This thesis is composed of three chapters that form two parts. The first part is composed of two chapters and studies problems related to the exotic option market. In the first chapter we are interested in a numerical problem. More precisely we derive closed-form approximations for the price of some exotic options in the Black and Scholes framework. The second chapter discusses the construction of multivariate Lévy processes with and without stochastic volatility. The second part is composed of one chapter. It deals with a completely different issue. There we will study the problem of individual and temporal aggregation in panel data models.

Wong, Shiu Fung. / "July 2010." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 36-38). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Participating policy --- p.4 / Chapter 3 --- Levy Process and its use in financial modelling --- p.8 / Chapter 3.1 --- Levy process in asset modelling --- p.8 / Chapter 3.2 --- Levy process in derivative pricing --- p.11 / Chapter 3.2.1 --- Review of FFT methods in option pricing --- p.12 / Chapter 3.2.2 --- Expectation using FFT --- p.13 / Chapter 4 --- Network methodology --- p.17 / Chapter 4.1 --- Asset dynamic: Network Approach --- p.17 / Chapter 4.1.1 --- Transition probability by FFT --- p.18 / Chapter 4.1.2 --- Example in American option pricing --- p.19 / Chapter 4.2 --- Extended Network for Participating Contract --- p.20 / Chapter 4.3 --- Practical network construction --- p.22 / Chapter 4.3.1 --- Modified network-drift offsetting --- p.23 / Chapter 4.3.2 --- Logarithmic scale network --- p.25 / Chapter 4.4 --- Incorporating surrender rights and mortality --- p.26 / Chapter 4.4.1 --- Surrender right --- p.26 / Chapter 4.4.2 --- Mortality --- p.27 / Chapter 4.5 --- Proof of convergence --- p.28 / Chapter 5 --- Numerical Results --- p.32 / Chapter 5.1 --- The Black and Scholes model --- p.33 / Chapter 5.2 --- The Merton's Jump diffusion model --- p.33 / Chapter 5.3 --- Variance gamma model --- p.34 / Chapter 6 --- Conclusion --- p.35 / Bibliography --- p.36

Lam, Ka Wai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 51-55). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Levy Processes --- p.6 / Chapter 2.1 --- Definition --- p.6 / Chapter 2.2 --- Levy-Khinchine formula --- p.7 / Chapter 2.3 --- Applications of Levy Processes in Finance --- p.10 / Chapter 2.4 --- Option pricing under Levy Processes --- p.12 / Chapter 2.4.1 --- Black-Scholes Formula with Characteristic Function --- p.12 / Chapter 2.4.2 --- Fast Fourier Transform --- p.14 / Chapter 2.4.3 --- Other Payoff Functions --- p.16 / Chapter 3 --- Dynamic Fund Protection --- p.19 / Chapter 3.1 --- Discrete Dynamic Fund Protection --- p.20 / Chapter 3.2 --- Link DFP to Discrete Lookback Options --- p.22 / Chapter 4 --- Spitzer´ةs Identity --- p.25 / Chapter 4.1 --- Applications of Spitzer's Identity --- p.25 / Chapter 4.2 --- Discrete Lookback Options --- p.29 / Chapter 5 --- Pricing Discrete DFP --- p.32 / Chapter 5.1 --- Girsanov´ةs Theorem --- p.32 / Chapter 5.2 --- Equivalent Martingale Measure in DFP --- p.34 / Chapter 5.3 --- Pricing DFP at any Time Points --- p.36 / Chapter 5.4 --- The Main Algorithm --- p.38 / Chapter 6 --- Numerical Results --- p.40 / Chapter 6.1 --- Simulation of Discrete DFP --- p.40 / Chapter 6.2 --- Numerical Implementation --- p.42 / Chapter 7 --- Conclusion --- p.50 / Bibliography --- p.51

