本文驗證台指選擇權在2001年12月到2008年5月間的選擇權價平 (at the money, ATM) 波動率、決定性波動函數 (deterministic volatility function, DVF) 的波動率與 ARFIMA 真實波動率 (realized volatility, RV)，進一步分析包含式迴歸模型對台指選擇權隱含波動率之預測能力，特別地，本文以預測值作為其相依變數。
    實證結果顯示，台股現貨指數之波動性不但有緩長記憶 (long memory)，且適用以ARFIMA模型來配適。互相比較三個預測變數之下，預測模型以DVF對全契約的隱含波動率預測能力最佳，此外，同時加入三個波動性預測值之後的包含式迴歸，其對全契約的隱含波動率預測能力最高，樣本外的預測誤差亦為最小。
    最後，為了驗證包含式迴歸模型的預測能力，本文須檢視其在台指選擇權市場能否獲利。透過風險中立 (delta neutral) 的選股策略，本文建構跨式 (straddle) 投資組合的交易策略。投資績效顯示，以同時加入ARFIMA真實波動率和決定性波動函數預測值之兩變數包含式迴歸，其一週投資績效為最佳；此外，上述預測誤差最小之包含式迴歸模型，不論在有無考慮交易成本下，皆可獲得正向報酬。

The paper estimates the implied volatilities of the at-the-money (ATM) option, deterministic volatility function (DVF) and realized volatility (RV) using ARFIMA model derived from TAIFEX options on Taiwan stock index during December 2001 to May 2008. We compare the predictive ability of encompassing regression model, especially, we use the predicted values as independent variables.
    The results indicate that we confirm not only the presence of long memory behavior in the TX volatility but also accurately fitted by ARFIMA. Comparing the different predictive variables, we find the DVF model has the highest forecasting ability of implied volatility for call and put options. Moreover, after including three predictive variables, the encompassing regression has the highest forecasting ability of the implied volatility in the sample and the smallest forecasting error out of the sample for call and put options.
    Finally, in order to examine the forecasting ability of the encompassing regression model, we need to tell whether implied volatility forecasts can be used to formulate profitable out-of-sample trading strategies in TXO market or not. By using  delta-neutral option, we construct straddle portfolios to estimate benefits of trading strategies. The results show that the encompassing regression with the realized volatility and DVF volatility of one week straddle portfolio has the best performance. Furthermore, regardless of the transaction cost, the encompassing regression with the smallest forecasting error can get positive return in TXO market.

中文摘要	I
英文摘要	II
目 次	III
表 次	IV
圖 次	VI
第一章、緒論	1
第一節、研究動機	1
第二節、研究目的	3
第三節、研究架構	5
第二章、文獻探討	6
第一節、真實波動率	6
第二節、隱含波動率	10
第三節、波動率的緩長記憶	18
第四節、波動率的預測	25
第五節、預測誤差的衡量	31
