In this paper, we estimate two stochastic volatility models applied to international equity markets. The two models are the log-normal stochastic volatility (SV) model and the two-regime switching model. Then based on the one-day-ahead forecasted volatility from each model, we calculate the Value-at-Risk (VaR) in each market. The estimated VaR measures from the SV are higher than those obtained from the regime-switching model for all markets and over all horizons. The exception is the Japanese market, where the stochastic volatility model generates low VaR estimates. Comparing those estimates with the unconditional return distribution, the two models generate smaller VaR measures; an evidence of the two models capturing volatility changes in international equity markets. Finally, we backtest each model and find that the performance of both models is the worst for the Canadian stock market, while the regime switching model does poorly for Germany. The results have significant implications for risk management, trading and hedging activities as well as in the pricing of equity derivatives.
KeywordsRegime SwitchingStochastic Volatility ModelValue at Risk
Black, F. (1976) Studies of Stock Price Volatility Changes. Proceedings of the 1976 Meetings of the American Statistical Association, Business and Economical Statistics Section, 177-181.
Engle, R., Lilien, D. and Robins, R. (1987) Estimating Time-Varying Risk Premia in the Term Structure. The ARCH-M Model. Econometrica, 55, 391-407. https://doi.org/10.2307/1913242
Andersen, T.G., Bollerslev, T. and Diebold, F.X. (2005) Parametric and Nonparametric Volatility Measurement. In: Hansen, L.P. and Ait-Sahalia, Y., Eds., Handbook of Financial Econometrics, North Holland, Amsterdam, 67-137.
Poon, S.H. and Granger, C.W.J. (2003) Forecasting Volatility in Financial Markets: A Review. Journal of Economic Literature, 41, 478-539. https://doi.org/10.1257/.41.2.478
Bollerslev, T., Chou, R.Y. and Kroner, K. (1992) ARCH Modelling in Finance: A Review of the Theory and Empirical Evidence. Journal of Econometrics, 52, 5-59.
Bera, A.K. and Higgins, M.L. (1993) ARCH Modes: Properties, Estimation and Testing. Journal of Economic Surveys, 7, 305-366. https://doi.org/10.1111/j.1467-6419.1993.tb00170.x
Bollerslev, T., Engle, R.F. and Nelson, D.B. (1994) ARCH Models. In: Engle, R.F. and McFadden, D.L., Eds., Handbook of Econometrics, Vol. 4, Elsevier Science, Amsterdam, 2959-3038.
Taylor, S.J. (1986) Modelling Financial Time Series. Wiley, Chichester.
Ghysels, E., Harvey, A.C. and Renault, E. (1996) Stochastic Volatility. In: Maddala, G.S. and Rao, C.R., Eds., Handbook of Statistics, Vol. 14, Statistical Methods in Finance, North-Holland, Amsterdam, 128-198.
Shephard, N. (1996) Statistical Aspects of ARCH and Stochastic Volatility. In: Cox, D.R., Hinkley, D.V. and Barndorff-Nielsen, O.E., Eds., Time Series Models in Econometrics, Finance and Other Fields, Monographs on Statistics and Applied Probability, Vol. 65, Chapman and Hall, 1-67. https://doi.org/10.1007/978-1-4899-2879-5_1
Broto, C. and Ruiz, E. (2004) Estimation Methods for Stochastic Volatility Models: A Survey. Journal of Economic Surveys, 18, 613-649. https://doi.org/10.1111/j.1467-6419.2004.00232.x
Hull, J. and White, A. (1987) The Pricing of Options on Assets with Stochastic Volatilities. The Journal of Finance, 42, 281-300. https://doi.org/10.1111/j.1540-6261.1987.tb02568.x
Campbell, J.Y., Lo, A.W. and Mackinlay, A.C. (1997) The Econometrics of Financial Markets. Princeton Press.
Koopman, S. and Hol Uspensky, E. (2002) The Stochastic Volatility in Mean Model: Empirical Evidence from International Stock Markets. Journal of Applied Econometrics, 17, 667-689. https://doi.org/10.1002/jae.652
Goldfeld, S.M. and Quandt, R.E. (1973) A Markov Model for Switching Regressions. Journal of Econometrics, 1, 3-16.
