指數選擇權BS評價模型與GARCH評價模型之比較

The Comparison of BS Option Pricing Model and GARCH Option Pricing Model in Index Options

羅善明(SHAN-MING LUO)

指導教授 ： 李孟峰

國立臺北大學/商學院/統計學系/碩士(2010年)

摘要

自有選擇權交易以來，選擇權交易在金融交易市場中始終扮演著重要的角色，然而選擇權的合理價格如何訂定，仍是重要的研究課題。Black & Scholes(1973)提出了有名的BS選擇權評價模型，自此選擇權訂價進入了一個重要的里程碑。然實務發現，BS模型之固定常數波動度之假設與交易實務之現況並不相符，於是後續有學者嘗試放寬其波動度假設。在離散時間的波動度模型，Duan(1995)導入經濟學之計量模型─廣義自我迴歸條件異質變異數模型(Generalized Autoregressive Conditional Heteroskedasticity, GARCH)用以修正波動度為固定常數之假設，惟該模型並不存在封閉解(closed-form)，後續Heston and Nandi(2000)則提出具有特殊封閉解之GARCH選擇權評價模型，利用積分技巧求算出選擇權之理論價格。本文即嘗試以Heston and Nandi(2000)所提出之GARCH選擇權評價模型，以台灣加權股價指數為資料，對台指選擇權作評價。
實證結果顯示，雖然Heston and Nandi(2000)模型放寬波動度為固定常數之假設，在樣本內(In-sample)與樣本外(Out-of-sample)模型比較，BS模型在評價效度上較HN模型略佳，亦即HN模型之誤差略高，推估原因應為變數定義導致資料型態與Heston and Nandi(2000)不盡相同。另外，參數的不良估計，使得HN模型最重要的兩個參數估計偏差過大，致最後評價誤差略高亦為原因之ㄧ。

關鍵字

選擇權評價 BS模型 GARCH

並列摘要

Options have been playing an important role in real financial market since the first option had traded. However, it is a major subject that how the rational price of options had been made. Black & Scholes (1973) set up the landmark of option pricing after they proposed the famous Black-Scholes option pricing model. In practice, one of the assumptions made in Black-Scholes (BS) option pricing model, namely volatility is a fixed constant, isn’t in accordance with the practices in real world. Scholars afterward try to release the assumption made by Black-Scholes and proposed so called stochastic volatility model which can categorized in discrete- and continuous-time model. Among discrete-time models, Duan (1995) introduced the model in quantitative economics, Generalized Autoregressive Conditional Heteroskedasticity (GARCH), to amend the assumption that volatility in Black-Scholes option pricing model is a fixed constant. Like most other option pricing models, the closed-form solutions do not exist. Heston & Nandi (2000) proposed a GARCH option pricing model with specific closed-form solution that can be directly derived by using numerical integration techniques. This research attempted to check out how the Heston & Nandi (2000) GARCH option pricing model perform in Taiwan Stock Exchange Capitalization Weighted Stock Index options (TXO) based on Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) and compared with BS option pricing model in option mispricing.
The results show that it is outperformed by BS option pricing model both in in-sample and out-of-sample valuation though Heston & Nandi (2000) released the assumption mentioned above. The significant mispricing could be caused by different data definitions. On the other hand, like Heston & Nandi (2000), the significant mispricing may also caused by the poor estimates of the parameters in the model.

並列關鍵字

Option pricing BS model GARCH

參考文獻

Bardia, K., and P. Ritchken,(1991) “Multinomial Approximating Model for Options with k State Variables,” Management Science, 37, 1640-1652.

Black, F. and Scholes, M.(1973) “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, 81, 637-659

Bollerslev, T.(1986) “Generalized Autoregressive Conditional Heteroscedasticity,” Journal of Econometrics, 31, 307-327.

Boyle, P., (1988) “A Lattice Framework for Option Pricing with Two State Variables,” The journal of Financial and Quantitative Analysis, 35,1.

Cox, J., Ross, S., and Rubinstein, M., (1979) “Option Pricing: A Simplified Approach,” Journal of Financial Economics, 3, 229-264.