Lévy與GARCH-Lévy過程之選擇權評價與實證分析：台灣加權股價指數選擇權為例

Abstract

根據過去實證指出，股價對數報酬率分配呈現高峰、偏態、厚尾及波動叢聚，而傳統Black-Scholes模型的缺點是無法捕捉這些現象。Lévy過程之優點爲能解決厚尾、高峰及偏態等問題，而GARCH-type優點爲能捕捉波動叢聚現象，本文結合兩者的優點提出GARCH-Lévy過程以捕捉負偏態、高峰、厚尾及波動叢聚等報酬分配特徵，並且以蒙地卡羅法估算歐式買權的報價；更進一步綜合文獻常採用選擇權評價模型，以台灣發行量加權股價指數與指數選擇權作爲研究對象，分別對GARCH-Lévy過程、布朗運動、Merton跳躍擴散過程、GARCH-Normal過程和Lévy過程等作實證分析比較，結果顯示GARCH-Lévy過程在樣本內對台股指數有較佳的配適，但是在樣本外，variance gamma選擇權評價模型對價平時的台指選擇權有最小評價誤差，價內外則是NGARCH-Normal選擇權評價模型的評價誤差最小。

The distribution of stock log-returns shows empirically some stylized facts, such as excess kurtosis, skewness, heavy tails and volatility clustering. The assumptions of traditional Black-Scholes model fail to capture the above phenomena well. Lévy processes can deal with the former three phenomena and GARCH type models can handle the final phenomena. In this research, we propose GARCH-Lévy processes combining both Lévy processes and GARCH processes, and then price European call option in risk-neutral world via Monte Carlo simulations. The empirical results show that the GARCH-Lévy processes fit well in samples. For out-of-sample performance, however, variance gamma option pricing model is the best at the money, but NGARCH-Normal option pricing model is best in the money or out of the money.