Forecasting realized volatility: a review

Bucci, Andrea (2017): Forecasting realized volatility: a review.

Abstract

Modeling financial volatility is an important part of empirical finance. This paper provides a literature review of the most relevant volatility models, with a particular focus on forecasting models. We firstly discuss the empirical foundations of different kinds of volatility. The paper, then, analyses the non-parametric measure of volatility, named realized variance, and its empirical applications. A wide range of realized volatility models, both univariate and multivariate, is presented, such as time series models, MIDAS and GARCH-MIDAS models, Realized GARCH, and HEAVY models. We further discuss forecasting evaluation methods specifically suited for volatility models.

• Item Type: MPRA Paper
• Language: English
• Keywords: Realized Volatility; Stochastic Volatility; Volatility Models
• Subjects: C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C53 - Forecasting and Prediction Methods ; Simulation Methods; G - Financial Economics > G1 - General Financial Markets > G10 - General

References

AÏT-SAHALIA, Y., P. A. MYKLAND, AND L. ZHANG (2006): “Comment,” Journal of Business and Economic Statistics, 24, 162–167.

ALEXANDER, C., AND A. M. CHIBUMBA (1996): “Multivariate orthogonal factor GARCH,” Discussion Paper in mathematics, University of Sussex.

AMIHUD, Y., AND H. MENDELSON (1987): “Trading mechanisms and stock returns: an empirical investigation,” Journal of Finance, 42, 533–553.

ANDERSEN, T. G. (2009): “Stochastic volatility,” in Encyclopedia of Complexity and Systems Science. Springer Verlag.

ANDERSEN, T. G., AND T. BOLLERSLEV (1998): “Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts,” International Economic Review, 39, 885–905.

ANDERSEN, T. G., T. BOLLERSLEV, P. CHRISTOFFERSEN, AND F. X. DIEBOLD (2006): “Volatility and correlation forecasting,” in Handbook of Economic Forecasting, pp. 778–878. Amsterdam: North-Holland.

ANDERSEN, T. G., T. BOLLERSLEV, F. X. DIEBOLD, AND H. EBENS (2001): “The distribution of realized stock return volatility,” Journal of Financial Economics, 61, 43–76.

ANDERSEN, T. G., T. BOLLERSLEV, F. X. DIEBOLD, AND P. LABYS (2000): “Exchange rate returns standardized by realized volatility are (nearly) Gaussian,” Multinational Finance Journal, 4, 159– 179.

(2001): “The distribution of exchange rate volatility,” Journal of American Statistical Association, 96, 42–55.

(2003): “Modeling and Forecasting Realized Volatility,” Econometrica, 71, 579–625.

ANDERSEN, T. G., T. BOLLERSLEV, AND N. MEDDAHI (2005): “Correcting the Errors: Volatility Forecast Evaluation Using High-Frequency Data and Realized Volatilities,” Econometrica, 73(1), 279–296.

ARTZNER, P., F. DELBAEN, J.-M. EBER, AND D. HEATH (1999): “Coherent measures of risk,” Mathematical Finance, 9, 203–228.

ASAI, M., M. MCALEER, AND J. YU (2006): “Multivariate stochastic volatility: A review,” Econometric Reviews, 25, 145–175.

ASGHARIAN, H., C. CHRISTIANSEN, AND A. J. HOU (2014): “Macro-Finance Determinants of the

Long-Run Stock-Bond Correlation: The DCC-MIDAS Specification,” Working paper.

ASGHARIAN, H., A. J. HOU, AND F. JAVED (2013): “The Importance of the Macroeconomic Variables in Forecasting Stock Return Variance: A GARCH-MIDAS Approach,” Journal of Forecasting, 32, 600–612.

BACK, K. (1991): “Asset pricing for general processes,” Journal of Mathematical Economics, 20, 371–395.

BAILLIE, R. T., T. BOLLERSLEV, AND H. O. MIKKELSEN (1996): “Fractionally integrated generalized autoregressive conditional heteroskedasticity,” Journal of Econometrics, 74, 3–30.