第六節、交易策略	33
第三章、研究設計與方法	35
第一節、取樣標準與研究設計	35
第二節、波動率的估計	46
第三節 包含式迴歸分析	56
第四節、預測能力的衡量指標	58
第五節、投資策略的設計	60
第四章、實證結果	62
第一節、波動率	62
第二節 包含式迴歸	79
第三節、樣本內與外的預測	103
第四節 投資績效的評估	109
第五章 結論	114
參考文獻	116
附錄表	123

表次
【表1】使用單一模型預測波動率的文獻整理	27
【表2】買賣權價內與價外程度分類表	38
【表3】包含式迴歸變數	45
【表4】買權隱含波動率依價性分類之敘述統計量	64
【表5】賣權隱含波動率依價性分類之敘述統計量	65
【表6】選擇權契約之隱含波動率估計值敘述統計表	67
【表7】價平選擇權之隱含波動率預測值敘述統計表	69
【表8】決定性波動函數之波動率預測值敘述統計表	71
【表9】ARFIMA(1, d, 0)真實波動率預測值之敘述統計表	77
【表10】單一ARFIMA真實波動率對TXO之隱含波動率預測能力迴歸表	81
【表11】單一ARFIMA真實波動率對TXO買權之隱含波動率預測能力迴歸表	82
【表12】單一ARFIMA真實波動率對TXO賣權之隱含波動率預測能力迴歸表	83
【表13】單一價平選擇權對TXO之隱含波動率預測能力迴歸表	84
【表14】單一決定性波動函數對TXO之隱含波動率預測能力迴歸表	84
【表15】對TXO買權隱含波動率之包含式迴歸	89
【表16】對TXO賣權隱含波動率之包含式迴歸	90
【表17】對TXO不分買賣權契約隱含波動率之包含式迴歸	91
【表18】ARFIMA真實波動率加入Jump對TXO買權契約隱含波動率之包含式迴歸	92
【表19】ARFIMA真實波動率加入Jump對TXO賣權契約隱含波動率之包含式迴歸	93
【表20】ARFIMA真實波動率加入Jump對TXO不分買賣權契約隱含波動率之包含式迴歸	94
【表21】ARFIMA真實波動率加入Leverage effect對TXO買權契約隱含波動率之包含式迴歸	95
【表22】ARFIMA真實波動率加入Leverage effect對TXO之賣權契約隱含波動率包含式迴歸	96
【表23】ARFIMA真實波動率加入Leverage effect對TXO之不分買賣權契約隱含波動率包含式迴歸	97
【表24】ARFIMA真實波動率同時加入Jump和Leverage effect對TXO之買權契約隱含波動率包含式迴歸	98
【表25】ARFIMA真實波動率同時加入Jump和Leverage effect對TXO之賣權契約隱含波動率包含式迴歸	99
【表26】ARFIMA真實波動率同時加入Jump和Leverage effect對TXO不分買賣權契約隱含波動率之包含式迴歸	100
【表27】買權情況下的預測誤差表	107
【表28】不分買賣權情況下的預測誤差表	108
【表30】近月份包含式迴歸預測的波動率用於模擬投資之績效	111
【附錄表2】對TXO賣權隱含波動率之包含式迴歸	124
【附錄表3】對TXO不分買賣權隱含波動率之包含式迴歸	125
【附錄表4】ARFIMA真實波動率加入Jump對TXO買權隱含波動率之包含式迴歸	126
【附錄表5】ARFIMA真實波動率加入Jump對TXO賣權隱含波動率之包含式迴歸	127
【附錄表6】ARFIMA真實波動率加入Jump對TXO不分買賣權隱含波動率之包含式迴歸	128
【附錄表7】ARFIMA真實波動率加入Leverage effect對TXO買權隱含波動率之包含式迴歸	129
【附錄表8】ARFIMA真實波動率加入Leverage effect對TXO賣權隱含波動率之包含式迴歸	130
【附錄表9】ARFIMA真實波動率加入Leverage effect對TXO不分買賣權隱含波動率之包含式迴歸	131
【附錄表10】ARFIMA真實波動同時加入Jump和Leverage effect對TXO買權隱含波動率之包含式迴歸	132
【附錄表11】ARFIMA真實波動率同時加入Jump和Leverage effect對TXO賣權隱含波動率之包含式迴歸	133
【附錄表12】ARFIMA波動率同時加入Jump和Leverage effect對TXO不分買賣權隱含波動率之包含式迴歸	134

圖次
圖1、決定性波動函數之隱含波動率預測值估計示意圖	42
圖2、台股指數與隱含波動率圖	62
圖3、樣本內全期間隱含波動率次數直方圖	68
圖4、近月份隱含波動率次數直方圖	69
圖5、樣本內全期間ATM隱含波動率次數直方圖	70
圖6、近月份ATM隱含波動率次數直方圖	70
圖7、樣本內全期間DVF波動率次數直方圖	72
圖8、近月份DVF波動率次數直方圖	72
圖9、不同樣本頻率下的年平均真實波動率	73
圖10、真實波動率在不同樣本頻率下之ACF圖	74
圖11、ARFIMA(1, d, 0)模型之各單位期間d值圖	75
圖12、RV、IV和ATM波動率預測值之對照圖	76

Alizadeh, S., Brandt, M. W., & Diebold, F. X. (2002). Range-based estimation of stochastic volatility models. The Journal of Finance, 57, 1047. 
