Lam, Ho Man. / "August 2010." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 42-43). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Levy Process --- p.5 / Chapter 2.1 --- Merton's Jump-Diffusion model (1976) --- p.8 / Chapter 2.2 --- Estimation of Levy processes --- p.9 / Chapter 3 --- Transform Martingale Estimation --- p.11 / Chapter 3.1 --- Maximum Likelihood Estimation --- p.11 / Chapter 3.2 --- Transform Martingale Estimating Functions --- p.13 / Chapter 3.2.1 --- Transform Quasi-Score Function --- p.15 / Chapter 3.2.2 --- Composite Quasi-Score Function --- p.17 / Chapter 3.2.3 --- Implementation Issue --- p.18 / Chapter 3.2.4 --- Transform Martingale Estimation on Levy process --- p.21 / Chapter 4 --- Structural Models of Credit Risk --- p.22 / Chapter 4.1 --- Overview --- p.22 / Chapter 4.2 --- Merton's structural credit risk model (1974) --- p.23 / Chapter 4.3 --- Estimation Methodologies --- p.24 / Chapter 4.4 --- Martingale Estimation with KMV's Method --- p.26 / Chapter 5 --- Simulation Study --- p.28 / Chapter 5.1 --- Equity Estimation --- p.28 / Chapter 5.2 --- Estimation of Structural Models --- p.37 / Chapter 6 --- Conclusion --- p.41 / Bibliography --- p.42

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

This thesis consists of several problems concerning numerical approximations for stochastic differential equations, and is divided into three parts. The first one is on the integrability and asymptotic stability with respect to a certain class of Lyapunov functions, and the preservation of the comparison theorem for the explicit numerical schemes. In general, those properties of the original equation can be lost after discretisation, but it will be shown that by some suitable modification of the Euler scheme they can be preserved to some extent while keeping the strong convergence rate maintained. The second part focuses on the approximation of iterated stochastic integrals, which is the essential ingredient for the construction of higher-order approximations. The coupling method is adopted for that purpose, which aims at finding a random variable whose law is easy to generate and is close to the target distribution. The last topic is motivated by the simulation of equations driven by Lévy processes, for which the main difficulty is to generalise some coupling results for the one-dimensional central limit theorem to the multi-dimensional case.

Processus de Lévy et leurs applications en finance / No English summary available.

M.Sc.

L'objet de cette thèse est l'étude de l'option américaine dans un modèle exponentiel de Lévy général. Dans le premier chapitre nous étudions la continuité des réduites dans le cadre des processus de Markov de Feller. Ensuite, nous introduisons les processus de Lévy multidimensionnels et nous montrons la continuité des réduites associées à ceux-ci. Dans le deuxième chapitre, nous clarifions les propriétés basiques de la frontière libre du put américain dans un modèle exponentiel de Lévy général avec dividendes. Nous commençons par caractériser le prix de l'option américaine comme l'unique solution d'une inéquation variationnelle au sens des distributions. Ce qui nous permettra de montrer la continuité de la frontière libre et de donner une caractérisation explicite de la limite du prix critique près de l'échéance. Dans le troisième chapitre, nous étudions la continuité de la dérivée de la fonction valeur du put américain à horizon fini et du put perpétuel. Nous donnons des conditions nécessaires et d'autres suffisantes pour la vérification du principe de smooth-fit. Dans le quatrième chapitre, nous étudions la vitesse de convergence du prix critique vers sa limite à l'échéance dans le cadre d'un modèle exponentiel de Lévy, dans le cas de diffusion avec sauts, puis dans le cas d'un processus de Lévy sans partie Brownienne. Après, nous donnons cette vitesse dans le cas où le terme de diffusion est absent. Enfin, dans le dernier chapitre, nous introduisons deux méthodes numériques pour le calcul des prix des options américaines : la méthode de l'arbre multinomial et celle des différences finies. Nous comparons les deux approches et nous améliorons la convergence de la première dans certains modèles exponentiels de Lévy / Pas de résumé en anglais

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第22823号 / 理博第4633号 / 新制||理||1666(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 矢野 孝次, 教授 泉 正己, 教授 日野 正訓 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

In this thesis we consider the calibration of models based on Lévy processes to option prices observed in some market. This means that we choose the parameters of the option pricing models such that the prices calculated using the models correspond as closely as possible to these option prices. We demonstrate the ability of relatively simple Lévy option pricing models to nearly perfectly replicate option prices observed in nancial markets. We speci cally consider calibrating option pricing models to barrier option prices and we demonstrate that the option prices obtained under one model can be very accurately replicated using another. Various types of calibration are considered in the thesis. We calibrate a wide range of Lévy option pricing models to option price data. We con- sider exponential Lévy models under which the log-return process of the stock is assumed to follow a Lévy process. We also consider linear Lévy models; under these models the stock price itself follows a Lévy process. Further, we consider time changed models. Under these models time does not pass at a constant rate, but follows some non-decreasing Lévy process. We model the passage of time using the lognormal, Pareto and gamma processes. In the context of time changed models we consider linear as well as exponential models. The normal inverse Gaussian (N IG) model plays an important role in the thesis. The numerical problems associated with the N IG distribution are explored and we propose ways of circumventing these problems. Parameter estimation for this distribution is discussed in detail. Changes of measure play a central role in option pricing. We discuss two well-known changes of measure; the Esscher transform and the mean correcting martingale measure. We also propose a generalisation of the latter and we consider the use of the resulting measure in the calculation of arbitrage free option prices under exponential Lévy models. / PhD (Risk Analysis), North-West University, Potchefstroom Campus, 2015

In this thesis we consider the calibration of models based on Lévy processes to option prices observed in some market. This means that we choose the parameters of the option pricing models such that the prices calculated using the models correspond as closely as possible to these option prices. We demonstrate the ability of relatively simple Lévy option pricing models to nearly perfectly replicate option prices observed in nancial markets. We speci cally consider calibrating option pricing models to barrier option prices and we demonstrate that the option prices obtained under one model can be very accurately replicated using another. Various types of calibration are considered in the thesis. We calibrate a wide range of Lévy option pricing models to option price data. We con- sider exponential Lévy models under which the log-return process of the stock is assumed to follow a Lévy process. We also consider linear Lévy models; under these models the stock price itself follows a Lévy process. Further, we consider time changed models. Under these models time does not pass at a constant rate, but follows some non-decreasing Lévy process. We model the passage of time using the lognormal, Pareto and gamma processes. In the context of time changed models we consider linear as well as exponential models. The normal inverse Gaussian (N IG) model plays an important role in the thesis. The numerical problems associated with the N IG distribution are explored and we propose ways of circumventing these problems. Parameter estimation for this distribution is discussed in detail. Changes of measure play a central role in option pricing. We discuss two well-known changes of measure; the Esscher transform and the mean correcting martingale measure. We also propose a generalisation of the latter and we consider the use of the resulting measure in the calculation of arbitrage free option prices under exponential Lévy models. / PhD (Risk Analysis), North-West University, Potchefstroom Campus, 2015

In this thesis we study the optimal stochastic control problem of the drift of a Lévy process. We show that, for a broad class of Lévy processes, the partial integro-differential Hamilton-Jacobi-Bellman equation for the value function admits classical solutions and that control policies exist in feedback form. We then explore the class of Lévy processes that satisfy the requirements of the theorem, and find connections between the uniform integrability requirement and the notions of the score function and Fisher information from information theory. Finally we present three different numerical implementations of the control problem: a traditional dynamic programming approach, and two iterative approaches, one based on a finite difference scheme and the other on the Fourier transform. / text