BANDI, F. M., AND J. R. RUSSELL (2005): “Realized covariation, realized beta and microstructure noise,” Unpublished paper, Graduate School of Business, University of Chicago.

(2006): “Volatility,” in Handbook of Financial Engineering. Elsevier.

(2007): “Microstructure noise, realized volatility, and optimal sampling,” Unpublished paper, Graduate School of Business, University of Chicago.

BARNDORFF-NIELSEN, O. E., P. R. HANSEN, A. LUNDE, AND N. SHEPHARD (2008): “Designing realized kernels to measure the ex post variation of equity prices in the presence of noise,” Econometrica, 76, 1481–1536.

BARNDORFF-NIELSEN, O. E., AND N. SHEPHARD (2002a): “Econometric analysis of realised volatility and its use in estimating stochastic volatility models,” Journal of the Royal Statistical Society, 64, 253–280.

(2002b): “Estimating Quadratic Variation Using Realised Variance,” Journal of Applied Econometrics, 17, 457–477.

(2004): “Power and Bipower Variation with Stochastic Volatility and Jumps,” Journal of Financial Econometrics, 2, 1–37.

BARR, D. G. (2013): “Value at Risk,” Bank of England, Centre for Central Banking Studies.

BARUNÍK, J., AND F. CECH (2016): “On the modelling and forecasting multivariate realized volatility: Generalized Heterogeneous Autoregressive (GHAR) model,” Journal of Forecasting.

BAUER, G. H., AND K. VORKINK (2011): “Forecasting multivariate realized stock market volatility,” Journal of Econometrics, 160, 93–101.

BAUWENS, L., C. HAFNER, AND S. LAURENT (2012): Handbook of Volatility Models and Their Applications. Wiley.

BECKER, R., A. CLEMENTS, AND R. O’NEILL (2010): “A Cholesky-MIDAS model for predicting stock portfolio volatility,” Working paper, Centre for Growth and Business Cycle Research Discussion Paper Series.

BLACK, F. (1976): “Noise,” Journal of Finance, 41, 529–543.

BLANCO, C., AND G. IHLE (1999): “How Good is Your VaR? Using Backtesting to Assess System Performance,” Financial Engineering News, 11(8), 1–2.

BOLLEN, B. E., AND B. INDER (2002): “Estimating daily volatility in financial markets utilizing intraday data,” Journal of Empirical Finance, 9, 551–562.

BOLLERSLEV, T. (1986): “Generalized autoregressive conditional heteroskedasticity,” Journal of Econometrics, 31, 307–327.

(1990): “Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH model,” The Review of Economics and Statistics, pp. 498–505.

(2009): “Glossary to ARCH (GARCH),” Working paper, Duke University.

BOLLERSLEV, T., R. F. ENGLE, AND D. B. NELSON (1994): “ARCH models,” in Handbook of Econometrics. Elsevier Science, Amsterdam.

BOLLERSLEV, T., R. F. ENGLE, AND J. M. WOOLDRIDGE (1988): “A capital asset pricing model with time-varying covariances,” The Journal of Political Economy, pp. 116–131.

BOLLERSLEV, T., AND J. M. WOOLDRIDGE (1992): “Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances,” Econometric Reviews, 11, 143–172.

BONATO, M. (2009): “Estimating the degrees of freedom of the Realized Volatility Wishart Autoregressive model,” Working paper.

BONATO, M., M. CAPORIN, AND A. RANALDO (2009): “Forecasting realized (co)variances with a block structure Wishart autoregressive model,” Working papers, Swiss National Bank.

BUCCI, A., G. PALOMBA, AND E. ROSSI (2017): “Modeling multivariate realized volatility with exogenous variables,” Working paper.

CAI, J. (1994): “A Markov Model of Switching-Regime ARCH,” Journal of Business and Economic Statistics, 12, 309–316.

CHAN, L. K., J. KARCESKI, AND J. LAKONISHOK (1999): “On Portfolio Optimization: Forecasting Covariances and Choosing the Risk Model,” The Review of Financial Studies, 12, 937–974.

