本文旨在由GARCH模型出發，類比Christoffersen et al (2013)文中所使用的U型定價核來推廣波動率期限結構模型，將波動率期限分成長期與短期的兩因子結構，從而給出選擇權價格的封閉解。使用U型定價核推廣之模型可以用便利的金融計算方法解釋市場上許多未解決的問題，並且能說明縮減因子在測度變換時的重要作用。模型的參數估計方面，本文將隨機變數視為遺失資料，因此對模型的概似函數使用EM演算法估計參數。對比模型在樣本內外的預測結果可知，推廣的波動率期限結構模型能更精準地刻畫股價報酬的波動變化，從而在使用樣本內外的資料來檢驗預測能力時表現得更好。

Based on GARCH models, the purpose of this paper is to extend the two-factor volatility components model by taking a U-shape pricing kernel which was developed by Christoffersen et al. (2013). By imposing the U-shaped pricing kernel, we surprisingly obtain a more elaborate model which can also explain some puzzles in the market. Furthermore, this article derives a closed-form solution for option valuation which is convenient and propitious for computational financing.
As for the methodology used in estimation, we chose the EM algorithm to modified the likelihood function because of the incomplete data.
The empirical result demonstrates the well ability of the generalized model when reconcile time series properties of stock returns with the option prices. Moreover, we also use the in-sample and out-of-sample to test the predictability of the generalized model. Under our specification, the U-shaped pricing kernel makes the new model somehow more predictable than before.

[01] Bakshi G., C. Cao, and Z. Chen (1997). Empirical Performance of Alternative Option Pricing Models. Journal of Finance 52, 2003-2049.
[02] Bates D. (1996). Testing Option Pricing Models. In Handbook of Statistics, Statistical Methods in Finance, G.S. Maddala and C.R. Rao (eds.), 567-611. Amsterdam: Elsevier.
[03] Black F., and M. Scholes (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81, 637-659.
[04] Brennan M. (1979). The Pricing of Contingent Claims in Discrete-Time Models. Journal of Finance 34, 53-68.
[05] Brown D., and J. Jackwerth (2001). The Pricing Kernel Puzzle: Reconciling Index Option Data and Economic Theory. Working Paper, University of Wisconsin.
[06] Christoffersen P., S. Heston and K. Jacobs (2006). Option Valuation with Conditional Skewness. Journal of Econometrics 131, 253-284.
[07] Christoffersen P., K. Jacobs, C. Ornthanalai, and Y. Wang (2008). Option Valuation with Long-Run and Short-Run Volatility Components. Journal of Financial Economics 90, 272-297.
[08] Christoffersen P., Heston S., and K. Jacobs (2013). Capturing Option Anomalies with a Variance-Dependent Pricing Kernel. Review of Financial Studies 26, 1963-2006.
[09] Engle R., and G. Lee (1999). A Permanent and Transitory Component Model of Stock Return Volatility. In: Engle, R., White, H. (Eds.), Cointegration, Causality, and Forecasting: A Festschrift in Honor of Clive W. J. Granger. Oxford University Press, New York, pp. 475-497.
[10] Heston S. and S. Nandi (2000). A Closed-Form GARCH Option Pricing Model. Review of Financial Studies 13, 585-626.
[11] Jackwerth J. (2000). Recovering Risk Aversion from Option Prices and Realized Returns. Review of Financial Studies 13, 433-451.
[12] Rubinstein M. (1976). The Valuation of Uncertain Income Streams and the Pricing of Options. Bell Journal of Economics 7, 407-425.
[13] A. P. Dempster, N. M. Laird and D. B. Rubin (1977). Maximum Likelihood from Incomplete Data via the EM Algorithm. Journal of the Royal Statistical Society. Series B(Methodological), Vol.39, No.1, 1-38.