摘要
選擇權價格的資訊內涵反映在其隱含波動率上，而選擇權的隱含波動率又具有微笑波幅的現象，因此文獻上多有探討如何建立能夠捕捉到微笑波幅現象的模型。另外，有學者認為選擇權的隱含波動率應該較標的資產的歷史波動率更能解釋標的資產的實際波動率。本文先利用S&P 500 期貨選擇權的賣權資料以及期貨資料，測試上述的命題；並使用四種不同的估計量來衡量實際波動率與歷史波動率，以檢驗其穩定性。另一方面，本文依據文獻建立數種有母數(隱含波動函數)以及無母數(隱含二元樹)模型來對未來選擇權價格做預測，並衡量其預測誤差。
根據實證結果，不同的實際波動率衡量方式會對結果造成影響；雖然隱含波動率和歷史波動率皆對實際波動率有解釋能力；但是隱含波動率對實際波動率的解釋能力略大於歷史波動率。在預測未來選擇權價格方面，各種有學術理論基礎的模型對於未來選擇權價格的預測反而沒有較市場人士所使用的經驗法則來的準，也就是說我們得到了越是簡單的模型，預測效果越好的結論。

Abstract
The information contents of option prices represent in their implied volatility. Generally speaking, implied volatility of options has the phenomenon of volatility smile. As a result, in the literature of option pricing, there are several models built in order to capture the shape of volatility smile. In addition, researches think implied volatility should have more strong explanatory power over historical volatility. In this study, we use S&P 500 put options on futures and S&P 500 futures data to test the topic stated above. We also test the stability by using four different measures of the realized volatility. On the other hand, several parametric (Implied Volatility Function) and non-parametric (Implied Binomial Tree) models are established to forecast future option prices and measure forecasting errors.
Empirical results reveal that implied volatility contains information in forecasting realized volatility but the results are not stable under different realized volatility measurements. On the other hand, historical volatility also has explanatory power to realized volatility. But implied volatility has higher explanatory power than historical volatility. As for option prices forecasting, several academic models have no better forecasting accuracy than ad hoc procedure. This results show that the simpler the model, the better the forecasting performance.

Anderson, T.G., Some reflections on analysis of high frequency data, Journal of Business & Economic Statistics 18 (2000), 146-153.
Barone-Adesi, G., and R.E. Whaley. “Efficient Analytic Approximation of American Option Values.” Journal of Finance, 42 (1987), pp. 301-320.
Black, F. “The Pricing of Commodity Contracts.” Journal of Financial Economics, 3 (January-March 1976), pp. 167-179.
Breeden, D.T., and R.H. Litzenberger. “Prices of Contingent Claims Implicit in Option Prices.” Journal of Business, 51 (1978), pp. 621-651.
Corrado, C., and T. Su. “Implied Volatility Skews and Stock Index Skewness and Kurtosis Implied by S&P 500 Index Option Prices.” Journal of Derivatives, 4, No. 4 (1997), pp. 8-19.
Campa, J., K. Chang, and R. Reider. “Implied Exchange Rate Distributions: Evidence from OTC Option Markets.” Journal of International Money and Finance, 17, No. 1 (1998), pp. 117-160.
Christensen, B.J., and Prabhala, N.R., “The relationship between implied and realized volatility.” Journal of Financial Economics, 50 (1998), pp. 125-150.
Derman, E., and I. Kani,. ”Riding on a Smile.” Risk, 7, No. 2 (1994), pp. 32-39.
Derman, E., I. Kani, and N. Chriss. “Implied Trinomial Trees of the Volatility Smile.” Journal of Derivatives, 3, No. 4 (1996), pp. 7-22.
Dumas, B., J. Fleming, and R. Whaley. “Implied Volatility Functions: Empirical Tests.” Journal of Finance, 53, No. 6 (1998), pp. 2059-2106.
Dupire, B. “Pricing With a Smile.” Risk, 7, No. 1 (1994), pp. 18-20.
Gemmill, G., and A. Saflekos “How Useful are Implied Distributions? Evidence from Stock Index Options.” Journal of Derivatives, 3, No. 1 (2000), pp. 83-98.
Hull, J. Options, Futures, and Other Derivatives 5th ed., Practice Hall, 2003
Jackwerth, J. “Generalized Binomial Trees. ” Journal of Derivatives, 5, No. 2 (1997). pp.7-17.
Jackwerth, J. “Option implied risk-neutral distributions and implied binomial trees:.A literature Review. ” Journal of Derivatives, 7, No. 2 (1999). pp.66-82.
Jackwerth, J. and M. Rubinstein. “Recovering Probability Distributions from Option Prices.” Journal of Finance, 51 (1996), pp. 1611-1631.
Jackwerth, J. and M. Rubinstein. “Recovering Stochastic Processes from Option Prices.” Working paper, 1998.
Jarrow, R., and A. Rudd. “Approximate Valuation for Arbitrary Stochastic Processes.” Journal of Financial Economics, 10, No. 3 (1982), pp. 347-369.
Longstaff, F. “Option Pricing and the Martingale Restriction.” Review of Financial Studies, 8, 4 (1995), pp. 1091-1124.
Malz, A. “Estimating the Probability Distribution of the Future Exchange Rate from Option Prices.” Journal of Derivatives, 5, No. 2 (1997), pp. 18-36.
Parkinson, M., “The Extreme Value Method for Estimating the Variance of the Rate of Return”, Journal of Business 53 (1980), pp. 61-68.
Rosenberg V. “Implied Volatility Functions: A Reprise” Journal of Derivatives, 7, No. 3 (2000), pp. 51-64.
Rubinstein, M. “Implied Binomial Trees.” Journal of Finance, 49, No. 3 (1994), pp. 771-818.
Shimko, D. “Bounds on Probability.” Risk, 6 (1993), pp. 33-37.
Shu, J. and J. Zhang. “The Relationship between Implied Volatility and Realized Volatility of S&P 500 Index.” Wilmott magazine (2003).
Silverman, B.W. Density Estimation for Statistics and Data Analysis. London: Chapman and Hull, 1986.
Tompkins, R. “Options Analysis: A State of the Art Guide to Options Pricing, Trading, and Portfolio Applications”. Burr Ridge, IL: Irwin, 1994
Yang, D., and Zhang Q., “Drift Independent Volatility Estimation Based on High, Low, Open and Close Prices”, Journal of Business 73 (2000), pp. 477-491.