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

Made available in DSpace on 2016-08-03T16:16:15Z (GMT). No. of bitstreams: 1 Giselda Geronymo Sanches Bretherick.pdf: 1094641 bytes, checksum: b21080b813650f47bb36a816721e177a (MD5) Previous issue date: 2010-03-26 / This study presents a research on the relationship between education and technology in the works of Pierre Lévy. The research focuses on the concepts of decentralization and deterritorialization of knowledge. The theme that the author develops about cognitive ecology, intelligence technology and collective intelligence, as well as the knowledge that permeate the education of the future, is the goal of this work. The object of research is the very production of the author. These concepts are problematized considering two cases: on the one hand, the institutionalized academic knowledge, and on the other hand, the proposal for a collective intelligence, which gives way to a broader dimension of human knowledge. Levy recognizes that every human being knows something, and that, because of this, understands that knowledge, rather than being restricted to specific areas, is present in humanity as a collective intelligence. He believes that widespread access to cyberspace through the use of intelligence and cyberculture result in an "anthropological space" where collective intelligence produces a "democratic space of knowledge" possible to all humans and, finally, a "production area of production of different kinds of knowledge. This research also considers the opposition to Lévy‟s proposal, and taking into account possible objections, wishes to critically investigate to a deeper point the idea of an education of the future.(AU) / Este estudo apresenta uma pesquisa nas obras de Pierre Lévy, sobre as relações entre educação e tecnologia. A investigação está centrada sobre os conceitos de desterritorialização do conhecimento e descentralização do saber. A temática que o referido autor desenvolve, a respeito dos conceitos de ecologia cognitiva, tecnologias da inteligência e inteligência coletiva, assim como o saber e o conhecimento que permeiam a educação do futuro, é o objetivo central deste trabalho. O objeto de pesquisa é a própria produção do autor. Esses conceitos são problematizados levando em conta duas instâncias: de um lado, o conhecimento acadêmico institucionalizado e de outro lado, a proposta de uma inteligência coletiva, que dá espaço para uma dimensão mais ampla do saber humano. Lévy reconhece que cada ser humano sabe alguma coisa, e que, em função disto, entende que o conhecimento no lugar de estar reservado a espaços específicos, está presente na humanidade na forma de uma inteligência coletiva. Acredita que a popularização do acesso ao ciberespaço através das tecnologias da inteligência e da cibercultura resulta em um espaço antropológico onde as inteligências coletivas produzem um espaço de saber democrático , possível a todos os seres humanos e, um espaço de produção de diferentes saberes. Esta pesquisa considera também as resistências à proposta de Pierre Lévy, e, aprofunda, de maneira crítica, a idéia enunciada acima, de uma educação do futuro.(AU)

Esta dissertação tem por objetivo compreender e discutir as abordagens e teorias que sustentam o uso das tecnologias da informação e comunicação na educação, em suas diferentes instâncias. Para tanto, o trabalho teve como objeto de análise as noções de conhecimento de Pierre Lévy, bem como as implicações destas no campo da educação. A escolha pelo referido autor se baseou na constatação da frequência significativa de citações de suas obras e suas ideias, sobre os processos de conhecimento no universo virtual, em trabalhos acadêmicos, na produção especializada e nos meios de comunicação de massa. A constância de suas reflexões e de seus conceitos, como por exemplo, inteligência coletiva, ecologia cognitiva, economia do saber, entre outros, demonstram sua influência na formação de discursos e visões sobre o uso dos computadores para fins educacionais. As noções do autor são expostas e debatidas à luz de teorias e pensadores que analisam o fenômeno das tecnologias da informação e comunicação e seu uso na educação, sob perspectivas distintas das de Pierre Lévy. Este trabalho tem a intenção de apresentar abordagens críticas e reflexões sobre o uso das tecnologias da informação e comunicação na educação, segundo um ponto de vista prudente e criterioso, contrapondo-se, dessa forma às posições entusiastas e apologéticas. / This research aims to understand and discuss the approaches and theories that support the use of information and communication technologies in education in its different instances. Thus, the work had as object of analysis the Pierre Lévys notions of knowledge as well as their implications in the education field. The choice for that author considered the finding of significant frequency of citations of his works and ideas about knowledge processes in the virtual universe present in academic papers, in specialized production and in mass media. The constancy of his thoughts and concepts, such as collective intelligence, cognitive ecology, economy of knowledge, among others, demonstrate his influence on the formation of discourses and views about the use of computers for educational purposes. The authors ideas are presented and debated in the light of theories and thinkers who analyze the phenomenon of information and communication technologies and their use in education, from different perspectives of those of Pierre Lévy. This research also intended to present critical approaches and reflections on the use of information and communication technologies in education, according to a prudent and judicious point of view and, that way, in stark contrast with the enthusiasts and apologetic positions.

