本文旨在比較不同模型預測波動率能力之優劣，文中以long short-term memory (LSTM)為深度學習模型的代表，另外則是以傳統時間序列模型generalized autoregressive conditional heteroscedasticity (GARCH) 為代表。分別以兩種模型對股價指數和加密貨幣進行波動率之預測，藉以分析不同模型對不同資產的優劣。
本文的研究設計方面，首先，為了能準確衡量波動率，並使深度學習模型與時間序列模型能在同一基準下進行比較，因此，本文所有波動率預測模型及波動率的衡量，皆以5分鐘日內高頻資料來計算。
結果發現：(1)LSTM在預測已實現波動率上優於GARCH模型；(2) LSTM在預測加密貨幣市場之波動率有更好的表現；(3) LSTM在預測高波動率資產時誤差更小

This paper aims to compare the pros and cons of different models in their ability to forecast volatility. In this paper, LSTM is used as the representative of the deep learning model, and the other is represented by the traditional time series model GARCH model. Two models are used to forecast the volatility of stock indexes and cryptocurrencies to analyze the advantages and disadvantages of different models for different assets.
In terms of the research design of this paper, first of all, in order to accurately measure volatility and make deep learning models and time series models can be compared under the same benchmark, all volatility prediction models and volatility measurements in this paper are based on 5-minute intraday high-frequency data to calculate.
The results show that: (1) LSTM is better than GARCH model in forecasting realized volatility. (2) LSTM has a better performance in forecasting the volatility of the cryptocurrency market. (3) LSTM has a smaller error in predicting high-volatility commodities.

一、 中文文獻
王毓敏(2009)。預測股價指數波動率－新 VIX 與長期記憶模型之比較。中山管理評論十七卷第一期
韓傳祥(2013)。金融中波動率的數學問題。數學傳播37卷1期, pp. 26-40
二、 英文文獻
Alizadeh, S., Brandt, M. W., and Diebold, F. X., 2002, “Range-Based Estimation of Stochastic Volatility Models,” Journal of Finance, Vol. 57, No. 3, 1047-1091.
Andersen, T. G. and Bollerslev, T., 1998, “Answering the Skeptics: Yes, Standard Volatility Models do Provide Accurate Forecasts,” International Economic Review, Vol. 39, No. 4, 885-905.
Andersen, T. G., Bollerslev, T., Diebold, F. X., and Ebens, H., 2001a, “The Distribution of Realized Stock Return Volatility,” Journal of Financial Economics, Vol. 61, No. 1, 43-76.
Andersen, T. G., Bollerslev, T., Diebold, F. X., and Labys, P., 2001b, “The Distribution of Realized Exchange Rate Volatility,” Journal of American Statistical Association, Vol. 96, No. 453, 42-57
Andersen, T. G., Bollerslev, T., Diebold, F. X., and Labys, P., 2003, “Modeling and Forecasting Realized Volatility,” Econometrica, Vol. 71, No. 2, 579-625.
Andersen, T.G., Bollerslev, T., Diebold, F.X., 2007. Roughing it up: Including jump components in the measurement, modeling, and forecasting of return volatility. Rev. Econ. Stat. 89 (4), 701–720.
Andersen, T.G., Bollerslev, T., Meddahi, N., 2004. Analytical evaluation of volatility forecasts. Internat. Econom. Rev. 45 (4), 1079–1110.
Andersen, T.G., Bollerslev, T., Meddahi, N., 2005. Correcting the errors: Volatility forecast evaluation using high-frequency data and realized volatilities. Econometrica 73 (1), 279–296.
Andersen, T.G., Bollerslev, T., Meddahi, N., 2011. Realized volatility forecasting and market microstructure noise. J. Econometrics 160 (1), 220–234.
Andersen, T.G., Dobrev, D., Schaumburg, E., 2012. Jump-robust volatility estimation using nearest neighbor truncation. J. Econometrics 169 (1), 75–93.
Bandi, F.M., Russell, J.R., 2006. Separating microstructure noise from volatility. J. Financ. Econ. 79 (3), 655–692.
Bandi, F.M., Russell, J.R., 2008. Microstructure noise, realized variance, and optimal sampling. Rev. Econom. Stud. 75 (2), 339–369
Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A., Shephard, N., 2008. Designing realized kernels to measure the ex post variation of equity prices in the presence of noise. Econometrica 76 (6), 1481–1536.
Barndorff-Nielsen, O.E., Shephard, N., 2004a. Econometric analysis of realized covariation: high frequency based covariance, regression and correlation in financial economics. Econometrica 72 (3), 885–925.
Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637–654.
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), 307–32
Bollerslev, T. (1987). A conditionally heteroskedastic time series model for speculative prices and rates of return. The Review of Economics and Statistics, 542-547
Bollerslev, T., Patton, A. J., & Quaedvlieg, R. (2016). Exploiting the errors: A simple approach for improved volatility forecasting. Journal of Econometrics, 192(1), 1–18.
