Crisis, Value at Risk and Conditional Extreme Value Theory via the NIG + Jump Model

Affiliation(s)

Department of Finance and Decision Sciences, Hong Kong Baptist University, Hong Kong, China.

Department of Finance and Decision Sciences, Hong Kong Baptist University, Hong Kong, China.

ABSTRACT

This study develops a new conditional extreme value theory-based model (EVT) combined with the NIG + Jump model to forecast extreme risks. This paper utilizes the NIG + Jump model to asymmetrically feedback the past realization of jump innovation to the future volatility of the return distribution and uses the EVT to model the tail distribution of the NIG + Jump-processed residuals. The model is compared to the GARCH-t model and NIG + Jump model to evaluate its performance in estimating extreme losses in three major market crashes and crises. The results show that the conditional EVT-NIG + Jump model outperforms the GARCH and GARCH-t models in depicting the non-normality and in pro- viding accurate VaR forecasts in the in-sample and out-sample tests. The EVT-NIG + Jump model, which can measure the volatility of extreme price movement in capital markets due to unexpected events, enhances the EVT-based model for measuring the tail risk.

This study develops a new conditional extreme value theory-based model (EVT) combined with the NIG + Jump model to forecast extreme risks. This paper utilizes the NIG + Jump model to asymmetrically feedback the past realization of jump innovation to the future volatility of the return distribution and uses the EVT to model the tail distribution of the NIG + Jump-processed residuals. The model is compared to the GARCH-t model and NIG + Jump model to evaluate its performance in estimating extreme losses in three major market crashes and crises. The results show that the conditional EVT-NIG + Jump model outperforms the GARCH and GARCH-t models in depicting the non-normality and in pro- viding accurate VaR forecasts in the in-sample and out-sample tests. The EVT-NIG + Jump model, which can measure the volatility of extreme price movement in capital markets due to unexpected events, enhances the EVT-based model for measuring the tail risk.

Cite this paper

S. Ze-To, "Crisis, Value at Risk and Conditional Extreme Value Theory via the NIG + Jump Model,"*Journal of Mathematical Finance*, Vol. 2 No. 3, 2012, pp. 225-237. doi: 10.4236/jmf.2012.23025.

S. Ze-To, "Crisis, Value at Risk and Conditional Extreme Value Theory via the NIG + Jump Model,"

References

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[23] J. M. Mahe, and T. H. McCurdy, “News Arrival, Jump Dynamics, and Volatility Components for Individual Stock Returns,” Journal of Finance, Vol. 59, No. 2, 2004, pp. 755-793. doi:10.1111/j.1540-6261.2004.00648.x

[24] A. C. Davison and R. L. Smith, “Models for Exceedences over High Thresholds,” Journal of the Royal Statistical Society, Vol. B52, No. 3, 1990, pp. 393-442.

[25] M. R. Leadbetter, “On a Basis for Peaks over Thresholds Modeling,” Statistics and Probability Letters, Vol. 12, No. 4, 1991, pp. 357-362.

[26] R. Smith, “Extreme Value Analysis of Environmental Time Series: An Application to Trend Detection in Ground- Level Ozone,” Statistical Science, Vol. 4, No. 4, 1989, pp. 367-393. doi:10.1214/ss/1177012400

[27] A. Balkema and L. de Haan, “Residual Life Time at Great Age,” Annals of Probability, Vol. 2, No. 5, 1974, pp. 792- 804. doi:10.1214/aop/1176996548

[28] J. Pickands, “Statistical Inference Using Extreme Order Statistics,” The Annals of Statistics, Vol. 3, No. 1, 1975, pp. 119-131. doi:10.1214/aos/1176343003

[29] C. H. Lin, and S. S. Shen, “Can the Student-t Distribution Provide Accurate Value at Risk?” The Journal of Risk Finance, Vol. 7, No. 3, 2006, pp. 292-300. doi:10.1108/15265940610664960

[30] M. Y. L. Li and H. W. W. Lin, “Estimating Value-at-Risk via Markov Switching ARCH Models: An Empirical Study on Stock Index Returns,” Applied Economics Letters, Vol. 11, 2004, pp. 679-691.

[1] P. Artzner, F. Delbaen, J. Eber and D. Heath, “Thinking Coherently,” Risk, Vol. 10, No. 11, 1997, pp. 68-71.

[2] P. Artzner, F. Delbaen, J. Eber and D. Heath, “Coherent Measures of Risk,” Mathematical Finance, Vol. 9, No. 3, 1999, pp. 203-228. doi:10.1111/1467-9965.00068

[3] T. G. Bali and Theodossiou, P, “A Conditional-SGT-VaR Approach with Alternative GARCH Models,” Annals of Operations Research, Forthcoming, Vol. 151, No. 1, 2005, pp. 241-267.

[4] F. M. Longin, “Optimal Margins Level in Future Markets: A Parametric Extreme-Based Method,” Journal of Futures Markets, Vol. 19, No. 2, 1999, pp. 127-152.

[5] S. C. Coles, “An introduction to statistical modeling of extreme values,” Springer, London, New York, 2001.