CHEN, L. (1996): “Stochastic Mean and Stochastic Volatility: A Four-Dimensional Term Structure of Interest Rates and Its Application to the Pricing of Derivative Securities,” Financial Markets, Institutions, and Instruments, 5, 1–88.

CHRISTENSEN, B., AND N. PRABHALA (1998): “The relation between implied and realized volatility,” Journal of Financial Economics, 37, 125–150.

CHRISTIANSEN, C., M. SCHMELING, AND A. SCHRIMPF (2012): “A comprehensive look at financial volatility prediction by economic variables,” Journal of Applied Econometrics, 27, 956–977.

CHRISTOFFERSEN, P. F. (1998): “Evaluating interval forecasts,” International Economic Review, 39, 841–862.

CLARK, T. E., AND K. D. WEST (2007): “Approximately normal tests for equal predictive accuracy in nested models,” Journal of Econometrics, 138(1), 291–311.

COLACITO, R., R. F. ENGLE, AND E. GHYSELS (2011): “A Component Model for Dynamic Correlations,” Journal of Econometrics, 164, 45–59.

CONRAD, C., AND K. LOCH (2014): “Anticipating Long-Term Stock Market Volatility,” Journal of Applied Econometrics, 30(7), 1090–1114.

CORSI, F. (2009): “A simple approximate long-memory model of realized volatility,” Journal of Financial Econometrics, 7, 174–196.

DE POOTER, M., M. MARTENS, AND D. VAN DIJK (2008): “Predicting the Daily Covariance Matrix for S&P 100 Stocks Using Intraday Data - But Which Frequency to Use?”, Econometric Reviews, 27, 199–229.

DIEBOLD, F. X., AND R. S. MARIANO (1995): “Comparing predictive accuracy,” Journal of Business and Economic Statistics, 13, 253–263.

EBENS, H. (1999): “Realized Stock Index Volatility,” Working Paper No. 420, Department of Economics, Johns Hopkins University, Baltimore.

ENGLE, R. F. (1982): “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation,” Econometrica, 50, 987–1007.

(2002): “Dynamic conditional correlation: a simple class of multivariate GARCH models,” Journal of Business and Economic Statistics, 20, 339–350.

ENGLE, R. F., AND G. M. GALLO (2006): “A multiple indicators model for volatility using intra-daily data,” Journal of Econometrics, 131, 3–27.

ENGLE, R. F., E. GHYSELS, AND B. SOHN (2013): “Stock Market Volatility and Macroeconomic Fundamentals,” Review of Economics and Statistics, 95, 776–797.

ENGLE, R. F., AND K. F. KRONER (1995): “Multivariate simultaneous generalized ARCH,” Econometric Theory, 11, 122–150.

ENGLE, R. F., D. M. LILIEN, AND R. P. ROBINS (1987): “Estimating time varying risk premia in

the term structure: The ARCH-M model,” Econometrica, 55, 391–407.

ENGLE, R. F., V. K. NG, AND M. ROTHSCHILD (1990): “Asset pricing with a factor ARCH covariance

structure: empirical estimates for treasury bills,” Journal of Econometrics, 45, 213–238.

ENGLE, R. F., AND J. G. RANGEL (2008): “The Spline-GARCH Model for Low-Frequency Volatility and Its Global Macroeconomic Causes,” Review of Financial Studies, 21, 1187–1222.

ENGLE, R. F., AND K. SHEPPARD (2001): “Theoretical and empirical properties of dynamical conditional correlation model multivariate GARCH,” UCSD Discussion.

EPPS, T. W. (1979): “Comovements in Stock Prices in the Very Short Run,” Journal of the American Statistical Association, 74, 291–298.

FLEMING, J., C. KIRBY, AND B. OSTDIEK (2003): “The economic value of volatility timing using realized volatility,” Journal of Financial Economics, 67, 473–509.