利率模型廣泛應用於利率衍生品的定價。為了吻合實證利率的分佈和隱含波動率，一種可能的辦法是用Lévy過程替換Hull- White模型中的布朗隨機變量的利率模型，但是這種方法很難實施。本文建立了一種有效的網絡數值方法對利率進行估測。利用Lévy過程的馬爾可夫性質， FFT網絡實質上是多項樹模型的擴展。這種數值方法的優勢在於一直固定不變的狀態點，對現時利率期限結構的超級校準以及基於對Lévy過程的特徵方程的快速傅裡葉變換(FFT) 去恢復概率密度函數以實現轉移概率的計算過程。這種網絡數值方法對利率衍生品的定價與利率樹類似。對利率上限期權和交換期權的解析解和數值解的比較表明網絡數值方法是準確和有效的。FFT網絡還可以對百慕達式利率交換期權以及美式期權進行定價。最後， FFT網絡被擴展去適應路徑依賴變量，因此，能對利率依賴的結構性票據進行定價，比如目標贖回票據和範團積息結構票據。 / Short rate models are widely used in valuing interest rate derivatives. To fit empirical distribution of interest rates and implied volatility, a possible way is to replace Brownian motion by a Lévy process in short rate models. However, this approach is difficult to implement. This thesis establishes an efficient network approach for interest rate valuation. The FFT-network is essentially an extension of multinomial tree model, taking advantage of the Markov property of Lévy processes. Its fixed and unchanged states at all time, super-calibration ability to the current term structure, and elegant computation procedure for transition probabilities using the fast Fourier transform (FFT) from the characteristic function of Lévy processes make it attractive and distinct from other numerical methods. The interest rate derivatives value is determined in a way similar to that of the tree approach. The comparison between the closed-form solution of interest rate caplets and swaptions and the numerical results under the network demonstrates that the proposed network is accurate and efficient. In addition, the FFT-network can also be used to pricing the Bermudan swaption and American-style option. Finally, the FFT-network is expanded to accommodate path-dependent variables, and hence can be used for pricing some path-dependent structured notes, such as the target redemption notes and range of accrual notes. / Detailed summary in vernacular field only. / Xu, Zhuolu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 91-93). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Preliminaries --- p.4 / Chapter 2.1 --- Elementary techniques --- p.4 / Chapter 2.1.1 --- Characteristic function --- p.4 / Chapter 2.1.2 --- Cumulant generating function --- p.5 / Chapter 2.1.3 --- Fourier Transform --- p.6 / Chapter 2.1.4 --- Fast Fourier Transform (FFT) --- p.8 / Chapter 2.2 --- Lévy Processes --- p.10 / Chapter 2.2.1 --- Definition --- p.10 / Chapter 2.2.2 --- Lévy-Khintchine --- p.11 / Chapter 2.2.3 --- Lévy Processes in Interest Rate --- p.13 / Chapter 2.3 --- Hull-White Model --- p.13 / Chapter 2.3.1 --- Model setup --- p.14 / Chapter 2.3.2 --- Interest rate caps --- p.15 / Chapter 2.3.3 --- European Swaptions --- p.16 / Chapter 2.3.4 --- A Tree-building procedure --- p.19 / Chapter 3 --- HW-Lévy Model --- p.20 / Chapter 3.1 --- Model Setup --- p.20 / Chapter 3.2 --- The Characteristic Function --- p.22 / Chapter 3.3 --- Analytic result on interest rate derivatives --- p.26 / Chapter 4 --- Valuation: FFT Network Model --- p.35 / Chapter 4.1 --- Drawbacks of Tree Approach --- p.35 / Chapter 4.2 --- FFT Network Setup --- p.37 / Chapter 4.3 --- Transition Probability Matrix --- p.38 / Chapter 4.4 --- Yield Curve Fitting --- p.42 / Chapter 4.5 --- Pricing Algorithm under FFT Network --- p.45 / Chapter 4.5.1 --- European Interest Rate Derivatives Pricing --- p.45 / Chapter 4.5.2 --- Bermudan Interest Rate Derivatives Pricing --- p.49 / Chapter 5 --- Extended FFT Network for Path-dependent Structured Notes --- p.55 / Chapter 5.1 --- Extended FFT-netwok --- p.55 / Chapter 5.2 --- Target Redemption Notes (TARN) --- p.61 / Chapter 6 --- Numerical Study --- p.69 / Chapter 6.1 --- Numerical Scheme --- p.69 / Chapter 6.2 --- Numerical Examples --- p.74 / Chapter 7 --- Conclusion --- p.89 / Bibliography --- p.91

