or before a specified date. In this thesis, a valuation model for a perpetual convertible bond is developed when the price dynamics of the underlying share are governed by Markovian regime-switching models. By making use of the relationship between the convertible bond and an American option, the valuation of a perpetual convertible bond can be transformed into an optimal stopping problem. A novel approach is also proposed to discuss an optimal inventory level of a retail product from a real option perspective in this thesis. The expected present value of the net profit from selling the product which is the objective function of the optimal inventory problem can be given by the actuarial value of a real option. Hence, option pricing techniques are adopted to solve the optimal inventory problem in this thesis. The goal of risk management is to eliminate or minimize the level of risk associated with a business operation. In the risk measurement literature, there is relatively little amount of work focusing on the risk measurement and management of interest rate instruments. This thesis concerns about building a risk measurement framework based on some modern risk measures, such as Value-at-Risk (VaR) and Expected Shortfall (ES), for describing and quantifying the risk of interest rate sensitive instruments. From the lessons of the recent financial turmoils, it is understood that maximizing profits is not the only objective that needs to be taken into account. The consideration for risk control is of primal importance. Hence, an optimal submission problem of bid and ask quotes in the presence of risk constraints is studied in this thesis. The optimal submission problem of bid and ask quotes is formulated as a stochastic optimal control problem. Portfolio management is a professional management of various securities and assets in order to match investment objectives and balance risk against performance. Different choices of time series models for asset price may lead to different portfolio management strategies. In this thesis, a discrete-time dynamic programming approach which is flexible enough to deal with the optimal asset allocation problem under a general stochastic dynamical system is explored. It’s also interesting to analyze the implications of the heteroscedastic effect described by a continuous-time stochastic volatility model for evaluating risk of a cash management problem. In this thesis, a continuous-time dynamic programming approach is employed to investigate the cash management problem under stochastic volatility model and constant volatility model respectively. / published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

Most recent empirical option valuation studies build on the affine square root (SQR) stochastic volatility model. The SQR model is a convenient choice, because it yields closed-form solutions for option prices. However, relatively little is known about the empirical shortcomings of this model. In the first essay, we investigate alternatives to the SQR model, by comparing its empirical performance with that of five different but equally parsimonious stochastic volatility models. We provide empirical evidence from three different sources. We first use realized volatilities to assess the properties of the SQR model and to guide us in the search for alternative specifications. We then estimate the models using maximum likelihood on a long sample of S& P500 returns. Finally, we employ nonlinear least squares on a time series of cross sections of option data. In the estimations on returns and options data, we use the particle filtering technique to retrieve the spot volatility path. The three sources of data we employ all point to the same conclusion: the SQR model is misspecified. Overall, the best of alternative volatility specifications is a model we refer to as the VAR model, which is of the GARCH diffusion type. / In the second essay, we estimate the Constant Elasticity of Variance (CEV) model in order to study the level of nonlinearity in the volatility dynamic. We also estimate a CEV process combined with a jump process (CEVJ) and analyze the effects of the jump component on the nonlinearity coefficient. Estimation is performed using the particle filtering technique on a long series of S&P500 returns and on options data. We find that both returns data and returns-and-options data favor nonlinear specifications for the volatility dynamic, suggesting that the extensive use of linear models is not supported empirically. We also find that the inclusion of jumps does not affect the level of nonlinearity and does not improve the CEV model fit. / The third essay provides an empirical comparison of two classes of option valuation models: continuous-time models and discrete-time models. The literature provides some theoretical limit results for these types of dynamics, and researchers have used these limit results to argue that the performance of certain discrete-time and continuous-time models ought to be very similar. This interpretation is somewhat contentious, because a given discrete-time model can have several continuous-time limits, and a given continuous-time model can be the limit for more than one discrete-time model. Therefore, it is imperative to investigate whether there exist similarities between these specifications from an empirical perspective. Using data on S&P500 returns and call options, we find that the discrete-time models investigated in this paper have the same performance in fitting the data as selected continuous-time models both in and out-of-sample.