Hamilton, J.D. (1989) A New Approach of the Economic Analysis of Non-Sta-tionary Time Series and the Business Cycle. Econometrica, 57, 357-384. https://doi.org/10.2307/1912559
LeBaron, B. (1992) Some Relationships between Volatility and Serial Correlations in Stock Market Returns. Journal of Business, 65, 199-219. https://doi.org/10.1086/296565
Timmerman, A. (2000) Moments of Markov Switching Models. Journal of Econometrics, 96, 75-111.
Ang, A. and Bekaert, G. (2002) International Asset Allocation with Regime Shifts. Review of Financial Studies, 15, 1137-1187. https://doi.org/10.1093/rfs/15.4.1137
Jorion, P. (1988) On Jump Processes in the Foreign Exchange and Stock Markets. Review of Financial Studies, 1, 427-445. https://doi.org/10.1093/rfs/1.4.427
Akgiray, V. and Booth, G. (1988) Mixed Diffusion-Jump Process Modelling of Exchange Rate Movements. Review of Economics and Statistics Studies, 70, 631-637. https://doi.org/10.2307/1935826
Bates, D. (1996) Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options. Review of Financial Studies, 9, 69-107. https://doi.org/10.1093/rfs/9.1.69
Bekaert, G, Erb, C., Harvey, C. and Viskanta, T. (1998) Distributional Characteristics of Emerging Market Returns and Asset Allocation. Journal of Portfolio Management, 24, 102-116. https://doi.org/10.3905/jpm.24.2.102
Asgharian, H. and Bengtsson, C. (2006) Jump Spillover in International Equity Markets. Journal of Financial Econometrics, 2, 167-203. https://doi.org/10.1093/jjfinec/nbj005
Ang, A. and Chen, J. (2002) Asymmetric Correlations of Equity Portfolios. Journal of Financial Economics, 63, 443-494.
Longin, F. and Solnik, B. (1995) Is the Correlation in International Equity Returns Constant? Journal of International Money and Finance, 14, 3-26.
Karolyi, A. and Stulz, R. (1996) Why Do Markets Move Together? An Investigation of U.S.-Japan Stock Return Movements. Journal of Finance, 51, 951-986. https://doi.org/10.1111/j.1540-6261.1996.tb02713.x
Chakrabarti, R. and Roll, R. (2000) East Asian and Europe during the 1997 Asian Collapse: A Clinical Study of a Financial Crisis. Working Paper, UCLA.
Chollete, L., de la Pena, V. and Lu, C.-C. (2011) International Diversification: A Copula Approach. Journal of Banking and Finance, 35, 403-417.
Buraschi, A., Porchia, P. and Trojani, F. (2010) Correlation Risk and Optimal Portfolio Choice. Journal of Finance, 65, 393-420. https://doi.org/10.1111/j.1540-6261.2009.01533.x
Kuester, K., Mittnik, S. and Paolella, M. (2006) Value-at-Risk Prediction: A Comparison of Alternative Strategies. Journal of Financial Econometrics, 1, 53-89.
Engle, R. (1982) Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation. Econometrica, 50, 987-1008. https://doi.org/10.2307/1912773
Andersen, T.G., Bollerslev, T., Diebold, F.X. and Labys, P. (2001) The Distribution of Realized Exchange Rate Volatility. Journal of the American Statistical Association, 96, 42-55. https://doi.org/10.1198/016214501750332965
Andersen, T.G., Bollerslev, T., Diebold, F.X. and Labys, P. (2003) Modeling and Forecasting Realized Volatility. Econometrica, 71, 579-626. https://doi.org/10.1111/1468-0262.00418
Melino, A. and Turnbull, S.M. (1990) Pricing Foreign Currency Options with Stochastic Volatility. Journal of Econometrics, 45, 239-265.
Harvey, A.C., Ruiz, E. and Shephard, N. (1994) Multivariate Stochastic Variance Models. Review of Economic Studies, 61, 247-264. https://doi.org/10.2307/2297980
Ruiz, E. (1994) Quasi-Maximum Likelihood Estimation of Stochastic Volatility Models. Journal of Econometrics, 63, 289-306.
Gallant, A.R., Hsieh, D.A. and Tauchen, G.E. (1997) Estimation of Stochastic Volatility Models with Diagnostics. Journal of Econometrics, 81, 159-192.
Jacquier, E., Polson, N.G. and Rossi, P.E. (1994) Bayesian Analysis of Stochastic Volatility Models (with Discussion) Journal of Business and Economics Statistics, 12, 371-389.
