除了買賣價上下限及資本適足性外，期貨保證金制度的設計更確保了期貨交易的安全性與期貨市場的完整性。如何在價格極端波動下設定一最適保證金水準，一方面考慮可以有效降低券商或期交所的違約風險；另一方面，又不會對市場交易機制或是市場流動性產生過度不利的影響是本研究訴求的核心。所以，本研究分別運用Generalized Extreme Value Distribution和Generalized Pareto Distribution去檢視當價格極端波動情況下之謹慎保證金水準為何及透過前向測試(Back testing)以確定不同模型成功率高低之比較。本文實證模擬結果發現: (1) 常態分配假設會造成保證金水準的低估。(2) 以損失期望值(Expected Shortfall)當做風險測量之檢視基準，Nikkei225在保證金水準的適足性最不足；S&P500、CAC40和DAX則較足夠。 (3) Generalized Pareto Distribution之損失期望值法對於補捉價格極端波動強況之能力來的比Generalized Pareto Distribution之保證金水準來的較佳。 (4) 運用Hillplot當做最適Threshold之指標並對於價格極端波動下之保證金水準設定為一個相當理想的風險測度工具。

Abstract
Along with a price limit and capital requirement, the existence of a margin decrease the likelihood of a customer defaulting, a broker going bankrupt and systemic instability of the futures market. This paper applies two sub-theories of generalized extreme value distribution and generalized Pareto distribution inherited from extreme value theory to examine the prudent margin policy for price extremal movements. The theoretical framework focuses explicitly on tail returns, thereby properly computing prudent margin level for large levels of risk, This paper finds： (1) the assumption of normality to impose a smaller margin level since the presence of a fat-tail. (2) on the basis of margin insolvency using an expected shortfall, the margin requirements of stock index futures across contracts with a Nikkei225 contract being the more risky, and S&P500, CAC40, and DAX futures indexes are the least risky. (3) the ability to capture extreme price movements using expected shortfall is more suitable than the approach of the VaR based on generalized extreme value distribution. (4) the proxy of the appropriate threshold using an expected shortfall can capture well the extreme price movements and can be an excellent risk measure instrument to set the prudent margin level.

Reference
Ackert, L., & Hunter, W. (1990)” A Sequential Test Methodology for Detecting Futures Market Disruption with Application to Futures Margin Management,” Review of Futures Markets,9,318 – 341.
Acerbi C., Nordio C.,Sirtori C., (2001) “Expected Shortfall as a Tool for Financial Risk Management,” working paper, Italy.
Basle Committee on Banking Supervision, (1996b)“Supervisory Framework for the Use of Backtesting in Conjuction with the Internal Approach to Market Risk Capital Requirement”. Basle, Switzerland.
Bertsimas, D., Lauprete, G.J., Samarov, A. (2000) “Shortfall as a Risk Measure: Properties, Optimization and Applications”. Working paper, Sloan School of Management, MIT, Cambridge.
Boudoukh, J., M. Richardson, & R. Whitelaw (1995) “Expect the Worst,” Risk, September, 100–101.
Booth, G. G., Broussard, J. P., Martikainen, T., & Puttonen, V. (1997). ”Prudent Margin Levels in the Finnish Stock Index Futures Market”. Management Science, 43, 1177–1188.
Booth, G. G., & Broussard, J. P. (1998). “Setting NYSE Circuit Breaker Triggers”. Journal of Financial Services Research, 13, 187–204.
Broussard, J. P., & Booth, G. G. (1998). ”The Behavior of Extreme Values in German Stock Index Futures: An Application to Margin Setting”. Journal of Operational Research, 104, 393–402.
Brenner, T. W. (1981). “Margin Authority: No Reason for a Change”. Journal of Futures Markets, 1, 487–490.
Brennan, M. J. (1986). ”A Theory of Price Limits in Futures Markets”. Journal of Financial Economics, 16, 213–233.
Broussard, J. P., & Booth, G. G. (1997). “The Behavior of Extreme-Values in Germany’s Stock Index Futures: An Application to Intra-daily Margin Setting”. European Journal of Operational Research, 104, 393–402.
Broissard, J. P.,(2001) “Extreme-Value and Margin Setting With and Without Price Limit”, The Quarterly Review of Economics and Finance,41,365-402.
Crouhy, M., D. Galai, & R. Mark (2000). ”A Comparative Analysis of Current Credit Risk Models," Journal of Banking and Finance, 24, 59-117.
Cotter,J., (2001), “Margin Exceedness for European Stock Index Futures Using Extreme Value Theory,” Journal of Banking and Finance 25,1475-1502.
Danielsson, J., & de Vries,C.G. ( 1997) , "Tail Index and Quantile Estimation with Very High Frequency Data", Journal of Empirical Finance, Vol.4, 241-257.
Danielsson ,J., & de Vries,C.G, (1997), “Value at Risk and Extreme Returns. Financial Market Group Discussion paper. London School of Economics , London.