GHYSELS, E., A. HARVEY, AND E. RENAULT (1996): “Stochastic volatility,” in Statistical models in

finance, pp. 119–191. Amsterdam: North-Holland.

GHYSELS, E., A. RUBIA, AND R. VALKANOV (2009): “Multi-Period Forecasts of Volatility: Direct, Iterated, and Mixed-Data Approaches,” Working paper.

GHYSELS, E., P. SANTA-CLARA, AND R. VALKANOV (2004): “The MIDAS touch: Mixed data sampling regression models,” Working paper, UNC and UCLA.

(2006): “Predicting volatility: getting the most out of return data sampled at different frequencies,” Journal of Econometrics, 131, 59–95.

GIACOMINI, R., AND H. WHITE (2006): “Tests of conditional predictive ability,” Econometrica, 74,

1545–1578.

GIOT, P., AND S. LAURENT (2004): “Modelling daily Value-at-Risk using realized volatility and ARCH type models,” Journal of Empirical Finance, 11, 379–398.

GLOSTEN, L. R., R. JAGANNATHAN, AND D. E. RUNKLE (1993): “On the Relation Between the Expected Value and the Volatility of the Nominal Excess Return on Stocks,” Journal of Finance, 48, 1779–1801.

GONZALO-RIVERA, G. (1998): “Smooth transition GARCH models,” Studies in Nonlinear Dynamics and Econometrics, 3, 61–78.

GOURIEROUX, C. S., J. JASIAK, AND R. SUFANA (2009): “The Wishart Autoregressive process of multivariate stochastic volatility,” Journal of Econometrics, 150, 167–181.

HAGERUD, G. (1997): “A New Non-Linear GARCH Model,” Ph.D. Thesis, Stockholm School of Economics.

HALBLEIB-CHIRIAC, R. (2007): “Nonstationary Wishart Autoregressive Model,” Working Paper, CoFE, University of Konstanz.

HALBLEIB-CHIRIAC, R., AND V. VOEV (2011): “Modelling and Forecasting Multivariate Realized Volatility,” Journal of Applied Econometrics, 26, 922–947.

HAMILTON, J. D., AND R. SUSMEL (1994): “Autoregressive conditional heteroskedasticity and changes in regime,” Journal of Econometrics, 64, 307–333.

HANOCH, G., AND H. LEVY (1969): “The Efficiency Analysis of Choices Involving Risk,” The Review of Economic Studies, 36, 335–346.

HANSEN, P. R., Z. HUANG, AND H. H. SHEK (2011): “Realized GARCH: A Joint Model for Returns and Realized Measures of volatility,” Journal of Applied Econometrics, 27, 877–906.

HANSEN, P. R., AND A. LUNDE (2004): “An unbiased measure of realized variance,” Unpublished manuscript, Stanford University.

(2005): “A realized variance for the whole day based on intermittent data,” Journal of Financial Econometrics, 3, 525–554.

(2006): “Consistent ranking of volatility models,” Journal of Econometrics, 131(1-2), 97–121.

HANSEN, P. R., A. LUNDE, AND J. M. NASON (2011): “The Model Confidence Set,” Econometrica, 79, 435–497.

HANSEN, P. R., A. LUNDE, AND V. VOEV (2014): “Realized Beta GARCH: A Multivariate GARCH Model With Realized Measures of Volatility,” Journal of Applied Econometrics, 29, 774–799.

HARRIS, L. (1990): “Estimation of stock variance and serial covariance from discrete observations,” Journal of Financial and Quantitative Analysis, 25, 291–306.

(1991): “Stock price clustering and discreteness,” Review of Financial Studies, 4, 389–415.

HARVEY, A., E. RUIZ, AND N. SHEPHARD (1994): “Multivariate stochastic variance models,” Review of Economic Studies, 61, 247–264.

HAUTSCH, N., L. M. KYJ, AND P. MALEC (2015): “Do High-Frequency Data Improve High Dimensional Portfolio Allocations?” Journal of Applied Econometrics, 30(2), 263–290.

