本篇論文主要為財務衍生性商品的定價與避險提出一個動態半參數的方法。此方法可應用於計算歐式、美式選擇權以及可轉換債券等衍生性商品的定價與避險。動態半參數法是一個在離散時間點上，由到期日向起始日逐步遞迴的方法，在每一個時間點上，首先應用迴歸的方法近似當期的衍生性商品價格函數，接著再計算此近似函數的一步條件期望值以求得前一期的價格函數。在一維度方面，本論文討論了幾何布朗運動(Geometric Brownian motion)跳躍擴散(jump-diffusion)以及非線性不對稱GARCH等標的資產模型的衍生性商品的定價。在多維度方面，本論文應用copula函數(如：Gaussian、Clayton與Gumbel等copula)聯結多資產間的相關性，對衍生性商品定價。當標的資產對數報酬率的轉移密度函數為已知的連續函數，本論文推導出此動態半參數法在一維度與多維度標的資產下的估計誤差階次及收斂速度，模擬研究亦支持理論上的結果。因此無論是一維度或是多維度的衍生性商品定價問題，動態半參數法都能提供精確的衍生性商品價格。
對於避險問題，由於實證研究顯示財務資料的報酬率具有條件厚尾分佈的情形，因此，本論文討論當報酬率服從條件厚尾分佈時的避險策略。首先我們應用 extended Girsanov測度變換建立對數報酬率與簡單報酬率的風險中立模型，其結果可以應用於簡單報酬率為GARCH-t模型(干擾項為t-分佈)的衍生性商品定價。接著並推廣動態半參數法以求得干擾項具有厚尾分佈的GARCH模型的衍生性商品價格。再進一步討論與extended Girsanov測度變換一致的避險策略，並證實在風險中立測度下，此避險策略比一般常用的 delta 避險有較小的平均避險成本變異。最後本論文探討當真實標的資產為GARCH-t模型，但卻以GARCH-normal模型配適時，一般選擇權(plain vanilla)與新奇選擇權(exotic option)的delta避險。模擬結果顯示一般選擇權在兩個模型下的delta避險值沒有顯著差異，但對於某些新奇選擇權(如：barrier option與lookback option)，GARCH模型中干擾項的不當配適則可能對delta避險帶來顯著的偏差。

A dynamic semiparametric pricing method is proposed for financial derivatives including European and American type options and convertible bonds. The proposed method is an iterative procedure which uses nonparametric regression to approximate derivative values and parametric asset models to derive the continuation values. Extension to higher dimensional option pricing is also developed, in which the dependence structure of financial time series is modeled by copula functions. In the simulation study, we valuate one dimensional American options, convertible bonds and multi-dimensional American geometric average options and max options. The considered one-dimensional underlying asset models include the Black-Scholes, jump-diffusion, and nonlinear asymmetric GARCH models and for multivariate case we study copula models such as the Gaussian, Clayton and Gumbel copulae. Convergence of the method is proved under continuity assumption on the transition densities of the underlying asset models. And the orders of the supnorm errors are derived. Both the theoretical findings and the simulation results show the proposed approach to be tractable for numerical implementation and provides a unified and accurate technique for financial derivative pricing.
The second part of this thesis studies the option pricing and hedging problems for conditional leptokurtic returns which is an important feature in financial data. The risk-neutral models for log and simple return models with heavy-tailed innovations are derived by an extended Girsanov change of measure, respectively. The result is applicable to the option pricing of the GARCH model with t innovations (GARCH-t) for simple eturn series. The dynamic semiparametric approach is extended to compute the option prices of conditional leptokurtic returns. The hedging strategy consistent with the extended Girsanov change of measure is constructed and is shown to have smaller cost variation than the commonly used delta hedging under the risk neutral measure. Simulation studies are also performed to show the effect of using GARCH-normal models to compute the option prices and delta hedging of GARCH-t model for plain vanilla and exotic options. The results indicate that there are little pricing and hedging differences between the normal and t innovations for plain vanilla and Asian options, yet significant disparities arise for barrier and lookback options due to improper distribution setting of the GARCH innovations.

1 Introduction 1
2 Valuation of One-dimensional Derivatives 9
2.1 Introduction 9
2.2 Methodology: A Dynamic Semiparametric Approach 10
2.2.1 Algorithm 2.1: Jump-diffusion Model 10
2.2.2 Algorithm 2.2: NGARCH(1,1) Model 14
2.3 Continuation Values and The Orders of Approximation Errors 19
2.3.1 Jump-diffusion Model 19
2.3.2 NGARCH(1,1) Model 20
2.3.3 Convertible Bond Pricing 22
2.4 Simulation Studies 23
2.5 Discussion 26
3 Valuation of Multidimensional Derivatives 28
3.1 Introduction 28
3.2 Copula: Modeling the Dependency Between Multidimensional Assets 29
3.3 Algorithm 3.1: Constant Transition Matrix 32
3.4 Algorithm 3.2: Constant Partitioned Number 37
3.5 Examples 42
3.6 Discussion 51
4 Option Pricing and Hedging for Conditional Leptokurtic Returns 52
4.1 Motivation 53
4.2 Risk-neutral Model and Option Valuation 54
4.2.1 Risk-neutral Model 54
4.2.2 Option Valuation 58
4.3 Hedging 59
4.3.1 Δ-hedging 60
4.3.2 η-hedging 62
4.3.3 η-hedging and Δ-hedging Under Complete and Incomplete Markets 66
4.3.4 Δ-hedging of Misspecified Conditional Less-leptokurtic Models 67
4.4 Simulation 69
4.5 Discussion 76
5 Conclusions and Future Works 78
Appendix 80
References 92

Bakshi, G., Cao, C. and Chen, Z. (1997). Empirical performance of alternaive option pricing models. J. Finance, 52, 2003-2049.
