Estimation of Short- and Long-Term VaR for Long-Memory Stochastic Volatility Models

  • Hwai-Chung Ho
  • Fang-I Liu


The phenomenon of long-memory stochastic volatility (LMSV) has been extensively documented for speculative returns. This research investigates the effect of LMSV for estimating the value at risk (VaR) or the quantile of returns. The proposed model allows the return’s volatility component to be short- or long-memory. We derive various types of limit theorems that can be used to construct confidence intervals of VaR for both short-term and long-term returns. For the latter case, the results are in particular of interest to financial institutions with exposure of long-term liabilities, such as pension funds and life insurance companies, which need a quantitative methodology to control market risk over longer horizons.


Long-memory stochastic volatility model Stochastic volatility model Value-at-risk Quantile 



The research is supported by NSC grant: NSC 97-2118-M-001-002-MY3.


  1. Adenstedt, R. K. 1974. “On large-sample estimation for the mean of a stationary sequence.” Annals of Statistics 2, 1095–1107.CrossRefGoogle Scholar
  2. Bollerslev, T. 1986. “Generalized autoregressive conditional heteroscedasticity.” Journal of Econometrics 31, 307–327.CrossRefGoogle Scholar
  3. Bollerslev, T. and J. H. Wright. 2000. “Semiparametric estimation of long-memory volatility dependencies: the role of high-frequency data.” Journal of Econometrics 98, 81–106.CrossRefGoogle Scholar
  4. Breidt, F. J., N. Crato, and P. De Lima. 1998. “The detection and estimation of long memory in stochastic volatility.” Journal of Econometrics 83, 325–348.CrossRefGoogle Scholar
  5. Brockwell, P. J. and R. A. Davis. 1991. Time series: theory and methods, Springer, New York.CrossRefGoogle Scholar
  6. Deo, R. S. and C. M. Hurvich. 2001. “On the log periodogram regression estimator of the memory parameter in long memory stochastic volatility models.” Econometric Theory 17, 686–710.CrossRefGoogle Scholar
  7. Ding, Z. and C. Granger. 1996. “Modeling volatility persistence of speculative returns: a new approach.” Journal of Econometrics 73, 185–215.CrossRefGoogle Scholar
  8. Ding, Z., C. W. J. Granger, and R. F. Engle. 1993. “A long memory property of stock market returns and a new model.” Journal of Empirical Finance 1, 83–106.CrossRefGoogle Scholar
  9. Dowd, K. 2001. “Estimating VaR with order statistics.” Journal of Derivatives 8, 23–70.CrossRefGoogle Scholar
  10. Engle, R. F. 1982. “Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflations.” Econometrica 50, 987–1007.CrossRefGoogle Scholar
  11. Granger, C. W. J. and R. Joyeux. 1980. “An introduction to long-memory time series models and fractional differencing.” Journal of Time Series Analysis 1, 15–29.CrossRefGoogle Scholar
  12. Hall, P., B.-Y. Jing, and S. N. Lahiri. 1998. “On the sampling window method for long-range dependent data.” Statistica Sinica 8, 1189–1204.Google Scholar
  13. Harvey, A. C., E. Ruiz, and N. Shephard. 1994. “Multivariate stochastic variance models.” Review of Economic Studies 61, 247–264.CrossRefGoogle Scholar
  14. Hjort, N. L. and D. Pollard. 1993. Asymptotics for minimisers of convex processes, Statistical Research Report, University of Oslo.Google Scholar
  15. Ho, H. C. 2006. “Estimation errors of the Sharpe ratio for long-memory stochastic volatility models,” in Time series and related topics, IMS Lecture Notes-Monograph Series 52, pp. 165–172.Google Scholar
  16. Ho, H. C. and T. Hsing. 1996. “On the asymptotic expansion of the empirical process of long-memory moving averages.” The Annals of Statistics 24, 922–1024.Google Scholar
  17. Ho, H. C. and T. Hsing. 1997. “Limit theorems for functional of moving averages.” The Annals of Probability 25, 1636–1669.CrossRefGoogle Scholar
  18. Ho, H. C. and F. I. Liu. 2008. Sample quantile analysis on long-memory stochastic volatility models and its applications in estimating VaR, Working paper.Google Scholar
  19. Ho, H. C., S. S. Yang, and F. I. Liu. 2008. A stochastic volatility model for reserving long-duration equity-linked insurance: long-memory, Working paper.Google Scholar
  20. Hosking, J. R. M. 1981. “Fractional differencing.” Biometrika 68, 165–176.CrossRefGoogle Scholar
  21. Koenker, R. 2005. Quantile regression, Cambridge University Press, New York.CrossRefGoogle Scholar
  22. Koenker, R. and G. Bassett. 1978. “Regression quantiles.” Econometrica 46, 33–50.CrossRefGoogle Scholar
  23. Lobato, I. N. and N. E. Savin. 1998. “Real and spurious long-memory properties of stock market data.” Journal of Business & Economic Statistics 16, 261–268.Google Scholar
  24. Mandelbrot, B. and J. W. van Ness. 1968. “Fractional Brownian motions, fractional noises and applications.” SIAM Review 10, 422–437.CrossRefGoogle Scholar
  25. Mosteller, F. 1946. “On some useful “inefficient” statistics.” Annals of Mathematical Statistics 17, 377–408.CrossRefGoogle Scholar
  26. Nelson, D. B. 1991. “Conditional heteroscedasticity in asset returns: a new approach.” Econometrica 59, 347–370.CrossRefGoogle Scholar
  27. Nordman, D. J. and S. N. Lahiri. 2005. “Validity of the sampling window method for long-range dependent linear processes.” Econometric Theory 21, 1087–1111.CrossRefGoogle Scholar
  28. Ray, B. K. and R. S. Tsay. 2000. “Long range dependence in daily stock volatilities.” Journal of Business & Economic Statistics 18(2), 254–262.Google Scholar
  29. Taylor, S. 1986. Modelling financial time series, Wiley, New York.Google Scholar
  30. Wishart, J. 1947. “The cumulants of the Z and of the logarithmic χ2 and t distributions.” Biometrika 34, 170–178.Google Scholar
  31. Yoshihara, K. 1995. “The Bahadur representation of sample quantiles for sequences of strongly mixing random variables.” Statistics and Probability Letters 24, 299–304.CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Institute of Statistical Science, Academia SinicaTaipeiROC
  2. 2.Department of FinanceNational Taiwan UniversityTaipeiROC

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