Although options have been traded for many centuries, it has remained a relatively thinly traded financial instrument. Paradoxically, the theory of option pricing has been studied extensively. This is due to the fact that many of the financial instruments that are traded in the market place have an option-like structure, and thus the development of a methodology for option-pricing may lead to a general methodology for the pricing of these derivative-assets. This thesis will focus on the development of the theory of option pricing. Initially, a fundamental principle that underlies the theory of option valuation will be given. This will be followed by a discussion of the different types of option pricing models that are prevalent in the literature. Special attention will then be given to a detailed derivation of both the Black-Scholes and the Binomial Option pricing models, which will be followed by a proof of the convergence of the Binomial pricing model to the Black-Scholes model. The Black-Scholes model will be adapted to take into account the payment of dividends, the possibility of a changing inter est rate and the possibility of a stochastic variance for the rate of return on the underlying as set. Several applications of the Black-Scholes model will finally be presented. / Thesis (M.Sc.)-University of Natal, 1992.

Thesis (M.B.A.)--University of Hong Kong, 1997.

Thesis (Ph. D.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.

No description available.

by Ka-kit Chan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1993. / Includes bibliographical references (leaves 71-73). / Chapter I. --- Introduction to Stochastic Calculus --- p.1 / Stochastic Processes --- p.2 / Stochastic Integration --- p.6 / Quadratic Variation Processes and Mutual Variation Process --- p.11 / The Ito Formula --- p.13 / Girsanov's Theorem --- p.16 / Stochastic Differential Equations --- p.18 / Chapter II. --- Pricing American Equity Options --- p.21 / A Representation Formula for European Put Option --- p.22 / The Free Boundary Formulation of American Put Option --- p.24 / A Representation Formula for American Put Option --- p.27 / An Alternative Representation Formula for American Put Option --- p.35 / The Optimal Exercise Boundary --- p.37 / Numerical Valuations of the Representation Formulae --- p.39 / Chapter III. --- The Effects of Margin Requirements on Option Prices --- p.42 / Pricing European Options --- p.44 / Pricing American Options --- p.46 / Chapter IV. --- General Pricing Theory --- p.49 / Transformations of Price Processes --- p.50 / No Arbitrage Condition and Completeness of Market --- p.52 / More on Market Completeness --- p.58 / Term Structure of Interest Rate and Interest Rate Options --- p.61 / Pricing Equity Options --- p.67 / Bibliography --- p.71