針對Lévy過程下的二維期權定價問題，本文提出了一種基於快速傅利葉變換(FFT)的解決方案，稱之為二維快速傅利葉變換網絡。不論是時間從屬還是線性組合，此方法適用於所有能取得聯合特徵函數的二維Lévy構建。快速傅利葉變換的種種優點使得比數值方法在不影響結果精確性的前提下，大大降低了所需計算時間。理論上，更高維的Lévy期權定價問題也可以通過擴展數值網絡解決。除此之外，我們還探究了資產波動性亦服從Lévy過程的單資產期權定價。這種資產價值和波動性由一組相關Lévy過程驅動的模型被稱為時間轉換Lévy過程。最後，關於美式及奇異期權定價的數值算例驗證了文中方法的準確性和有效性。 / We propose a two-dimensional network to retrieve the price of two-asset option under Lévy processes by using the fast Fourier transform (FFT). It can be applied to different multivariate Lévy constructions such as subordination and linear combination provided that the joint characteristic function is obtainable. With the prevalent implementation of FFT, the network approach results in significant computational time reduction while maintaining satisfactory accuracy. In general, multi-dimensional option pricing problems are also solvable by extending this network. Furthermore, we investigate option pricing on a single asset where the asset return and its volatility are driven by a pair of dependent Lévy processes. Such a model is also called the random time-changed Lévy process. Numerical examples are given to demonstrate the efficiency and accuracy of FFT-network applied to exotic and American-style options. / Detailed summary in vernacular field only. / Wang, Weiyin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 41-43). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese. / List of Tables --- p.ii / List of Figures --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Literature Review --- p.4 / Chapter 2.1 --- Lévy Process --- p.4 / Chapter 2.1.1 --- Definition and Properties --- p.4 / Chapter 2.1.2 --- Multivariate Lévy Construction --- p.6 / Chapter 2.2 --- Fast Fourier Transform (FFT) in Option Pricing --- p.9 / Chapter 2.2.1 --- European Option on One Asset --- p.9 / Chapter 2.2.2 --- European Option on Two Assets --- p.11 / Chapter 3 --- Two-dimensional FFT-network Model --- p.13 / Chapter 3.1 --- Two-dimensional FFT-network --- p.15 / Chapter 3.2 --- Two-asset Option Pricing --- p.22 / Chapter 3.2.1 --- General Model --- p.22 / Chapter 3.2.2 --- Specific Models --- p.23 / Chapter 3.3 --- Random Time-changed Lévy Process --- p.25 / Chapter 3.3.1 --- Model --- p.26 / Chapter 3.3.2 --- Correlation Adjustment --- p.28 / Chapter 4 --- Numerical Examples --- p.31 / Chapter 4.1 --- Two-asset Option --- p.31 / Chapter 4.1.1 --- Spread Option Pricing --- p.31 / Chapter 4.1.2 --- Pricing under Diffierent Multivariate Lévy Constructions --- p.36 / Chapter 4.2 --- One-asset Option under Random Time-changed Lévy Process --- p.37 / Chapter 5 --- Conclusion --- p.40 / Bibliography --- p.41

