极值理论在期货风险管理中的应用
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摘要
本文以极值理论为基础,探讨期货的极端价格行为,并以CARR模型描述期货日内价格的趋势,且与McNeil和Frey(2000)所提出的条件极值理论比较。本文提出两阶段法结合极值理论与CARR模型,以S&P500指数期货及轻质原油期货为样本,通过数据实证指出,无论在样本内回溯测试,或是样本外预期损失率估计,都优于McNeil和Frey(2000)的条件极值理论。
同时,本文采用大贩证券交易所(OSE)推出日经225股指期货与新加坡国际金融交易所(SIMEX)推出的日本的日经225指数期货为样本,来分析以极值理论为基础,结合CARR模型是否能避免期货价格涨跌幅限制的影响。实证结果发现,McNeil和Frey(2000)所采用的GARCH模型与极值理论常会估计出超过涨跌幅限制的价格变化,较为不合理。相对而言,极值理论结合CARR模型所估计的价格变化较为合理。此结果支持了极值理论结合CARR模型在期货保证金设定方面应用的优越性。
最后,基于上述的结果,本文建议期货交易所在设定保证金时应该考虑期货日内变幅信息,并采用极值理论结合CARR模型以捕捉实时的价格信息,来制定动态保证金制度,以实时反应当下的风险变化。
The article discusses future's extreme price behavior and uses the range-based CARR model to estimate the intra-day'price's heteroskedasticity. It also compared with the McNeil &Frey's (2000) result which combines with the GARCH model and extreme theory. S&P 500Stock Index Futures traded on CME, and Sweet Crude Oil Futures traded on NYMEX are used. Both the in-sample back- testing and out-of-sample expect ed loss indicate that the CARR model and extreme value theory performs better than McNeil & Frey (2000) conditional extreme value theory.
Furthermore, we use Osaka Stock Exchange (OSE) launched the Nikkei 225 stock index futures and the Singapore International Monetary Exchange (SIMEX) launched Japan's Nikkei 225 index futures as a sample to analyze the extreme value theory, combined with CARR model is able to avoid the futures price impact of price limits. Empirical results, McNeil and Frey (2000) used GARCH model and estimate the extreme value theory often exceed the price limit of price changes, the more unreasonable. In contrast, extreme value theory combined CARR model is more reasonable estimate of price changes. The results support the CARR model with extreme value theory to set the application in the futures margin superiority.
Finally, based on the above results, this article suggests the futures should be considered when setting bond futures days of amplitude information and the use of extreme value theory combined CARR model to capture real-time price information, to develop dynamic margin system, the immediate reaction in real time changes in risk.
同时,本文采用大贩证券交易所(OSE)推出日经225股指期货与新加坡国际金融交易所(SIMEX)推出的日本的日经225指数期货为样本,来分析以极值理论为基础,结合CARR模型是否能避免期货价格涨跌幅限制的影响。实证结果发现,McNeil和Frey(2000)所采用的GARCH模型与极值理论常会估计出超过涨跌幅限制的价格变化,较为不合理。相对而言,极值理论结合CARR模型所估计的价格变化较为合理。此结果支持了极值理论结合CARR模型在期货保证金设定方面应用的优越性。
最后,基于上述的结果,本文建议期货交易所在设定保证金时应该考虑期货日内变幅信息,并采用极值理论结合CARR模型以捕捉实时的价格信息,来制定动态保证金制度,以实时反应当下的风险变化。
The article discusses future's extreme price behavior and uses the range-based CARR model to estimate the intra-day'price's heteroskedasticity. It also compared with the McNeil &Frey's (2000) result which combines with the GARCH model and extreme theory. S&P 500Stock Index Futures traded on CME, and Sweet Crude Oil Futures traded on NYMEX are used. Both the in-sample back- testing and out-of-sample expect ed loss indicate that the CARR model and extreme value theory performs better than McNeil & Frey (2000) conditional extreme value theory.
Furthermore, we use Osaka Stock Exchange (OSE) launched the Nikkei 225 stock index futures and the Singapore International Monetary Exchange (SIMEX) launched Japan's Nikkei 225 index futures as a sample to analyze the extreme value theory, combined with CARR model is able to avoid the futures price impact of price limits. Empirical results, McNeil and Frey (2000) used GARCH model and estimate the extreme value theory often exceed the price limit of price changes, the more unreasonable. In contrast, extreme value theory combined CARR model is more reasonable estimate of price changes. The results support the CARR model with extreme value theory to set the application in the futures margin superiority.
