本文類比Heston and Nandi (2000) 所使用的Garch模型，結合了不對稱效果以及Christoffersen et al (2013) 中所使用的U型定價核與兩因子結構，進而得到選擇權價格的封閉解。使用兩因子模型可將 Black and Scholes (1973) 中所假設的波動率為定值一般化為使波動率可以具有長短期的波動，使用U型定價核能夠解釋市場上隱含波動率的訂價疑問。在估計方面，本文使用EM演算法結合貝式估計法與Particle filtering對模型中的參數進行估計。實證部分，採用新模型與原模型進行配適度比較，從而驗證新模型對於波動率與股價的配適度較原模型高。

This paper based on a Stochastic volatility model in Heston and Nandi (2000) incorporating U-shaped pricing kernel and two-factor volatility model used by Christoffersen et al (2013). We can obtain a closed form solution for option valuation.
Using two-factor components model can be more generalized since Black and Scholes model assume that volatility is a constant. Using U-shaped pricing kernel can explain implied volatility puzzle in option market. Moreover, under our specification, the new model is more predictable than before.

Contents
Abstract ii
Acknowledgement iii
Chapter1 Introduction…………...……………………..………………1
Chapter 2 Model…………………………..…………………………….3
2.1Two-FactorComponentsModel………………..………………………..3
2.2 Option Valuation………………………..……………………………...6
Chapter 3 Estimation and Fitting………………………………………8
3.1 Parameter estimation Constrains……………………….………………8
3.2 EM Algorithm……………………………………………….…………9
3.3 Particle Filtering………………………………..….…………………10
Chapter 4 Empirical Analysis………………………...……………….11
4.1 Data………………………...…………………………………………11
4.2 Estimation result………………………………...……………………13
Chapter 5 Conclusion……………………….…………………………16
Appendix…………………….……………………………..…………..17
References………………….…………………………….……….…….26

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