Kim, S., Shephard, N. and Chib, S. (1998) Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models. Review of Economics Studies, 65, 361-393. https://doi.org/10.1111/1467-937X.00050
Shephard, N. and Pitt, M. (1997) Likelihood Analysis of Non-Gaussian Measurement Time Series. Biometrika, 84, 653-667. https://doi.org/10.1093/biomet/84.3.653
Durbin, J. and Koopman, S. (1997) Monte Carlo Maximum Likelihood Estimation for Non-Gaussian State Space Models. Biometrika, 84, 669-684. https://doi.org/10.1093/biomet/84.3.669
Ripley, B. (1987) Stochastic Simulation. Wiley, New York. https://doi.org/10.1002/9780470316726
De Jong, P. and Shephard, N. (1995) The Simulation Smoother for Time Series Models. Biometrika, 82, 339-350. https://doi.org/10.1093/biomet/82.2.339
Doornik, J. (1998) Object-Oriented Matrix Programming Using Ox 2.0. Timberlake Consultants Ltd., London. http://www.nuff.ox.ac.uk/Users/Doornik
Koopman, S., Shephard, N. and Doornik, J. (1999) Statistical Algorithms for Models in State Space Using Ssfpack 2.2. Econometrics Journal, 2, 113-166. http://www.ssfpack.com https://doi.org/10.1111/1368-423X.00023
French, K.R., Schwert, G.W. and Stanbaugh, R.F. (1987) Expected Stock Returns and Volatility. Journal of Financial Economic, 19, 3-29.
Poon, S. and Taylor, S.J. (1992) Stock Returns and Volatility: An Empirical Study of the UK Stock Market. Journal of Banking and Finance, 16, 37-59.
Kim, D. and Kon, S. (1994) Alternative Models for the Conditional Heteroscedasticity of Stock Returns. Journal of Business, 67, 563-598. https://doi.org/10.1086/296647
Lamoureux, C.G. and Lastrapes, W.D. (1990) Persistence in Variance, Structural Change and the GARCH Model. Journal of Business and Economic Statistics, 8, 225-243. https://doi.org/10.1080/07350015.1990.10509794
Hamilton, J.D. and Susmel, R. (1994) Autoregressive Conditional Heteroskedasticity and Changes in Regime. Journal of Econometrics, 64, 307-333.
Hansen, B. (1992) The Likelihood Ratio Test under Nonstandard Conditions: Testing the Markov Switching Model of GNP. Journal of Applied Econometrics, 7, S61-S82. https://doi.org/10.1002/jae.3950070506
Kobayashi, M. and Shi, X. (2003) Testing for EGARCH against Stochastic Volatility Models. Journal of Time Series Analysis, 26, 135-150. https://doi.org/10.1111/j.1467-9892.2005.00394.x
Barone-Adesi, G., Burgoin, F. and Giannopoulos, K. (1998) Don’t Look Back. Risk, 11, 100-104.
Christoffersen, P. (2003) Elements of Financial Risk Management, Academic Press, San Diego.
Dowd, K. (1998) Beyond Value at Risk: The New Science of Risk Management. Wiley, New York.
Prisker, M. (2006) The Hidden Dangers of Historical Simulation. Journal of Banking and Finance, 30, 561-582.
Christoffersen, P. (1998) Evaluating Interval Forecasts. International Economic Review, 39, 841-862. https://doi.org/10.2307/2527341
Christoffersen, P. and Diebold, F. (2000) How Relevant Is Volatility Forecasting for Financial Risk Management? Review of Economics and Statistics, 82, 12-22. https://doi.org/10.1162/003465300558597
Christoffersen, P. and Pelletier, D. (2003) Backtesting Portfolio Risk Measures: A Duration-Based Approach, Manuscript, McGill University and CIRANO.
McNeil, A. and Frey, R. (2000) Estimation of Tail-Related Risk Measures for Heteroskedastic Financial Time Series: An Extreme Value Approach. Journal of Empirical Finance, 7, 271-300.
Diebold, F.X., Gunther, T. and Tsay, A. (1998) Evaluating Density Forecasts, with Applications to Financial Risk Management. International Economic Review, 39, 863-883. https://doi.org/10.2307/2527342
Diebold, F.X., Hahn, J. and Tsay, A. (1999) Multivariate Density Forecasts Evaluation and Calibration in Financial Risk Management: High Frequency Returns on Foreign Exchange. Review of Economics and Statistics, 81, 661-673. https://doi.org/10.1162/003465399558526