[6] F. M. Longin, “From Value at Risk to Stress Testing: The Extreme Value Approach,” Journal of Banking & Finance, Vol. 24, No. 7, 24, 2000, pp. 1097-1130. doi:10.1016/S0378-4266(99)00077-1

[7] J. Cotter, “Downside Risk for European Equity Markets,” Applied Financial Economics, Vol. 14, No. 10, 2004, pp. 707-716. doi:10.1080/0960310042000243547

[8] T. G. Bali, “A Generalized Extreme Value Approach to Financial Risk Measurement,” Journal of Money, Credit and Banking, Vol. 39, No. 7, 2006, pp. 1613-1649.

[9] T. G. Bali, “An Extreme Value Approach to Estimating Volatility and Value-at-Risk,” Journal of Business, Vol. 76, No. 1, 2003, pp. 83-108. doi:10.1086/344669

[10] T. G. Bali and S. N. Neftci, “Disturbing Extremal Behavior of Spot Rate Dynamics,” Journal of Empirical Finance, Vol. 10, No. 4, 2003, pp. 455-477. doi:10.1016/S0927-5398(02)00070-1

[11] J. Beirlant, J. Teugels and P. Vynckier, “Practical Analysis of Extreme Values,” Leuven University Press, Leuven. 1996.

[12] P. Embrechts, C. Klüppelberg and T. Mikosch, “Modelling Extremal Events for Insurance and Finance,” Springer- Veriag, New York, 1997.

[13] R. Reiss and M. Thomas, “Statistical Analysis of Extreme Values: From Insurance, Finance, Hydrology, and Other Fields,” Birkh?user Verlag, Basel, Boston, 2001.

[14] H. N. E. Bystr?m, “Managing Extreme Risks in Tranquil and Volatile Markets Using Conditional Extreme Value Theory,” International Review of Financial Analysis, Vol. 13, No. 2, 2004, pp. 133-152. doi:10.1016/j.irfa.2004.02.003

[15] R. Smith, “Measuring Risk with Extreme Value Theory,” Extremes and Integrated Risk Management, Vol. 2, 1996, pp. 19-36.

[16] A. J. McNeil and Frey, R. “Estimation of Tail-Related Risk Measures for Heteroscedastic Financial Time Series: An Extreme Value Approach,” Journal of Empirical Finance, Vol. 7, No. 3-4, 2000, pp. 271-300. doi:10.1016/S0927-5398(00)00012-8

[17] T. G. Bali and Weinbaum, D. “A Conditional Extreme Value Volatility Estimator Based on High-Frequency Returns,” Journal of Economic Dynamics and Control, Vol. 31, No. 2, 2007, pp. 361-397.

[18] C. Brooks, A. D. Clare, J. W. D. Molle and G. Persand, “A Comparison of Extreme Value Theory Approaches for Determining Value at Risk,” Journal of Empirical Finance, Vol. 12, No. 2, 2005, pp. 339-352. doi:10.1016/j.jempfin.2004.01.004

[19] R. F. Engle, “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of UK,” Econometrica, Vol. 50, No. 4, 1982, pp. 987-1008. doi:10.2307/1912773

[20] T. P. Bollerslev, “Generalized Autoregressive Conditional Heteroscedasticity,” Journal of Econometrics, Vol. 31, No. 3, 1986, pp. 309-328. doi:10.1016/0304-4076(86)90063-1

[21] S. Taylor, “Modelling Financial Time Series,” In: H. Akaike, Ed., A New Look at the Statistical Model Identification, John Wiley & Sons, New York, 1986.

[22] R. Susmel, “Switching Volatility in Private International Equity Markets,” International Journal of Finance & Economics, Vol. 5, No. 4, 2000, pp. 265-283. doi:10.1002/1099-1158(200010)5:4<265::AID-IJFE132>3.0.CO;2-H

[23] J. M. Mahe, and T. H. McCurdy, “News Arrival, Jump Dynamics, and Volatility Components for Individual Stock Returns,” Journal of Finance, Vol. 59, No. 2, 2004, pp. 755-793. doi:10.1111/j.1540-6261.2004.00648.x

[24] A. C. Davison and R. L. Smith, “Models for Exceedences over High Thresholds,” Journal of the Royal Statistical Society, Vol. B52, No. 3, 1990, pp. 393-442.

[25] M. R. Leadbetter, “On a Basis for Peaks over Thresholds Modeling,” Statistics and Probability Letters, Vol. 12, No. 4, 1991, pp. 357-362.

[26] R. Smith, “Extreme Value Analysis of Environmental Time Series: An Application to Trend Detection in Ground- Level Ozone,” Statistical Science, Vol. 4, No. 4, 1989, pp. 367-393. doi:10.1214/ss/1177012400

[27] A. Balkema and L. de Haan, “Residual Life Time at Great Age,” Annals of Probability, Vol. 2, No. 5, 1974, pp. 792- 804. doi:10.1214/aop/1176996548

[28] J. Pickands, “Statistical Inference Using Extreme Order Statistics,” The Annals of Statistics, Vol. 3, No. 1, 1975, pp. 119-131. doi:10.1214/aos/1176343003

[29] C. H. Lin, and S. S. Shen, “Can the Student-t Distribution Provide Accurate Value at Risk?” The Journal of Risk Finance, Vol. 7, No. 3, 2006, pp. 292-300. doi:10.1108/15265940610664960

[30] M. Y. L. Li and H. W. W. Lin, “Estimating Value-at-Risk via Markov Switching ARCH Models: An Empirical Study on Stock Index Returns,” Applied Economics Letters, Vol. 11, 2004, pp. 679-691.