資產報酬財務現象與time-changed Lévy過程下美式選擇權定價

Stylized Empirical Features of Asset Return and American Option pricing under time-changed Lévy processes

過去實證研究發現，資產的動態過程存在不連續的跳躍與大波動伴隨大波動的波動度叢聚現象而造成資產報酬分配呈現出厚偉與高狹峰的情況，然而，此現象並不能完全被傳統所使用幾何布朗運動模型與跳躍擴散模型給解釋。因此，本文設定資產模型服從Lévy過程中Generalized Hyperbolic（GH）的normal inverse Gaussian（NIG）和variance gamma（VG）兩個模型，然而，Lévy過程是一個跳躍過程，是屬於一個不完備的市場，這將使得平賭測度並非唯一，因此，本文將採用Gerber和Shiu（1994）所提的Esscher轉換來求得平賭測度。關於美式選擇權將採用LongStaff and Schwartz（2001）所提的最小平方蒙地卡羅模擬法來評價美式選擇權。實證結果發現對於所比較的模型NIG、VG、JDM和GBM的評價績效並無顯著的差異，然而卻發現市場價格與理論價格有明顯的差距，因此，本研究從交易量與交易比數的觀點發現，樣本中的選擇權交易量與交易筆數都是偏低的，因此，缺乏流動性，根據Chen et al., 2013發現流動性與價值（Moneyness）對於評價誤差有重大的影響，因此，本文推論在此研究中評價誤差較大的原因，可能因於選擇權流動性低與過於價外，此外，模型間沒有顯著的差異，可能是模型配適度相似，且流動性低導致無法產生價格發現的效果。

This paper evaluates the American put options under the assumptions the underlying stock return is non-normally distributed. The main idea comes from the fact of that the distributions of return for financial securities always have heavy tail and leptokurtic phenomena due to price jumps or changing return volatilities over time. In addition, the phenomena described above cannot be fully explained by the traditional GBM model or the Merton jump diffusion model. Hence, we adopt normal inverse Gaussian (NIG) and variance gamma (VG) two time-changed Lévy processes to model the asset dynamics which are proposed respectively by Barndorff-Nielsen(1995,1998) and Madan and Senata (1990). Regarding to the pricing methodology, we use the Esscher transform proposed by Geber et al., 1994 to find a martingale measure. Furthermore, we adopt the Least-squared Monte Carlo Simulation (LSM) proposed by LongStaff and Schwartz (2001) to deal with the early-exercised properties of the American options. The empirical results show that there is no big difference in pricing performance among GBM, JDM, NIG and VG models; however, there is significant difference between price and theoretical prices. According to Chen et al., 2013, they find the liquidity and moneyness have influence on pricing error, hence, the price error is huge in this study, we may infer the difference from the issue of liquidity and moneyness, Meanwhile, no distinct difference among models may result from the price discovery deficiency under illiquidity and the fitness of model.