Finally, based on the above results, this article suggests the futures should be considered when setting bond futures days of amplitude information and the use of extreme value theory combined CARR model to capture real-time price information, to develop dynamic margin system, the immediate reaction in real time changes in risk.
引文
3 Figlewski, S., 1984, Margins and Market Integrity:Margin Setting for Stock Index Futures and Options, Journal of Futures Markets,4,385-416
4 Brennan, M.J.,1986, A Theory of Price Limits in Futures Markets. Journal of Finance Economics.16,213-233. Markets,19,127-152.
5 Longin, F., 1999, Optimal Margin Levels in Futures Markets: Extreme Price Movements, Journal of Futures
6 Alexander J. McNeil, Rudiger Frey, Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach, Journal of Empirical Finance,Volume 7, Issues 3-4, November 2000, Pages 271-300
7 Cotter, J.,2001, Margin Excessdences for European Stock Index Futures Using Extreme Value Theory, Journal of Banking and Finance, 25, 1474-1502.
8 Parkinson, M.,1980, The Extreme Value Method for Estimating the Variance of the Rate of Return, Journal of Business,53,61-65
9 Fenn, G., and P. Kupiec,1993. Prudential Margin Policy in a Future Style Settlements System, Journal of Futures Markets.13.389-408.
10 Figlewski, S.,1984, Margins and Market Integrity: Margin Setting for Stock Index Futures and Options, Journal of Futures Markets, 4, 385-416.
" Gay, G.D., W.C. Hunter, and R.W. Kolb, 1986, A Comparative Analysis of Futures Contract Margins, Journal of Futures Markets,6,307-324.
12 Kofman, P., 1993, Optimizing Futures Margins with Distribution Tails, Advances in Futures and Options Research,6,263-278.
13 Booth, G.G., J. P. Broussard, T. Martikainen, and V. Puttonen, 1997, Prudent Margin Levels in the Finnish Stock Index Futures Markets, Management Science,43,1177-1188.
14 Broussard, J. P., and G.G. Booth, 1998, The Behavior of Extreme Values in German Stock Index Futures:An Application to Margin Setting, Journal of Operational Research, 104, 393-402.
15 Greenward, B.C. and J.C. Stein,1991, Transaction Risk, Market Crashes, and the Role of Circuit Breakers, Journal of Business,64,443-462.
16 Arak, M and R.E. Cook, 1997, Do Daily Price Limits Act as Magnets The case of Treasury Bond Futures, Journal of Financial Services Research, 12,5-20.
17 Ma, C.K. R.P. Rao, and R.S. Sears, 1989, Volatility, Prices Resolution, and the Effectiveness of Price Limits, Journal of Financial Services Research,3,165-199.
18 Moser, J.T., 1990, Circuit Breakers, Economic Perspectives, Federal Reserve Bank of Chicago,14,2-13.
19 Ackert, L. and W. Hunter, 1994, Tests of a Simple Optimizing Model of Daily Price Limits on Futures Contracts, Review of Financial Economics,4,93-108.
20 Fama, E.F.,1989, Perspectives on October 1987, or What Did We Learn from the Crash in Black Monday and the futures of financial markets,Dow Jones-lrwin, Inc., Homewood, IL, 71-82.
21 Kim, K.A. and S.G. Rhee,1997, Price Limits Performance: Evidence from the Tokyo Stock Exchange, Journal of Finance, 52,885-901.
22 Jeremy Berkowitz,James O'Brien.How Accurate Are Value-at-Risk Models at Commercial Banks?,The Journal of Finance, Volume 57, Issue 3, pages 1093-1111, June 2002
3 Koenker. R. and G. S. Bassett, 1978, Regression Quantiles. Econometrica, 46, pp.33-50.
24 Brandt, M.W., and C.S. Jones (2002), "Volatility Forecasting with Ranged-Based EGARCH Models," working paper, University of Pennsylvania, USA.
25 Alizadeh, S., M. Brandt, and F. Diebold, (2002)," Range-Based Estimation of Stochastic Volatility Models," Journal of Finance, Vol.57, pp.1047-1091.
[1]Alizadeh, S., M. Brandt, and F. Diebold, (2002), Range-Based Estimation of Stochastic Volatility Models, Journal of Finance, Vol.57, pp.1047-1091.
[2]Artzner, P., Delbaen, J.Eber, and D.Heath(1997), Thinking Coherently, Risk, Vol.10, Issue 11, pp.68-71.