Andersen, T. G., & Bollerslev, T. (1997). Intraday periodicity and volatility persistence in financial markets. Journal of Empirical Finance, 4, 115.
Andersen, T. G., Bollerslev, T., Diebold, F. X., & Labys, P. (2001). The distribution of realized exchange rate volatility. Journal of the American Statistical Association, 96, 42. 
Andersen, T. G., Bollerslev, T., Diebold, F. X., & Labys, P. (2003). Modeling and forecasting realized volatility. Econometrica, 71, 579. 
Andersen, T., Bollerslev, T., Diebold, F., & Ebens, H. (2001). The distribution of realized stock return volatility. Journal of Financial Economics, 61, 43. 
Areal, N. M. P. C., & Taylor, S. J. (2002). The realized volatility of FTSE-100 futures prices. The Journal of Futures Markets, 22, 627. 
Backus, D. K., & Zin, S. E. (1993). Long-memory inflation uncertainty: Evidence from the term structure of interest rates. Journal of Money, Credit, and Banking, 25, 681. 
Baillie, R. T., & Bollerslev, T. (1994). Cointegration, fractional cointegration, and exchange rate dynamics. The Journal of Finance, 49, 737. 
Baillie, R. T., & Chung, C. (1992). Estimation of fractionally integrated processes with ARCH innovations. manuscript.
Baillie, R. T., Chung, C., & Tieslau, M. (1992). The long memory and variability of inflation: A reappraisal of the Friedman hypothesis. CentER Discussion Paper 9246, Tilburg University.
Bailie, R. T., & DeGennaro, R. P. (1990). Stock Returns and Volatility. Journal of Financial and Quantitative Analysis, 25, 203.
Bakshi, G., Cao, C., & Chen, Z. (1997). Empirical performance of alternative option pricing models. The Journal of Finance, 52, 2003. 
Barkoulas, J. T., Baum, C.F., & Oguz, G.S. (1997). Fractional Dynamics in a System of Long Term International Interest Rates. Working Paper.
Beckers, S. (1981). Standard deviations implied in option prices as predictors of future stock price variability. Journal of Banking & Finance, 5, 363. 
Beran, J. (1994). Statistics for Long-Memory Processes. New York: Chapman and Hall.
Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. The Journal of Political Economy, 81, 637. 
Blair, B. J., Poon, S., & Taylor, S. J. (2001). Forecasting S&P 100 volatility: The incremental information content of implied volatilities and high-frequency index returns. Journal of Econometrics, 105, 5. 
Bollen, N. P. B., & Whaley, R. E. (2004). Does net buying pressure affect the shape of implied volatility functions? The Journal of Finance, 59, 711. 
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31, 307. 
Bollerslev, T., Engle, R. F., & Wooldridge, J. M. (1988). A capital asset pricing model with time-varying covariances. The Journal of Political Economy, 96, 116. 
Britten-Jones, M., & Neuberger, A. (2000). Option prices, implied price processes, and stochastic volatility. The Journal of Finance, 55, 839. 
Brockwell, P. J., & Davis, R. A. (1991). Time Series: Theory and Methods. New York: Springer-Verlag.
Brunetti, C., & Gilbert, C. L. (2000). Bivariate FIGARCH and Fractional Cointegration.  Journal of Empirical Finance, 7, 509
Campbell, J. Y., & Mankiw, N. G. (1987). Permanent and transitory components in macroeconomic fluctuations. The American Economic Review, 77, 111. 
Canina, L., & Figlewski, S. (1993). The informational content of implied volatility. The Review of Financial Studies, 6, 659. 