by Fu Yiu-Hang. / Thesis (M.B.A.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 66-67). / ABSTRACT --- p.ii / TABLE OF CONTENTS --- p.ii / LIST OF TABLES --- p.vi / LIST OF EXHIBITS --- p.vii / PREFACE --- p.viii / ACKNOWLEDGMENTS --- p.x / Chapter / Chapter I. --- INTRODUCTION --- p.1 / What is an Option? --- p.1 / Options Market --- p.2 / Uses of Options --- p.2 / Value of Options --- p.3 / Index Options --- p.4 / Hang Seng Index Options --- p.4 / Chapter II. --- BASIC PROPERTIES OF OPTIONS --- p.5 / Assumptions --- p.5 / Notation --- p.5 / Option Prices at Expiration --- p.6 / Call Option Prices at Expiration --- p.6 / Put Option Prices at Expiration --- p.6 / Upper Bounds for Option Prices --- p.6 / Upper Bounds for Call Option Prices --- p.6 / Upper Bounds for Put Option Prices --- p.6 / Lower Bounds for European Option Prices --- p.7 / Lower Bounds for European Call Option Prices --- p.7 / Lower Bounds for European Put Option Prices --- p.8 / Put-Call Parity --- p.8 / Chapter III. --- FACTORS AFFECTING OPTION PRICES --- p.10 / Price of Underlying Instrument --- p.10 / Exercise Price of the Option --- p.10 / Volatility of the Price of Underlying Instrument --- p.11 / Time to Expiration --- p.11 / Risk-free Rate --- p.11 / Dividends --- p.12 / Chapter IV. --- OPTION PRICING MODEL --- p.13 / Assumptions --- p.13 / The Price of Underlying Instrument Follows a Lognormal Distribution --- p.13 / The Variance of the Rate of Return of Underlying Instrument is a Constant --- p.17 / The Risk-free Rate is a Constant --- p.19 / No Dividends are Paid --- p.20 / There are No Transaction Costs and Taxes --- p.20 / The Black-Scholes Option Pricing Model --- p.21 / Notation --- p.21 / The Formulas --- p.21 / The Variables --- p.22 / Properties of the Black-Scholes Formulas --- p.22 / Implied Volatility --- p.23 / Bias of the Black-Scholes Option Pricing Model --- p.26 / Other Option Pricing Models。……………… --- p.27 / Chapter V. --- SENSITIVITIES OF OPTION PRICE TO ITS FACTORS --- p.29 / Delta --- p.29 / Vega --- p.30 / Theta --- p.31 / Rho --- p.32 / Gamma --- p.33 / Managing the Change in the Value of Option --- p.34 / Sensitivities of Portfolio Value to the Factors --- p.34 / Chapter VI. --- TRADING STRATEGIES OF OPTIONS --- p.35 / Methodology --- p.35 / Limitations --- p.36 / Basic Strategies --- p.37 / Long Call --- p.37 / Short Call --- p.39 / Long Put --- p.40 / Short Put --- p.42 / Spread Strategies --- p.43 / Money Spread --- p.43 / Ratio Spread --- p.46 / Box Spread --- p.46 / Butterfly Spread --- p.46 / Condor --- p.49 / Calendar Spread --- p.49 / Diagonal Spread --- p.52 / Combination Strategies --- p.52 / Straddle --- p.52 / Strap --- p.54 / Strip --- p.54 / Strangle --- p.54 / Selecting Trading Strategies Intelligently --- p.56 / Chapter VII. --- CONCLUSIONS --- p.57 / APPENDICES --- p.60 / BIBLIOGRAPHY --- p.66

Lau, Ka Yung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 78-79). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Background --- p.5 / Chapter 2.1 --- Jump Diffusion Models --- p.6 / Chapter 2.2 --- Double Exponential Jump Diffusion Model --- p.8 / Chapter 3 --- Option Pricing with DEJD --- p.10 / Chapter 3.1 --- Laplace Transform --- p.10 / Chapter 3.2 --- European Option Pricing --- p.13 / Chapter 3.3 --- Barrier Option Pricing --- p.14 / Chapter 3.4 --- Lookback Options --- p.16 / Chapter 3.5 --- Turbo Warrant --- p.17 / Chapter 3.6 --- Numerical Examples --- p.26 / Chapter 4 --- Quanto Options under DEJD --- p.30 / Chapter 4.1 --- Domestic Risk-neutral Dynamics --- p.31 / Chapter 4.2 --- The Exponential Copula --- p.33 / Chapter 4.3 --- The moment generating function --- p.36 / Chapter 4.4 --- European Quanto Options --- p.38 / Chapter 4.4.1 --- Floating Exchange Rate Foreign Equity Call --- p.38 / Chapter 4.4.2 --- Fixed Exchange Rate Foreign Equity Call --- p.40 / Chapter 4.4.3 --- Domestic Foreign Equity Call --- p.42 / Chapter 4.4.4 --- Joint Quanto Call --- p.43 / Chapter 4.5 --- Numerical Examples --- p.45 / Chapter 5 --- Path-Dependent Quanto Options --- p.48 / Chapter 5.1 --- The Domestic Equivalent Asset --- p.48 / Chapter 5.1.1 --- Mathematical Results on the First Passage Time of the Mixture Exponential Jump Diffusion Model --- p.50 / Chapter 5.2 --- Quanto Lookback Option --- p.54 / Chapter 5.3 --- Quanto Barrier Option --- p.57 / Chapter 5.4 --- Numerical results --- p.61 / Chapter 6 --- Conclusion --- p.64 / Chapter A --- Numerical Laplace Inversion for Turbo Warrants --- p.66 / Chapter B --- The Relation Among Barrier Options --- p.69 / Chapter C --- Proof of Lemma 51 --- p.71 / Chapter D --- Proof of Theorem 5.4 and 5.5 --- p.74 / Bibliography --- p.78