Bollerslev, T., Tauchen, G., & Zhou, H. (2009). Expected stock returns and variance risk premia. Review of Financial Studies, 22(11), 4463–4492
Brownlees, C. T., & Gallo, G. M. (2006). Financial econometric analysis at ultra-high frequency: Data handling concerns. Computational Statistics & Data Analysis, 51(4), 2232–2245.
Chiriac, R., Voev, V., 2010. Modeling and forecasting multivaraite realized volatility. J. Appl. Econometrics 26 (6), 922–947.
Christie, A. A. (1982). The stochastic behavior of common stock variances: Value, leverage and interest rate effects. Journal of Financial Economics, 10(4), 407–432.
Corsi, F., 2009. A simple approximate long-memory model of realized volatility. J. Financ. Econ. 7 (2), 174–196.
Corsi, F., Mittnik, S., Pigorsch, C., & Pigorsch, U. (2008). The volatility of realized volatility. Econometric Reviews, 27(1–3), 46–78.
Creal, D., Koopman, S. J., & Lucas, A. (2013). Generalized autoregressive score models with applications. Journal of Applied Econometrics, 28(5), 777–795.
Cybenko, G. (1989). Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals and Systems, 2(4), 303–
D. B. Nelson, “Conditional heteroskedasticity in asset returns: a new approach,” Econometrica, vol. 59, no. 2, pp. 347–370, 1991.
Dickey, D. A., & Fuller, W. A. (1979). Distribution of the estimators for autoregressive time series with a unit root. Journal of the American statistical association, 74(366a), 427–431.
Diebold, F. X., & Mariano, R. S. (1995). Comparing predictive accuracy. Journal of Business & Economic Statistics, 20(1), 134–144.
E. Hajizadeh, A. Seifi, M. H. Fazel Zarandi, and I. B. Turksen, “A hybrid modeling approach for forecasting the volatility of S&P 500 index return,” Expert Systems with Applications, vol. 39, no. 1, pp. 431–436, 2012.
E. Ramos-Perez, P. J. Alonso-Gonzalez, and J. J. Nuñez- ´ Velazquez, “Forecasting volatility with a stacked model based ´ on a hybridized Artificial Neural Network,” Expert Systems with Applications, vol. 129, pp. 1–9, 2019.
Engle, R. F., & Lee, G. (1999). A long-run and short-run component model of stock return volatility. In Cointegration, causality, and forecasting: A festschrift in honour of Clive WJ Granger (pp. 475–497). Oxford: Oxford University Press.
Engle, R. F., & Ng, V. K. (1993). Measuring and testing the impact of news on volatility. The Journal of Finance, 48(5), 1749–1778
Fama, E. F. (1970). Efficient capital markets: A review of theory and empirical work. The Journal of Finance, 25(2), 383–417.
Francis, X., & Roberto, S. (1995). Comparing Predictive Accuracy. Journal of Business & Economic Statistics, 13(3).
Gallant, A. R., Hsu, C.-T., & Tauchen, G. (1999). Using daily range data to calibrate volatility diffusions and extract the forward integrated variance. The Review of Economics and Statistics, 81(4), 617–631.
Ghysels, E., Sinko, A., 2011. Volatility forecasting and microstructure noise. J. Econometrics 160 (1), 257–271.
Glosten, L. R., Jagannathan, R., & Runkle, D. E. (1993). On the relation between the expected value and the volatility of the nominal excess return on stocks. The Journal of Finance, 48(5), 1779–1801.
Gonçalves, S., Meddahi, N., 2009. Bootstrapping realized volatility. Econometrica 77 (1), 283–306.
Granger, C. W., & Joyeux, R. (1980). An introduction to long-memory time series models and fractional differencing. Journal of Time Series Analysis, 1(1), 15–29.
H. Markowitz, “Portfolio selection,” The Journal of Finance, vol. 7, no. 1, pp. 77–91, 1952.
Hansen, P. R., Huang, Z., & Shek, H. H. (2012). Realized garch: a joint model for returns and realized measures of volatility. Journal of Applied Econometrics, 27(6), 877–906
Hansen, P.R., Lunde, A., 2006. Realized variance and market microstructure noise. J. Bus. Econom. Statist. 24 (2), 127–161.
Hansen, P. R., Janus, P., & Koopman, S. J. (2016). Realized wishart-garch: A score-driven multi-asset volatility model.
Hansen, P. R., Lunde, A., & Nason, J. M. (2011). The model confidence set. Econometrica, 79(2), 453–497.
Hansen, P.R., Lunde, A., Voev, V., 2014. Realized beta garch: A multivariate garch model with realized measures of volatility. J. Appl. Econometrics 29 (5), 774–799.
Harvey, A. C. (2013). Dynamic models for volatility and heavy tails: with applications to financial and economic time series (Vol. 52). Cambridge University Press.
Harvey, A., & Luati, A. (2014). Filtering with heavy tails. Journal of the American Statistical Association, 109(507), 1112–1122.