本研究主要探討附保證最低提 (Guaranteed Minimum Withdrawal Benefits, GMWB)的變額(Variable Annuity, VA) 在隨機模型下之定價。保證最低提是變額的一種附加約 (rider) 並在市場下跌的情況下為變額持有人提供保障。它保證持有人在合約期內的總提少於一個預先訂的額，而變額的投資表現。一般，這個保證額相等於變額的初始投資額。本研究的融模型假設投資標的基價格符合對維過程 (exponential Lévy process)，而隨機則符合由維過程驅動的瓦西克模型 (Vasiček model)。融模型中的個維過程的相依結構 (dependence structure) 會由維關結構 (Lévy Copula) 描述。這個方法的好處是可描述同型的相依結構。用一個配合維關結構而有效的蒙地卡模擬方法，我們研究在同相依結構及模型下保證最低提的價值變化。在固定的特別情況下，保證最低提的價值能夠透過卷積方法 (convolution method) 而得到半解析解 (semi-analytical solution) 。最後，我們將本研究中的學模型擴展以研究近期出現由保證最低提演化而成的一種保證產品。這個產品名稱為保證終身提 (Guaranteed Lifelong Withdrawal Benefit, GLWB)，而此產品的到期日則與持有人的壽命相關。 / In this thesis, we study the problem of pricing the variable annuity(VA) with the Guaranteed Minimum Withdrawal Benefits (GMWB) under the stochastic interest rate framework. The GMWB is a rider that can be elected to supplement a VA. It provides downside protection to policyholders by guaranteeing the total withdrawals throughout the life of the contract to be not less than a pre-specied amount, usually the initial lump sum investment, regardless of the investment performance of the VA. In our nancial model, we employ an exponential L´evy model for the underlying fund process and a Vasiček type model driven by a L´evy process for the interest rate dynamic. The dependence structure between the two driving L´evy processes is modeledby the L´evy copula approach whichis exible to model a wide range of dependence structure. An effcient simulation algorithm on L´evy copula is then used to study the behavior of the value of the GMWB when the dependence structure of the two L´evy processes and model parameters Vry. When the interest rate is deterministic, the value of the GMWB can be solved semi-analytically by the convolution method. Finally, we extend our model to study a recent variation of GMWB called Guaranteed Life long Withdrawal Benefits (GLWB) in which the maturity of the GLWB depends on the life of the policyhodler. / Detailed summary in vernacular field only. / Chan, Wang Ngai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 115-121). / Abstracts also in Chinese. / Abstract --- p.i / Acknowledgement --- p.iv / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Variable Annuity & Guaranteed Minimum Withdrawal Benefit --- p.1 / Chapter 1.2 --- Literature Review --- p.4 / Chapter 1.3 --- Financial Model for GMWB --- p.7 / Chapter 2 --- L´evy Copulas and the Simulation Algorithm --- p.12 / Chapter 2.1 --- Definitions and Theorem --- p.15 / Chapter 2.2 --- Examples of L´evy Copulas --- p.19 / Chapter 2.2.1 --- Independence case --- p.19 / Chapter 2.2.2 --- Complete Dependence --- p.20 / Chapter 2.2.3 --- The Clayton L´evy Copula --- p.21 / Chapter 2.3 --- Simulation algorithm for two-dimensional dependent L´evy process --- p.22 / Chapter 3 --- Model Formulation for GMWB --- p.26 / Chapter 3.1 --- Financial Model for GMWB --- p.27 / Chapter 3.2 --- Underlying Fund of VA and the Interest Rate --- p.30 / Chapter 3.3 --- A Special Case of Deterministic Interest Rate --- p.34 / Chapter 4 --- Numerical Implementation --- p.38 / Chapter 4.1 --- The Clayton L´evy Copula --- p.39 / Chapter 4.2 --- The Underlying Fund and the Interest Rate Processes --- p.42 / Chapter 4.3 --- Kendall’s Tau Coefficient --- p.47 / Chapter 4.4 --- The GMWB Option Value --- p.49 / Chapter 4.4.1 --- Control Variate for Simulation --- p.49 / Chapter 4.4.2 --- Simulation Results --- p.51 / Chapter 4.5 --- Deterministic Interest Rate --- p.52 / Chapter 5 --- GMWB Pricing Behavior --- p.56 / Chapter 5.1 --- L´evy model for the underlying fund --- p.57 / Chapter 5.1.1 --- The Skewness --- p.57 / Chapter 5.1.2 --- The Kurtosis --- p.65 / Chapter 5.2 --- The Vasiček model driven by L´evy process --- p.73 / Chapter 5.2.1 --- The Volatility Parameter ôV --- p.73 / Chapter 5.2.2 --- The Mean Reverting Parameter aV --- p.77 / Chapter 5.3 --- Dependence between the underlying fund and rate processes --- p.81 / Chapter 5.3.1 --- The jump direction dependence parameter n{U+1D9C} --- p.83 / Chapter 5.3.2 --- The jump magnitude dependence parameter θ{U+1D9C} --- p.90 / Chapter 6 --- GMWB for Life --- p.96 / Chapter 6.1 --- Model Formulation --- p.98 / Chapter 6.1.1 --- Mortality model --- p.99 / Chapter 6.1.2 --- Financial Model for GLWB --- p.101 / Chapter 6.2 --- GLWB product from John Hancock --- p.103 / Chapter 6.3 --- GLWB Pricing Behavior --- p.104 / Chapter 6.3.1 --- The correlation effect --- p.106 / Chapter 7 --- Conclusion --- p.108 / A Proofs --- p.113 / Chapter A.1 --- Proof of Equation 3.1 --- p.113 / Chapter A.2 --- Proof of Equation 3.3 --- p.114 / Bibliography --- p.115

An estimation methodology to deal with fractionally integrated processes of Ornstein- Uhlenbeck type is proposed. The methodology is based on the continuous Whittle contrast. A simulation study is performed by driving this process with a symmetric CGMY background Lévy process.