Carlin, J. B., & Dempster, A. P. (1989). Sensitivity analysis of seasonal adjustments: Empirical case studies. Journal of the American Statistical Association, 84, 6-20. 
Chan, K. C., Cheng, L. T. W., & Lung, P. P. (2004). Net buying pressure, volatility smile, and abnormal profit of hang seng index options. The Journal of Futures Markets, 24, 1165.
Chaput, J. S., & Ederington, L. (2005b). Vertical spread design. Journal of Derivatives, 12, 28.
Cheung, Y. (1993). Long memory in foreign-exchange rates. Journal of Business & Economic Statistics, 11, 93. 
Cheung, Y., & Diebold, F. X. (1994). On maximum likelihood estimation of the differencing parameter of fractionally-integrated noise with unknown mean. Journal of Econometrics, 62, 301. 
Cheung, Y., & Lai, K. S. (1993). A fractional cointegration analysis of purchasing power parity. Journal of Business & Economic Statistics, 11, 103. 
Chiras, D. P., & Manaster, S. (1978). The information content of option prices and a test of market efficiency. Journal of Financial Economics, 6, 213. 
Choi, K., & Zivot, E. (2007). Long memory and structural changes in the forward discount: An empirical investigation. Journal of International Money and Finance, 26, 342. 
Choudhry, T. (1999). Re-examining forward market efficiency: Evidence from fractional and harris-inder cointegration tests. International Review of Economics & Finance, 8, 433. 
Christensen, B. J., & Prabhala, N. R. (1998). The relation between implied and realized volatility. Journal of Financial Economics, 50, 125. 
Christoffersen, P., & Jacobs, K. (2004). The importance of the loss function in option valuation. Journal of Financial Economics, 72, 291. 
Cox, J. C., Ross, S. A., & Rubinstein, M. (1979). Option pricing: A simplified approach. Journal of Financial Economics, 7, 229. 
Dahlhaus, R. (1988). Small Sample Effects in Time Series Analysis : A New Asymptotic Theory and a New Estimate. Annals of Statistics, 16, 808
Dahlhaus, R. (1989). Efficient parameter estimation for self-similar processes. The Annals of Statistics, 17, 1749. 
Day, T. E., & Lewis, C. M. (1988). The behavior of the volatility implicit in the prices of stock index options. Journal of Financial Economics, 22, 103. 
Day, T. E., & Lewis, C. M. (1992). Stock market volatility and the information content of stock index options. Journal of Econometrics, 52, 267. 
Deo, R., Hurvich, C., & Lu, Y. (2006). Forecasting realized volatility using a long-memory stochastic volatility model: Estimation, prediction and seasonal adjustment. Journal of Econometrics, 131, 29. 
Derman, E., Kamal, M., Kani, I., McClure J., & Zou, J. (1998). Investing in Volatility. Journal of Derivatives, 7.
Diebold, F. X., Husted, S., & Rush, M. (1991). Real exchange rates under the gold standard. The Journal of Political Economy, 99, 1252. 
Diebold, F. X., & Rudebusch, G. D. (1989). Long memory and persistence in aggregate output. Journal of Monetary Economics, 24, 189. 
Diebold, F. X., & Rudebusch, G. D. (1991). Is consumption too smooth? long memory and the deaton paradox. The Review of Economics and Statistics, 73, 1. 
Ding, Z., Granger, C. W. J., & Engle, R. F. (1993). A long memory property of stock market returns and a new model. Journal of Empirical Fianace1, 83.
Dumas, B., Fleming, J., & Whaley, R. E. (1998). Implied volatility functions: Empirical tests. The Journal of Finance, 53, 2059. 
Ebens, H. (1999). Realized Stock Volatility. unpublished manuscript. Johns Hopkins University.
Engle, R. F. (1982). Autoregressive Conditional Heteroskedasticity with Estimates of The Variance of UK Inflation. Econometrica, 50, 987.
