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