Barone-Adesi, G. and Whaley, R. (1987). Efficient analytic approximation of American option values. J. Finance, 42, 301-320.
Barraquand, J. and Martineau, D. (1995). Numerical valuation of high dimensional multivariate American securities. J. Fin. Quant. Anal., 30, 383-405.
Bates, D. S. (1991). The Crash of '87: What is expected? The evidence from options markets. J. Finance, 46, 1009-1044.
Bates, D. S. (1996). Jumps and stochastic volatility: exchange rate processes implicit in Deutsche Mark options. Rev. Fin. Stud., 9, 69-107.
Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. J. Political Econ., 81, 637-654.
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Econometrics, 31, 307-327.
Bollerslev, T. (1987). A conditionally heteroskedastic time series model for speculative prices and rates of return. Rev. Econ. Stat., 69, 542-547.
Brennan, M. J. and Schwartz, E. S. (1977). Convertible bonds: valuation and optimal strategies for call and conversion. J. Finance, 32, 1699-1715.
Broadie, M. and Glasserman, P. (2004). A stochastic mesh method for pricing high-dimensional American options. J. Comput. Finance, 7, 35-72.
Broadie, M. and Yamamoto, Y. (2003). Application of the fast Gauss transform to option pricing. Management Science, 49, 1071-1088.
Cherubini, U., Luciano, E. and Vecchiato, W. (2004). Copula Methods in Finance, Wiley, West Sussex.
Chesney, M. and Jeanblanc, M. (2004). Pricing American currency options in an exponential Levy model. Appl. Mathematical Finance, 11, 207-225.
Chiarella, C. and Ziogas, A. (2005). Pricing American options on jump-diffusion processes using Fourier Hermite series expansions. Quantitative Finance Research Centre Research Paper, No. 145. Available at SSRN: http://ssrn.com/abstract=893090
Christoffersen, P. F., Heston, S. and Jacobs, K. (2006). Option valuation with conditional skewness. J. Econometrics, 131, 253-284.
Corhay, A. and Rad, A. T. (1994). Statistical properties of daily returns: evidence from European stock markets. J. Business Finance and Accounting, 21, 271-282.
Deng, S. J. and Lee, S. (2004). American-style options pricing by adaptive simulation. Manuscript, Georgia Tech.
Duan, J. C. (1995). The GARCH option pricing model. Math. Finance, 5, 13-32.
Duan, J. C. (1999). Conditionally fat-tailed distributions and the volatility smile in options. Manuscript, University of Toronto.
Duan, J. C., Gauthier, G. and Simonato, J. G. (1999). An analytical approximation for the GARCH option pricing model. J. Comput. Finance, 2, 75-116.
Duan, J. C., Ritchken, P. H. and Sun, Z. (2006). Jump starting GARCH: pricing and hedging options with jumps in returns and volatilities. FRB of Cleveland Working Paper No. 06-19. Available at SSRN: http://ssrn.com/abatract=479483.
Duan, J. C. and Simonato, J. G. (1998). Empirical martingale simulation for asset prices. Management Science, 44, 1218-1233.
Duan, J. C. and Simonato, J. G. (2001). American option pricing under GARCH by a Markov chain approximation. J. Econ. Dynamics Control, 25, 1689-1718.
Dumas, B., Fleming, J. and Whaley, R. (1998). Implied volatility functions: empirical tests. J. Finance, 53, 2059-2106.
Elliott, R. J. and Madan, D. B. (1998). A discrete time equivalent martingale measure. Math. Finance, 8, 127-152.
Engle, R. F. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50, 987-1007.
Engle, R. F. and Ng, V. (1982). Measuring and testing of the impact of news on volatility. J. Finance, 48, 1749-1778.
Engle R. F. and Rosenberg, J. V. (1994). Hedging options in a GARCH environment: testing the term structure of stochastic volatility models. NBER Working Paper No. W4958. Available at SSRN: http://ssrn.com/abstract=226557
Engle, R. F. and Rosenberg, J. V. (1995). GARCH gamma. J. Derivatives, Summer 1995, 47-59.
Fan, J. and Yao, Q. (2003). Nonlinear Time Series: Nonparametric and Parametric Models. Springer, New York.
Franke, J., Hardle, W. and Hafner,C. M. (2004). Statistics of Financial Markets. Springer, Berlin.