HESTON, S. L. (1993): “A closed form solution for options with stochastic volatility with applications to bond and currency options,” The Review of Financial Studies, 6, 327–343.

HIGGINS, M. L., AND A. K. BERA (1992): “A Class of Nonlinear ARCH Models,” International Economic Review, 33, 137–158.

JACOD, J., AND P. PROTTER (1998): “Asymptotic error distributions for the Euler method for stochastic differential equations,” Annals of Probability, 26, 267–307.

JAGANNATHAN, R., AND T. MA (2003): “Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps,” Journal of Finance, 58, 1651–1683.

JIN, X., AND J. M. MAHEU (2012): “Modelling Realized Covariances and Returns,” Journal of Financial Econometrics, 11, 335–369.

KUESTER, K., S. MITTNIK, AND M. S. PAOLELLA (2006): “Value-at-risk prediction: a comparison of alternative strategies,” Journal of Financial Econometrics, 4, 53–89.

KUPIEC, P. (1995): “Techniques for verifying the accuracy of risk measurement models,” Journal of Derivatives, 2, 73–84.

KYJ, L., B. OSTDIEK, AND K. ENSOR (2009): “Realized Covariance Estimation in Dynamic Portfolio Optimization,” Working Paper.

LAURENT, S., J. V. ROMBOUTS, AND F. VIOLANTE (2013): “On Loss Functions and Ranking Fore-casting Performances of Multivariate Volatility Models,” Journal of Econometrics, 173(1), 1–10.

LAURENT, S., AND F. VIOLANTE (2012): “Volatility forecasts evaluation and comparison,” Wiley

Interdisciplinary Reviews: Computational Statistics, 4(1), 1–12.

LOPEZ, J. A. (1998): “Regulatory evaluation of Value-at-Risk models,” Federal Reserve Bank of New York Economic Policy Review, 4(3), 119–124.

MAHEU, J. M., AND T. H. MCCURDY (2002): “Nonlinear features of realized volatility,” Review of Economics and Statistics, 84, 668–681.

MARKOWITZ, H. (1952): “Portfolio selection,” Journal of Finance, 7, 77–91.

MARTENS, M., M. DE POOTER, AND D. J. VAN DIJK (2004): “Modeling and Forecasting S&P 500 Volatility: Long Memory, Structural Breaks and Nonlinearity,” Tinbergen Institute Discussion Paper No. 04-067/4.

MCALEER, M., AND M. MEDEIROS (2008a): “A multiple regime smooth transition Heterogeneous Autoregressive model for long memory and asymmetries,” Journal of Econometrics, 147, 104–119.

(2008b): “Realized Volatility: a Review,” Econometric Reviews, 27, 10–45.

MERTON, R. C. (1980): “On estimating the expected return on the market: An explanatory investigation,” Journal of Financial Economics, 8, 323–361.

MINCER, J. A., AND V. ZARNOWITZ (1969): “The Evaluation of Economic Forecasts,” in Economic Forecasts and Expectations: Analysis of Forecasting Behavior and Performance, NBER Chapters, pp. 3–46. National Bureau of Economic Research, Inc.

NELSON, D. B. (1990): “Stationarity and persistence in the GARCH (1,1) model,” Econometric Theory, 6, 318–334.

NOURELDIN, D., N. SHEPHARD, AND K. SHEPPARD (2011): “Multivariate high-frequency-based volatility (HEAVY) models,” Journal of Applied Econometrics, 27, 907–933.

OOMEN, R. C. (2001): “Using High Frequency Stock Market Index Data to Calculate, Model & Forecast Realized Return Variance,” Working paper.

(2005a): “Properties of bias-corrected realized variance under alternative sampling schemes,” Journal of Financial Econometrics, 3, 555–577.

(2005b): “Properties of realized variance under alternative sampling schemes,” Journal of Business and Economic Statistics, 24, 219–237.

PATTON, A. J. (2011): “Volatility forecast comparison using imperfect volatility proxies,” Journal of Econometrics, 160(1), 246 – 256.

