| Summary: | 博士 === 國立中山大學 === 應用數學系研究所 === 96 === 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.
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