Fu, M., Laprise, S., Madan, D., Su, Y. and Wu, R. (2001). Pricing American options: A comparison of Monte Carlo approaches. J. Comput. Finance, 4, 39-88.
Giacomini, E., Hardle, W., Ignatieva, E. and Spokoiny, V. (2007). Inhomogeneous dependency modelling with time varying copulae. SFB 649 Economic Risk Berlin.
Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. Springer-Verlag, New York.
Hagerud G. E. (1996). Discrete time hedging of OTC options in a GARCH environment: a simulation experiment. Working Paper Series in Economics and Finance No. 165.
Hamilton, J. D. (1994). Time Series Analysis. Princeton, New Jersey.
Hardle, W., Muller, M., Sperlich, S. and Werwatz, A. (2004). Nonparametric and Semiparametric Models. Springer, Berlin.
Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Fin. Stud., 6, 327-343.
Heston, S. L. and Nandi, S. (2000). A closed-form GARCH option valuation model. Rev. Fin. Stud., 13, 585-625.
Ho, T. S. Y. and Pfeffer, D. M. (1996). Convertible bonds: model, value, attribution and analytics. Fin. Anal. J., 52, 35-44.
Huang, S. F. and Guo, M. H. (2008a). Valuation of multidimensional Bermudan options. In "Applied Quantitative Finance 2nd Edition" (Edited by W. Hardle), Springer, Berlin.
Huang, S. F. and Guo, M. H. (2008b). Financial derivative valuation - a dynamic semiparametric approach. Statistica Sinica, to appear.
Hull, J. C. and Suo, W. (2002). A methodology for assessing model risk and its application to the implied volatility function model. J. Fin. Quant. Anal., 37, 297-318.
Ju, N. (1998). Pricing an American option by approximating its early exercise boundary as a multipiece exponential function. Rev. Fin. Stud., 11, 627-646.
Ju, N. and Zhong, R. (1999). An approximate formula for pricing American options. J. Derivatives, 7, 31-40.
Judd, K. (1998). Numerical Methods in Economics. MIT Press.
Khuri, A. I. (2003). Advanced Calculus with Applications in Statistics. Wiley, New Jersey.
Kim, I. J. (1990). The analytical valuation of American options. Rev. Fin. Stud., 3, 547-572.
Kou, S. G. (2002). A jump-diffusion model for option pricing. Management Science, 48, 1086-1101.
Lai, T. L. and AitSahalia, F. (2001). Exercise boundaries and efficient approximations to American option prices and hedge parameters. J. Comput. Finance, 4, 85-103.
Longstaff, F. A. and Schwartz, E. S. (2001). Valuing American options by simulation: a simple least-squares approach. Rev. Fin. Stud., 14, 113-147.
McNeil, A. J., Frey, R. and Embrechts, P. (2005). Quantitative Risk Management. Princeton University Press, New Jersey.
Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. J. Financial Economics, 3, 125-144.
Mills, T. C. (1999). The Econometric Modeling of Financial Time Series. Cambridge University Press.
Nelsen, R. B. (2006). An Introduction to Copulas, Springer, Berlin.
Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: a new approach. Econometrica, 59, 347-370.
Ritchken, P. and Trevor, R. (1999). Pricing option under generalized GARCH and stochastic volatility process. J. Finance, 54, 377-402.
Rust, J. (1997). Using randomization to break the curse of dimensionality. Econometrica, 65, 487-516.
Schulmerich, M. and Trautmann, S. (2003). Local expected shortfall-hedging in discrete time. European Finance Review, 7, 75-102.
Shephard, N. (1996). Statistical aspects of ARCH and stochastic volatility, In "Time Series Models: In Econometrics, Finance and Other Fields." (Edited by D.R. Cox, O.E. Barndorff-Nielsen and D.V. Hinkley), Chapman and Hall, New York.
Shreve, S. E. (2004). Stochastic Calculus for Finance I. Springer, New York.
Siu, T. K., Tong, H. and Yang, H. (2004) On pricing derivatives under GARCH models: a dynamic Gerber-Shiu approach. North American Actuarial J., 8, 17-31.
Sklar, A. (1959). Fonctions de repartition an dimensions et leurs marges. Pub. Inst. Statist. Univ. Paris, 8, 229-231.
Tong, H. (1990). Nonlinear Time Series Analysis: A Dynamical System Approach. Oxford University Press.
Tsay, R. S. (2005). Analysis of Financial Time Series, 2nd edn, Wiley, New York.
Tsitsiklis, J. N. and Van Roy, B. (1999). Optimal stopping of Markov processes: Hilbert space theory, approximation algorithms, and an application to pricing high-dimensional financial derivatives. Automatic Control, IEEE Transactions on, 44, 1840-1851.
Wagner, N. and Marsh, T. A. (2005). Measuring tail thickness under GARCH and an application to extreme exchange rate changes. J. Empirical Finance, 12, 165-185.
Yigitbasioglu, A. B., Lvov, D. and EI-Bachir, N. (2004). Pricing convertible bonds by simulation. ISMA Centre Discussion Papers in Finance 2004-15.