Engle, R. F., & Granger, C. W. J. (1987). Co-integration and error correction: Representation, estimation, and testing. Econometrica, 55, 251. 
Fitzgerald, D. (1999). Trading Volatility. Risk Mangement and Analysis, 2, 261
Fleming, J. (1998). The quality of market volatility forecasts implied by S&P 100 index option prices. Journal of Empirical Finance, 5, 317.
Fox, R., & Taqqu, M. (1986). Large sample properties of parameter estimates for strongly dependend stationary Gaussian time series. The Annals of Statistics, 14, 517.
French, K. R., Schwert, G. W., & Stambaugh, R. F. (1987). Expected stock returns and volatility. Journal of Financial Economics, 19, 3. 
Gemmill, G. (1986). The forecasting performance of stock options on the london traded options market. Journal of Business Finance & Accounting, 13, 535. 
Granger, C. W. J., & Joyeux, R. (1980). An Introduction to Long-Memory Time Series Models and Fractional Differencing. Journal of Time Series Analysis, 1, 15.
Granger, C. W. J., & Newbold, P. (1974). Spurious Regression in Econometrics.  Journal of Econometrics, 2, 111.
Geweke, J., & Porter, S. H. (1983). The estimation and application of long memory time series models. Journal of Time Series Analysis, 4, 221.
Granger, C. W. J. (1986). Developments in the study of cointegrated economic variables. Oxford Bulletin of Economics and Statistics, 48, 213. 
Granger, C. W. J., & Ding, Z. (1996). Varieties of long memory models. Journal of Econometrics, 73, 61. 
Greene, M. T., & Fielitz, B. D. (1977). Long-term dependence in common stock returns. Journal of Financial Economics, 5, 339.
Gwilym, O., & Buckle, M. (1999). Volatility forecasting in the framework of the option expiry cycle. The European Journal of Finance, 5, 73.
Hauser, M.A., P‥otscher, B. M. & Reschenhofer, E. (1992). Measuring persistence in aggregate output: ARMA models, fractionally integrated ARMA models and nonparametric procedures. manuscript.
Hosking, J. R. M. (1981). Fractional Differencing. Biometrika, 68, 165.
Hurst, H. E. (1951). Long--term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers, 116, 700.
Janacek, G. (1982). Determining the degree of differencing for time series via the long spectrum. Journal of Time Series Analysis, 3, 177.
Jiang, G. J., & Tian, Y. S. (2005). The model-free implied volatility and its information content. The Review of Financial Studies, 18, 1305. 
Jorion, P. (1995). Predicting volatility in the foreign exchange market. The Journal of Finance, 50, 507. 
Koop, G. (1991). Intertemporal properties of real output: A bayesian analysis. Journal of Business & Economic Statistics, 9, 253. 
Koop, G., Ley, E., Osiewalski, J., & Steel, M. F. J. (1997). Bayesian analysis of long memory and persistence using ARFIMA models. Journal of Econometrics, 76, 149. 
Koopman, S. J., Jungbacker, B., & Hol U. E. (2005). Forecasting Daily Variability of the S&P 100 Stock Index Using Historical, Realised and Implied Volatility Measurements. Structural Breaks and Nonlinearity. Tinbergen Institute Discussion Paper.
Lamoureux, C. G., & Lastrapes, W. D. (1993). Forecasting stock-return variance: Toward an understanding of stochastic implied volatilities. The Review of Financial Studies, 6, 293. 
Latane, H. A., & Rendleman, R. J.,Jr. (1976). Standard deviations of stock price ratios implied in option prices. The Journal of Finance, 31, 369. 
Li, W. K., & McLeod, A. I. (1986). Fractional time series modelling. Biometrika, 73, 217-221. 
Lobato, I. N., & Savin, N. E. (1998). Real and spurious long-memory properties of stock-market data: Reply. Journal of Business & Economic Statistics, 16, 280. 
Lobato, I. N., & Velasco, C. (2000). Long memory in stock-market trading volume. Journal of Business & Economic Statistics, 18, 410. 