PATTON, A. J., AND K. SHEPPARD (2009): Evaluating Volatility and Correlation Forecasts, Springer Berlin Heidelberg.

PAYE, B. S. (2012): “Dèjà vol: Predictive regressions for aggregate stock market volatility using macroeconomic variables,” Journal of Financial Economics, 106, 527–546.

POON, S.-H., AND C. W. GRANGER (2003): “Forecasting Volatility in Financial Markets: A Review,” Journal of Economic Literature, 41(2), 478–539.

POTERBA, J. M., AND L. H. SUMMERS (1986): “The persistence of volatility and stock market fluctuations,” The American Economic Review, 76, 1142–1151.

PROTTER, P. E. (1992): Stochastic Integration and Differential Equations: A New Approach. New

York: Springer-Verlag.

RICHARDSON, M., AND J. H. STOCK (1989): “Drawing inference from statistics based on multiyear asset returns,” Journal of Financial Economics, 25, 323–348.

ROSS, S. A. (1976): “The arbitrage theory of capital asset pricing,” Journal of Economic Theory, 13, 341–360.

SCHWERT, W. G. (1989): “Why does stock market volatility change over time,” Journal of Finance, 44, 1115–1153.

(1990): “Indexes of U.S. stock prices from 1802 to 1987,” Journal of Business, 63, 399–426.

SHARPE, W. F. (1964): “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk,” Journal of Finance, 19, 425–442.

SHEPHARD, N. (1996): “Statistical aspects of ARCH and stochastic volatility,” in Time Series Models in Econometrics, Finance and Other Fields.

SHEPHARD, N., AND K. SHEPPARD (2010): “Realising the future: forecasting with high frequency based volatility (HEAVY) models,” Journal of Applied Econometrics, 25, 197–231.

SHIRYAEV, A. N. (1999): Essentials of Stochastic Finance: Facts, Models, Theory. World Scientific.

SILVENNOINEN, A., AND T. TERÄSVIRTA (2008): “Multivariate GARCH Models,” Working paper.

SOWELL, F. (1992): “Maximum likelihood estimation of fractionally integrated time series models,” Journal of Econometrics, 53, 165–188.

TAYLOR, S. J. (1986): Modeling Financial Time Series. John Wiley and Sons.

(1994): “Modeling Stochastic Volatility: A Review and Comparative Study,” Mathematical Finance, 4, 183–204.

TAYLOR, S. J., AND X. XU (1997): “The incremental volatility information in one million foreign exchange quotations,” Journal of Empirical Finance, 4, 317–340.

THEIL, H., AND J. C. G. BOOT (1962): “The final form of econometric equation systems,” Review of International Statistical Institute, 30, 136–152.

VAN DER WEIDE, R. (2002): “GO-GARCH: a multivariate generalized orthogonal GARCH model,” Journal of Applied Econometrics, 17, 549–564.

VON NEUMANN, J., AND O. MORGENSTERN (1947): “Theory of Games and Economic Behavior”, Princeton University Press, Princeton, New Jersey, 2a ed.

WEST, K. (1996): “Asymptotic inference about predictive ability,” Econometrica, 64, 1067–1084.

WOLD, S. (1976): “Spline Functions in Data Analysis,” Technometrics, 16(1), 1–11.

ZAKOIAN, J. M. (1994): “Threshold heteroskedastic,” Journal of Economics Dynamics and Control, 18, 931–955.

ZELLNER, A. (1962): “An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests of Aggregation Bias,” Journal of the American Statistical Association, 57(298), 348–368.

ZHANG, L. (2006): “Estimating Covariation: Epps Effect, Microstructure Noise,” Working paper.

ZHANG, L., P. A. MYKLAND, AND Y. AÏT¡SAHALIA (2005): “A tale of two time scales: determining integrated volatility with noisy high frequency data,” Journal of the American Statistical Association, 100, 1394–1411.

ZHOU, B. (1996): “High frequency data and volatility in foreign-exchange rates,” Journal of Business and Economic Statistics, 14, 45–52.