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

Nous considérons deux modèles d’arbres aléatoires continus, à savoir les arbres de Lévy et les arbres inhomogènes. Les arbres de Lévy, introduits par Le Gall et Le Jan (1998) comme extension de l’arbre brownien d’Aldous (1991), décrivent les structures généalogiques des processus de branchement. Nous donnons une description de la loi d’un arbre de Lévy conditionné par son diamètre, ainsi qu’une décomposition de l’arbre le long de ce diamètre, qui est décrite à l’aide d’une mesure ponctuelle de Poisson. Dans le cas particulier d’un mécanisme de branchement stable, nous caractérisons la loi jointe du diamètre et de la hauteur d’un arbre de Lévy conditionné par sa masse totale. Dans le cas brownien nous obtenons une formule explicite de cette loi jointe, ce qui permet de retrouver par un calcul direct sur l’excursion brownienne, un résultat de Szekeres (1983) et Aldous (1991) concernant la loi du diamètre. Dans les cas stables, nous obtenons également des développements asymptotiques pour les lois de la hauteur et du diamètre. Les arbres inhomogènes sont introduits par Aldous et Pitman (2000), Camarri et Pitman (2000). Ce sont des généralisations de l’arbre brownien d’Aldous. Pour un arbre inhomogène, nous étudions une fragmentation de cet arbre qui généralise celle introduite par Aldous et Pitman pour l’arbre brownien. Nous construisons un arbre généalogique de cette fragmentation. En utilisant des arguments de convergence, nous montrons qu’il y a une dualité́ en loi entre l’arbre initial et l’arbre généalogique de fragmentation. Pour l’arbre brownien, nous trouvons aussi une façon de reconstruire l’arbre initial à partir de l’arbre généalogique. / We consider two models of random continuous trees: Lévy trees and inhomogeneous continuum random trees. Lévy trees are scaling limits of Galton-Watson trees. They describe the genealogical structures of continuous-state branching processes. The class of Lévy trees is introduced by Le Gall and Le Jan (1998) as an extension of Aldous’ notion of Brownian Continuum Random Tree (1991). For a Lévy tree, we give a description of its law conditioned to have a fixed diameter that is expressed in terms of a Poisson point measure. In the special case of a stable branching mechanism, we characterize the joint law of the diameter and the height of a Lévy tree conditioned on its total mass. From this, we deduce explicit distributions for the diameter in the Brownian case, as well as tail estimates in the general case.Inhomogeneous continuum random trees are introduced by Aldous and Pitman (2000), Camarri and Pitman (2000). They are also generalizations of Aldous’ Brownian Continuum Random Tree (and of Lévy trees). For an inhomogeneous continuum random tree, we consider a fragmentation which generalizes the one introduced by Aldous and Pitman on the Brownian tree. We construct a genealogical tree for this fragmentation. With weak limit arguments, we show that there is a duality in distribution between the initial tree and the genealogical tree. For the Brownian tree, we also present a way to reconstruct the initial tree from the genealogical tree.

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

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

El trabajo consiste en el estudio de un modelo de valuación de opciones europeas de compra, el cual asume que la dinámica del precio del activo financiero subyacente está caracterizada por un proceso de Lévy simétrico. El modelo busca capturar la evidencia empírica mostrada por los precios de los activos financieros. Este modelo es trabajado en [12], artículo que será seguido de cerca. La particularidad del modelo consiste en incorporar procesos estocásticos de salto con distribuciones marginales simétricas, lo cual reproduce de manera más fiel la realidad. En este trabajo, primero se revisa en detalle los principales resultados obtenidos en [12], más precisamente, se revisa la definición de medida martingala equivalente natural en el contexto del modelo. Se estudia la existencia y unicidad de la medida martingala equivalente natural (MMEN). Luego, se usa esta medida para obtener el precio de la opción y calcular los parámetros de la distribución simétrica bajo esta medida MMEN y así obtener una fórmula generalizada tipo Black-Scholes. Además, se realizan aplicaciones con procesos de Lévy específicos tales como Varianza Gamma Simétrico, Normal Inverso Gaussiano Simétrico. Segundo, para extender las aplicaciones proporcionadas en [12], se propone una aplicación adicional. Así, se elige el proceso de Meixner Simétrico (MS) para describir la dinámica del activo subyacente y obtener el precio de la opción de compra europea en el contexto del modelo MS. Finalmente, se realiza simulaciones numéricas del precio de las opciones europeas bajo los tres modelos estudiados, para luego comparar dichos precios con el precio obtenido en el modelo clásico de Black-Scholes. / Tesis

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