Harvey, A. C., & Shephard, N. (1996). Estimation of an asymmetric stochastic volatility model for asset returns. Journal of Business & Economic Statistics, 14(4), 429–434
Harvey, A., & Sucarrat, G. (2014). Egarch models with fat tails, skewness and leverage. Computational Statistics & Data Analysis, 76, 320–338.
H. Y. Kim and C. H. Won, “Forecasting the volatility of stock price index: a hybrid model integrating LSTM with multiple GARCH-type models,” Expert Systems with Applications , vol. 103, pp. 25–37, 2018.
Jacquier, E., Polson, N. G., & Rossi, P. E. (2004). Bayesian analysis of stochastic volatility models with fat-tails and correlated errors. Journal of Econometrics, 122(1), 185–212.
M. Bildirici and O. ¨ O. Ersin, “Improving forecasts of GARCH ¨ family models with the artificial neural networks: an application to the daily returns in Istanbul Stock Exchange,” Expert Systems with Applications, vol. 36, no. 4, pp. 7355–7362, 2009.
McAleer, M., & Medeiros, M. C. (2008). Realized volatility: A review. Econometric Reviews, 27(1-3), 10–45.
McAleer, M., & Medeiros, M. C. (2011). Forecasting realized volatility with linear and nonlinear univariate models. Journal of Economic Surveys, 25(1), 6–18.
Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica: Journal of the Econometric Society, 347–370
Ormoneit, D., & Neuneier, R. (1996, March). Experiments in predicting the German stock index DAX with density estimating neural networks. In Proceedings of the IEEE/IAFE 1996 conference on computational intelligence for financial engineering, (pp. 66–71). IEEE.
Park, H., Kim, N., & Lee, J. (2014). Parametric models and non-parametric machine learning models for predicting option prices: Empirical comparison study over KOSPI 200 Index options. Expert Systems with Applications, 41(11), 5227–5237.
Park, J. I., Lee, D. J., Song, C. K., & Chun, M. G. (2010). TAIFEX and KOSPI 200 forecasting based on two-factors high-order fuzzy time series and particle swarm optimization. Expert Systems with Applications, 37(2), 959–967
R. F. Engle, “Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation,” Econometrica, vol. 50, no. 4, pp. 987–1007, 1982.
Ryu, D. (2012). Implied volatility index of KOSPI200: Information contents and properties. Emerging Markets Finance and Trade, 48(sup2), 24-39.
S. Lahmiri, “Modeling and predicting historical volatility in exchange rate markets,” Physica A: Statistical Mechanics and Its Applications, vol. 471, pp. 387–395, 2017.
Simon, D. P., 1997, “Implied Volatility Asymmetries in Treasury Bond Futures Options,” Journal of Futures Markets, Vol. 17, No. 8, 837-885.
Simon, D. P., 2003, “The Nasdaq Volatility Index During and After the Bubble,” Journal of derivatives, Vol. 11, No. 2, 9-24
T. H. Roh, “Forecasting the volatility of stock price index,” Expert Systems with Applications, vol. 33, no. 4, pp. 916–922, 2007
Tseng, C. H., Cheng, S. T., Wang, Y. H., & Peng, J. T. (2008). Artificial neural network model of the hybrid EGARCH volatility of the Taiwan stock index option prices. Physica A: Statistical Mechanics and its Applications, 387(13), 3192–3200
W. Kristjanpoller, A. Fadic, and M. C. Minutolo, “Volatility forecast using hybrid Neural Network models,” Expert Systems with Applications, vol. 41, no. 5, pp. 2437–2442, 20
W. Kristjanpoller and E. Hernndez, “Volatility of main metals forecasted by a hybrid ANN-GARCH model with regressors,” Expert Systems with Applications, vol. 84, pp. 290–300, 2017.
W. Kristjanpoller and M. C. Minutolo, “Gold price volatility: a forecasting approach using the Artificial Neural Network GARCH model,” Expert Systems with Applications, vol. 42, no. 20, pp. 7245–7251, 2015.
W. Kristjanpoller and M. C. Minutolo, “A hybrid volatility forecasting framework integrating GARCH, Artificial Neural Network, technical analysis and principal components analysis,” Expert Systems with Applications, vol. 109, pp. 1–11, 2018.
Wilhelmsson, A. (2006). GARCH forecasting performance under different distribution assumptions. Journal of Forecasting, 25(8), 561–578.
Y. Peng, P. H. M. Albuquerque, J. M. Camboim de Sa, ´ A. J. A. Padula, and M. R. Montenegro, “ The best of two worlds: forecasting high frequency volatility for cryptocurrencies and traditional currencies with support vector regression,” Expert Systems with Applications, vol. 97, pp. 177–192, 2018.
Y. Hu, J. Ni, and L. Wen, “A hybrid deep learning approach by integrating LSTM-ANN networks with GARCH model for copper price volatility prediction,” Physica A: Statistical Mechanics and Its Applications, vol. 557, pp. 1–14, 2020.
Yao, Y., Zhai, J., Cao, Y., Ding, X., Liu, J., & Luo, Y. (2017). Data analytics enhanced component volatility model. Expert Systems with Applications, 84, 232–241