McLeod, A.I. & Hipel, K.W. (1978). Preservation of the Rescaled Adjusted Range 1: a Reassessment of the Hurst Phenomenon. Water Resources Research, 14, 491.
Mandelbrot, B. B. (1968). Fractional Brownian motions, fractional Brownian noises and applications. SIAM Review, 10, 422.
Martens, M., D. van Dijk & Pooter de M. (2004). Modeling and Forecasting S&P 500 Volatility: Long Memory. Structural Breaks and Nonlinearity. Tinbergen Institute Discussion Paper
Martens, M., & Zein, J. (2004). Predicting financial volatility: High-frequency time-series forecasts vis-à-vis implied volatility. The Journal of Futures Markets, 24, 1005. 
Mayhew, S. (1995). Implied volatility. Financial Analysts Journal, 51, 8. 
Mayhew, S., & Stivers, C. (2003). Stock return dynamics, option volume, and the information content of implied volatility. The Journal of Futures Markets, 23, 615. 
Merton, R. C. (1980). On estimating the expected return of the market. Journal of Financial Economics, 8, 323. 
Nelson, D. B. (1992). Filtering and forecasting with misspecified ARCH models I: Getting the right variance with the wrong model. Journal of Econometrics, 51, 61. 
Odaki, M. (1993). On the invertibility of fractionally differenced ARIMA processes. Biometrika, 80, 703.
Osiewalski, J., & Steel, M.F.J. (1993). Robust Bayesian inference in elliptical regression models. Journal of Econometrics, 57, 345.
Oomen, R. C. A. (2002). Modelling Realized Variance when Returns are Serially Correlated. working paper.
Poon, S., & Granger, C. W. J. (2003). Forecasting volatility in financial markets: A review. Journal of Economic Literature, 41, 478. 
Pong, S., Shackleton, M. B., Taylor, S. J. & Xu, X. (2004). Forecasting Currency Volatility:A Comparison of Implied Volatilities and AR(FI)MA Models. Journal of Banking and Finance, in press.
Poterba, J. M., & Summers, L. H. (1986). The persistence of volatility and stock market fluctuations. The American Economic Review, 76, 1142. 
Sadique, S., & Silvapulle, P. (2001). Long-term memory in stock market returns: International evidence. International Journal of Finance & Economics, 6, 59. 
Schotman, P.C., & van Dijk H.K. (1991). On Bayesian routes to unit roots. Journal of Applied Econometrics, 6, 387.
Schwert, G. W. (1989). Tests for unit roots: A monte carlo investigation. Journal of Business & Economic Statistics, 7, 147. 
Schwert, G. W., & Seguin, P. J. (1990). Heteroskedasticity in stock returns. The Journal of Finance, 45, 1129. 
Shea, G.S. (1990). Uncertainty and Implied Variance Bounds in Long–Memory Model of the Interest Rate Term Structure. Empirical Economics, 16, 387.
So, M. K. P. (2000). Long-term memory in stock market volatility. Applied Financial Economics, 10, 519. 
Sowell, F. (1992). Maximum likelihood estimation of stationary univariate fractionally integrated time series models. Journal of Econometrics, 53, 165. 
Szakmary, A., Ors, E., Kim, J. K., & Davidson, W. N.,III. (2003). The predictive power of implied volatility: Evidence from 35 futures markets. Journal of Banking & Finance, 27, 2151. 
Taylor, S. J., & Xu, X. (1997). The incremental volatility information in one million foreign exchange quotations. Journal of Empirical Finance, 4(4), 317-340. 
Thomakos, D. D., & Wang, T. (2003). Realized volatility in the futures markets. Journal of Empirical Finance, 10, 321-353. 
Tsay, W. J. (2000). The long memory story of the real interest rate. Economics Letters, 67, 325.
Zhou, B. (1996). High-frequency data and volatility in foreign-exchange rates. Journal of Business & Economic Statistics, 14, 45.