[3]Artzner, P., Delbaen, J.Eber, and D.Heath (1999), Conerent Measures of Risk, Mathematical Finance, Vol.9, Issue 3, pp.203-228.
[5]Balkema,A.A. and de Haan, L. (1974), Residual Lifetime at Great Age, Annals of Probability, Vol.2, pp.792-804.
[6]Bates, D. and R. Craine (1999), Valuing the Futures Market Clearinghouse's Default Exposure During the 1987 Crash, Journal of Money, Credit and Banking, Vol.31, pp.248-272.
[7]Bollerslev, T., (1986), Generalized Autoregressive Conditional Heteroskedasticity, Journal of Econometrica, Vol.31, pp.3037-327.
[8]Bollerslev, T., R. Chou, and K. Kroner (1992), ARCH Modeling in Finance:A Review of the Theory and Empirical Evidence, Journal of Econometrics. Vol.52. pp.5-59.
[9]Brandt, M.W., and C.S. Jones (2002), Volatility Forecasting with Ranged-Based EGARCH Models, working paper, University of Pennsylvania, USA.
[10]Broussard, J. P. (2001), Extreme-Value and Margin Setting with and without Price Limits, The Quarterly Review of Economics and Finance, Vol.41, pp.365-385.
[11]Chou R.Y. (2005), Forecasting Financial Volatilities with Extreme Values:the Conditional Autoregressive Range (CARR) Model, Journal of Money,Credit and Banking, Vol.37, Issue 3,pp. 561-582.
[12]Cotter, J. (2001), Margin Exceedences for European Stock Index Futures Using Extreme Value Theory, Journal of Banking & Finance, Vol.25, pp.1474-1502.
[13]De Haan, L. and S. I. Resnick (1980), "A Simple Asymptotic Estimate for the Index of a Stable Distribution," Journal of the Royal Stat. Soc. B, Vol.42, pp.83-87.
[14]Danielsson, J. and C.G. de Vries (1997), Tail Index and Quantile Estimation with Very High Frequency Data, Journal of Empirical Finance, Vol.4, pp.241-257.
[15]Engle, R., and J. Russell (1998) Autoregressive Conditional Duration:A New Model for Irregular Spaced Transaction Data, Econometrica.Vol.66, pp.1127-1162.
[16]Figlewski, S. (1984), Margin and Market Integrity:Margin Setting for Stock Index Futures and Options, The Journal of Future Markets, Vol.43, pp.385-416.
[17]Hahn, M. and D.C. Weiner(1991), On Joint Estimation of an Exponent of Regular Variation and an Asymmetry Parameter for Tail Distributions,Sums, Trimmed Sums, and Extremes," Progress in Probability, Vol.30, pp.82-111.
[18]Hall, P. (1990), Using the Bootstrap to Estimate Mean Squared Error and Select Smoothing Parameter in Nonparametric Problems, Journal of Multivariate Analysis, Vol.32, pp.177-203.
[19]Hols, M.C. and G. de Vries(1991), The Limiting Distribution of Extreme Exchange Rate Return, Journal of Applied econometrics, Vol.6, pp.287-302
[20]Longin, F. (1999), Optimal Margin Levels in Futures Markets:Extreme Price Movements, Journal of Futures Markets, Vol.19, pp.127-152.
[21]Loretan, M. and P. C.B. Philips (1994), Testing the Covariance Stationarity of Heavy-tailed Time Series, Journal of Empirical Finance,Vol.1, pp.211-248.
[22]McNeil, A.J. (1997), Estimating the Tails of Loss Severity Distributions Using Extreme Value Theory, ASTIN Bulletin, Vol.27, pp.117-137.
[23]McNeil, A.J. (1999), Extreme Value Theory for Risk Managers, in Internal Modelling and CAD Ⅱpublished by RISK Books, pp.93-113.
[24]McNeil, A.J. and R. Frey (2000), Estimation of Tail-Related Risk Measures for Heteroscedastic Financial Time Series:An Extreme Value Approach, Journal of Empirical Finance, Vol.7, pp.271-300.
[25]Parkinson, M. (1980), The Extreme Value Method for Estimating the Variance of the Rate of Return, Journal of Business, Vol.53, pp.61-65.
[26]King Jack L陈剑,柳克俊,陈剑锋译Operational Risk Measurement and Modeling,北京：中国人民大学出版社,2005.
[27]McNeil A J. Extreme Value Theory for risk managers. Internal Modeling and CAD Ⅱ, Published by RISK Books,1999.93-113.