La présente thèse couvre deux principaux thèmes de recherche qui seront présentés dans deux parties et précédés par un prolegomenon commun. Dans ce dernier nous introduisons les concepts essentiels et nous exploitons aussi le lien entre les deux parties.<p><p>Dans la première partie, le principal objet d’intérêt est la soi-disant fonctionnelle exponentielle de processus de Lévy. La loi de cette variable aléatoire joue un rôle primordial dans de nombreux domaines divers tant sur le plan théorique que dans des domaines appliqués. Doney dérive une factorisation de la loi arc-sinus en termes de suprema de processus stables indépendants et de même index. Une factorisation similaire de la loi arc-sinus en termes de derniers temps de passage au niveau 1 de processus de Bessel peut aussi être établie en utilisant un résultat dû à Getoor. Des factorisations semblables d’une variable de Pareto en termes des mêmes objets peut également être obtenue. Le but de cette partie est de donner une preuve unifiée et une généralisation de ces factorisations qui semblent n’avoir aucun lien à première vue. Même s’il semble n’y avoir aucune connexion entre le supremum d’un processus stable et le dernier temps de passage d’un processus de Bessel, il peut être montré que ces variables aleatoires sont liées à des fonctionnelles exponentielles de processus de Lévy spécifiques. Notre contribution principale dans cette partie et aussi au niveau de caractérisations de la loi de la fonctionnelle exponentielle sont des factorisations de la loi arc-sinus et de variables de Pareto généralisées. Notre preuve s’appuie sur une factorisation de Wiener-Hopf récente de Patie et Savov.<p>Dans la deuxième partie, motivée par le fait que la dérivée fractionnaire de Caputo et d’autres opérateurs fractionnaires classiques coïncident avec le générateur de processus de Markov auto-similaires positifs particuliers, nous introduisons des opérateurs généralisés de Caputo et nous étudions certaines propriétés. Nous nous intéressons particulièrement aux conditions sous lesquelles ces opérateurs coïncident avec les générateurs infinitésimaux de processus de Markov auto-similaires positifs généraux. Dans ce cas, nous étudions les fonctions invariantes de ces opérateurs qui admettent une représentation en termes de séries entières. Nous précisons que cette classe de fonctions contient les fonctions de Bessel modifiées, les fonctions de Mittag-Leffler ainsi que plusieurs fonctions hypergéométriques. Nous proposons une étude unifiant et en profondeur de cette classe de fonctions. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

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

Wong, Tat Wing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (p. 53-55). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Problem Formulation --- p.5 / Chapter 2.1 --- Phase-type distribution --- p.5 / Chapter 2.1.1 --- A generalization of the exponential distribution --- p.5 / Chapter 2.1.2 --- Properties of the phase-type distribution --- p.6 / Chapter 2.2 --- Phase-type jump diffusion model --- p.8 / Chapter 2.2.1 --- Jump diffusion model --- p.8 / Chapter 2.2.2 --- The stock price model --- p.9 / Chapter 2.3 --- Stock Loans --- p.10 / Chapter 3 --- General Properties of Stock Loans --- p.12 / Chapter 3.1 --- Preliminary results --- p.12 / Chapter 3.2 --- Characterization of the function V(x) --- p.15 / Chapter 4 --- Valuation / Chapter 4.1 --- Hyperexponential jumps --- p.25 / Chapter 4.1.1 --- Solution of the linear system --- p.29 / Chapter 4.1.2 --- Solution of the optimal exercise boundary --- p.30 / Chapter 4.2 --- Phase-type jumps --- p.33 / Chapter 4.3 --- The case for G'(1)≥ 0 --- p.36 / Chapter 5 --- Future Research Direction --- p.38 / Chapter 5.1 --- The fast mean-reverting stochastic volatility model --- p.38 / Chapter 5.2 --- Asymptotic expansion of stock loan --- p.39 / Chapter 5.2.1 --- The zeroth order term --- p.41 / Chapter 5.2.2 --- The first order term --- p.43 / Chapter 6 --- Conclusion --- p.52 / Bibliography --- p.53