[28]Embrechts Paul, Sidney I, Resnick and Gennady Samorodnitsky.Extreme value theory as a risk management tool. Cornell University
[29]魏宇.金融市场的收益分布与EVT风险测度.数量经济技术经济研究,2006(4)：101110.
[30]余炜彬,范英,魏一鸣.基于极值理论的原油市场价格风险VaR的研究.系统工程理论与实践,2007(8)：12-20.
[31]Kellezi Evis,Gilli Manfred. Extreme Value Theory for Tail-Related Risk Measures. University of Geneva.
[32]Artzner P,F Delbaen,J Eber,and D Heath. Coherent Measures of Risk. Mathematical Finance, 1999(9):203-228.
[33]Acerbi C,D Tasche. On the coherence of Expected Shortfall. http://www. schoolargoogle. com,2002.
[34]Szego Giorgio. Measures of risk. European Journal of Operational Research,2005(163):5-19.
[35]Adam Alexandre,Houkari Mohamed,Laurent Jean-Paul. Spectral Risk measures and portfolio selection. Journal of Banking&Finance,2008(32):1870-1882.
[36]Hubner Robert著,李雪莲,万志宏译.金融机构操作风险新论.天津：南开大学出版社,2005.
[37]Chavez-Demoulin V,Embrechts P,Neslehova J.Quantitative models for operational risk,Extremes,dependence and aggregation.Journal of Banking&Finance,2006(30):2635-2658.
[38]Rockafellar, R Tyrrell,Stanislav Uryasev.Conditional Value-at-Risk for General Loss Distribution. Journal of Banking&Finance,2002(26):1443 — 1471.
[39]Quaranta Anna Grazia, Zaffaroni Alberto. Robust optimization of conditional value at risk and portfolio selection. Journal of Banking&Finance,2008(32):2046-2056.
[40]Angelelli Enrico,Mansini Renata,Speranza M Grazia. A comparison of MAD and CVaR models with real features.Journal of Banking&Finance,2008(32):1188-1197.
[41]田新时,郭海燕.极值理论在风险度量中的应用—基于上证180指数.运筹与管理,2004(1)：108-111.
[42]陈守东,孔繁利,胡铮洋.基于极值分布理论的VaR与ES度量.数量经济技术经济研究,2007(3)：119-125+134.
[43]Jorion P. Value at Risk.New York: McGraw-Hill,1997.19.
4 Brennan, M.J.,1986, A Theory of Price Limits in Futures Markets. Journal of Finance Economics.16,213-233. Markets,19,127-152.
5 Longin, F., 1999, Optimal Margin Levels in Futures Markets: Extreme Price Movements, Journal of Futures
6 Alexander J. McNeil, Rudiger Frey, Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach, Journal of Empirical Finance,Volume 7, Issues 3-4, November 2000, Pages 271-300
7 Cotter, J.,2001, Margin Excessdences for European Stock Index Futures Using Extreme Value Theory, Journal of Banking and Finance, 25, 1474-1502.
8 Parkinson, M.,1980, The Extreme Value Method for Estimating the Variance of the Rate of Return, Journal of Business,53,61-65
9 Fenn, G., and P. Kupiec,1993. Prudential Margin Policy in a Future Style Settlements System, Journal of Futures Markets.13.389-408.
10 Figlewski, S.,1984, Margins and Market Integrity: Margin Setting for Stock Index Futures and Options, Journal of Futures Markets, 4, 385-416.
" Gay, G.D., W.C. Hunter, and R.W. Kolb, 1986, A Comparative Analysis of Futures Contract Margins, Journal of Futures Markets,6,307-324.
12 Kofman, P., 1993, Optimizing Futures Margins with Distribution Tails, Advances in Futures and Options Research,6,263-278.
13 Booth, G.G., J. P. Broussard, T. Martikainen, and V. Puttonen, 1997, Prudent Margin Levels in the Finnish Stock Index Futures Markets, Management Science,43,1177-1188.
14 Broussard, J. P., and G.G. Booth, 1998, The Behavior of Extreme Values in German Stock Index Futures:An Application to Margin Setting, Journal of Operational Research, 104, 393-402.
15 Greenward, B.C. and J.C. Stein,1991, Transaction Risk, Market Crashes, and the Role of Circuit Breakers, Journal of Business,64,443-462.
16 Arak, M and R.E. Cook, 1997, Do Daily Price Limits Act as Magnets The case of Treasury Bond Futures, Journal of Financial Services Research, 12,5-20.
17 Ma, C.K. R.P. Rao, and R.S. Sears, 1989, Volatility, Prices Resolution, and the Effectiveness of Price Limits, Journal of Financial Services Research,3,165-199.
18 Moser, J.T., 1990, Circuit Breakers, Economic Perspectives, Federal Reserve Bank of Chicago,14,2-13.
19 Ackert, L. and W. Hunter, 1994, Tests of a Simple Optimizing Model of Daily Price Limits on Futures Contracts, Review of Financial Economics,4,93-108.
20 Fama, E.F.,1989, Perspectives on October 1987, or What Did We Learn from the Crash in Black Monday and the futures of financial markets,Dow Jones-lrwin, Inc., Homewood, IL, 71-82.
21 Kim, K.A. and S.G. Rhee,1997, Price Limits Performance: Evidence from the Tokyo Stock Exchange, Journal of Finance, 52,885-901.
22 Jeremy Berkowitz,James O'Brien.How Accurate Are Value-at-Risk Models at Commercial Banks?,The Journal of Finance, Volume 57, Issue 3, pages 1093-1111, June 2002
3 Koenker. R. and G. S. Bassett, 1978, Regression Quantiles. Econometrica, 46, pp.33-50.
24 Brandt, M.W., and C.S. Jones (2002), "Volatility Forecasting with Ranged-Based EGARCH Models," working paper, University of Pennsylvania, USA.
25 Alizadeh, S., M. Brandt, and F. Diebold, (2002)," Range-Based Estimation of Stochastic Volatility Models," Journal of Finance, Vol.57, pp.1047-1091.
[1]Alizadeh, S., M. Brandt, and F. Diebold, (2002), Range-Based Estimation of Stochastic Volatility Models, Journal of Finance, Vol.57, pp.1047-1091.
[2]Artzner, P., Delbaen, J.Eber, and D.Heath(1997), Thinking Coherently, Risk, Vol.10, Issue 11, pp.68-71.
[3]Artzner, P., Delbaen, J.Eber, and D.Heath (1999), Conerent Measures of Risk, Mathematical Finance, Vol.9, Issue 3, pp.203-228.
[5]Balkema,A.A. and de Haan, L. (1974), Residual Lifetime at Great Age, Annals of Probability, Vol.2, pp.792-804.
[6]Bates, D. and R. Craine (1999), Valuing the Futures Market Clearinghouse's Default Exposure During the 1987 Crash, Journal of Money, Credit and Banking, Vol.31, pp.248-272.
[7]Bollerslev, T., (1986), Generalized Autoregressive Conditional Heteroskedasticity, Journal of Econometrica, Vol.31, pp.3037-327.
[8]Bollerslev, T., R. Chou, and K. Kroner (1992), ARCH Modeling in Finance:A Review of the Theory and Empirical Evidence, Journal of Econometrics. Vol.52. pp.5-59.
[9]Brandt, M.W., and C.S. Jones (2002), Volatility Forecasting with Ranged-Based EGARCH Models, working paper, University of Pennsylvania, USA.
[10]Broussard, J. P. (2001), Extreme-Value and Margin Setting with and without Price Limits, The Quarterly Review of Economics and Finance, Vol.41, pp.365-385.
[11]Chou R.Y. (2005), Forecasting Financial Volatilities with Extreme Values:the Conditional Autoregressive Range (CARR) Model, Journal of Money,Credit and Banking, Vol.37, Issue 3,pp. 561-582.
[12]Cotter, J. (2001), Margin Exceedences for European Stock Index Futures Using Extreme Value Theory, Journal of Banking & Finance, Vol.25, pp.1474-1502.
[13]De Haan, L. and S. I. Resnick (1980), "A Simple Asymptotic Estimate for the Index of a Stable Distribution," Journal of the Royal Stat. Soc. B, Vol.42, pp.83-87.
[14]Danielsson, J. and C.G. de Vries (1997), Tail Index and Quantile Estimation with Very High Frequency Data, Journal of Empirical Finance, Vol.4, pp.241-257.
[15]Engle, R., and J. Russell (1998) Autoregressive Conditional Duration:A New Model for Irregular Spaced Transaction Data, Econometrica.Vol.66, pp.1127-1162.
[16]Figlewski, S. (1984), Margin and Market Integrity:Margin Setting for Stock Index Futures and Options, The Journal of Future Markets, Vol.43, pp.385-416.
[17]Hahn, M. and D.C. Weiner(1991), On Joint Estimation of an Exponent of Regular Variation and an Asymmetry Parameter for Tail Distributions,Sums, Trimmed Sums, and Extremes," Progress in Probability, Vol.30, pp.82-111.
[18]Hall, P. (1990), Using the Bootstrap to Estimate Mean Squared Error and Select Smoothing Parameter in Nonparametric Problems, Journal of Multivariate Analysis, Vol.32, pp.177-203.
[19]Hols, M.C. and G. de Vries(1991), The Limiting Distribution of Extreme Exchange Rate Return, Journal of Applied econometrics, Vol.6, pp.287-302
[20]Longin, F. (1999), Optimal Margin Levels in Futures Markets:Extreme Price Movements, Journal of Futures Markets, Vol.19, pp.127-152.
[21]Loretan, M. and P. C.B. Philips (1994), Testing the Covariance Stationarity of Heavy-tailed Time Series, Journal of Empirical Finance,Vol.1, pp.211-248.
[22]McNeil, A.J. (1997), Estimating the Tails of Loss Severity Distributions Using Extreme Value Theory, ASTIN Bulletin, Vol.27, pp.117-137.
[23]McNeil, A.J. (1999), Extreme Value Theory for Risk Managers, in Internal Modelling and CAD Ⅱpublished by RISK Books, pp.93-113.
[24]McNeil, A.J. and R. Frey (2000), Estimation of Tail-Related Risk Measures for Heteroscedastic Financial Time Series:An Extreme Value Approach, Journal of Empirical Finance, Vol.7, pp.271-300.
[25]Parkinson, M. (1980), The Extreme Value Method for Estimating the Variance of the Rate of Return, Journal of Business, Vol.53, pp.61-65.
[26]King Jack L陈剑,柳克俊,陈剑锋译Operational Risk Measurement and Modeling,北京：中国人民大学出版社,2005.
[27]McNeil A J. Extreme Value Theory for risk managers. Internal Modeling and CAD Ⅱ, Published by RISK Books,1999.93-113.
[28]Embrechts Paul, Sidney I, Resnick and Gennady Samorodnitsky.Extreme value theory as a risk management tool. Cornell University
[29]魏宇.金融市场的收益分布与EVT风险测度.数量经济技术经济研究,2006(4)：101110.
[30]余炜彬,范英,魏一鸣.基于极值理论的原油市场价格风险VaR的研究.系统工程理论与实践,2007(8)：12-20.
[31]Kellezi Evis,Gilli Manfred. Extreme Value Theory for Tail-Related Risk Measures. University of Geneva.
[32]Artzner P,F Delbaen,J Eber,and D Heath. Coherent Measures of Risk. Mathematical Finance, 1999(9):203-228.
[33]Acerbi C,D Tasche. On the coherence of Expected Shortfall. http://www. schoolargoogle. com,2002.
[34]Szego Giorgio. Measures of risk. European Journal of Operational Research,2005(163):5-19.
[35]Adam Alexandre,Houkari Mohamed,Laurent Jean-Paul. Spectral Risk measures and portfolio selection. Journal of Banking&Finance,2008(32):1870-1882.
[36]Hubner Robert著,李雪莲,万志宏译.金融机构操作风险新论.天津：南开大学出版社,2005.
[37]Chavez-Demoulin V,Embrechts P,Neslehova J.Quantitative models for operational risk,Extremes,dependence and aggregation.Journal of Banking&Finance,2006(30):2635-2658.
[38]Rockafellar, R Tyrrell,Stanislav Uryasev.Conditional Value-at-Risk for General Loss Distribution. Journal of Banking&Finance,2002(26):1443 — 1471.
[39]Quaranta Anna Grazia, Zaffaroni Alberto. Robust optimization of conditional value at risk and portfolio selection. Journal of Banking&Finance,2008(32):2046-2056.
[40]Angelelli Enrico,Mansini Renata,Speranza M Grazia. A comparison of MAD and CVaR models with real features.Journal of Banking&Finance,2008(32):1188-1197.
[41]田新时,郭海燕.极值理论在风险度量中的应用—基于上证180指数.运筹与管理,2004(1)：108-111.
[42]陈守东,孔繁利,胡铮洋.基于极值分布理论的VaR与ES度量.数量经济技术经济研究,2007(3)：119-125+134.
[43]Jorion P. Value at Risk.New York: McGraw-Hill,1997.19.