Danielsson, J., & de Vries, C.G., (1997), “Beyond the Sample: Extreme Quantile and Probability Estimation.” Mimeo., Tinbergen Institute Rotterdam.
Danielsson, J., & de Vries C.G., ( 2000), "Value-at-Risk and Extreme Returns", Extremes and Integrated Risk Management, Risk Books, 85-106.
Dimson, E., & Marsh, P. (1995) “Capital Requirement for Securities Firms”. Journal of Finance, 50 ,800-851.
Duffie, D., & J. Pan. ( 1997), "An Overview of Value at Risk", the Journal of Derivatives, Spring, 7-49
Edwards, F.R., & Neftci, S. N. (1988), “ Extreme Price Movements and Margin Levels in Futures Markets,” Journal of Futures Markets ,13,p.639-655.
Embrechts, P., Kl¨uppelberg, C. and T. Mikosch, (1999). Modelling Extremal Events for Insurance and Finance. 2nd ed., Springer-Verlag, Berlin. (1st ed., 1997).
Fenn, G.D. & Kupiec, P. (1993), “Prudent Margin Policy in a Futures Style Settlement System,” Journal of Futures Markets, 13, p.389-408.
Figlewski, S. (1984). “Margins and Market Integrity: Margin Setting for Stock Index Futures and Options”. Journal of Futures Markets, 4, 385–416.
Fishe, R.., Goldberg, L., Gonsell, T., & Sinha, S. (1990). ” Margin Requirements in Future Markets: Their Relationship to Price Volatility,” Journal of Future Markets.10,541-554.
Gay, G. D., Hunter, W. C., & Kolb, R. W. (1986). ”A Comparative Analysis of Futures Contract Margins”. Journal of Futures Markets, 6, 307–324.
Goldberg, L., & Hachey, A. (1992). ”Price Volatility and Margin Requirements in Foreign Exchange Future Markets,” Journal of International Money and Finance. 11, 328-339.
Haas, M. and Kondratyev ,A. (2000), “Value-at-Risk and Expected shortfall whit Confidence Bands: an Extreme Value Theory Approach”, working paper, Group of Financial Engineering Research Center CAESAR, Bonn, Germany.
Ho ,L.C.,Burridge, P., Cadle, J., Theobald, M. (2000), “Value-at-risk: Applying the Extreme Value Approach to Asian Markets in the Recent Financial Turmoil” Pacific-Basin Finance Journal,249-275.
Hull, J., & A. White. ( 1998), "Incorporating Volatility Updating into the Historical Simulation Method for Value at Risk", Journal of Risk, Fall.
J.A., Brorsen, B.W., Irwin, S.H. (1989), “The Distribution of Futures Prices: A Test of the Stable Paretian and Mixture of Normal Hypothesis”, Journal of Financial and Quantitative Analysis 24, 105-116.
Jorion, P. (1997): Value-at-Risk. Irvin.
Kusuoka, S. (2001) “On Law Invariant Coherent Risk Measures”. In Advances in Mathematical Economics, volume 3, 83–95.
Lo, A. W., Mackinlay, A.C., (1990) ”An Econometric Analysis of Nonsynchornous Trading”. Journal of Econometrics 45,181-211.
Longin, F. (1995). “Optimal Margins in Futures Markets: A Parametric Extreme-based Approach”. Proceedings, Ninth Chicago Board of Trade Conference on Futures and Options, Bonn.
Longin, F. (1997). “The Threshold Effect in Expected Volatility: A Model Based on Asymmetric Information”. The Review of Financial Studies 10,837-869.
Longin, F. (1999). “Optimal Margin Level in Futures Markets: Extreme Price Movements”. Journal of Futures Markets, 19, 127–152.
Longin, F., (2000), “From Value at Risk to Stress Testing: The Extreme Value Approach,” Journal of Banking and Finance 24(7),1097-1130.
M., Phillips, P.C.B. (1994), “Testing the Covariance Stationary of Heavy-tailed Time Series”, Journal of Empirical Finance 1, 211-248.
Ma, C. K., Mercer, J. M., & Walker, M. A. (1992). “Rolling Over Futures Contracts: A Note”. Journal of Futures Markets, 12, 203–217.
Merton, R.C., & Perold,A. F., (1993) “Theory of Risk Capital in Financial Firms. Journal of Applied Corporate Finance 6, 16-32.
McNeil, A. J. ( 1997), "The Peaks Over Thresholds Method for Estimating High Quantiles of Loss Distributions", in XXVIIth International ASTIN Colloquium, 23-43.
McNeil, A. (1998), “Calculating Quantile Risk Measures for Financial Return Series Using Extreme Value Theory”, Working paper.
Mcneil, A.J. & Saladin, T. (1998) ,“Developing Scenarios for Future Extreme Losses Using the POT Model, working paper Zenturm, Zurich.
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.
Miller, H. M. (1988) “Who Should Set Futures Margins?” Review of Futures Markets ,7, 398-404.
Rockafellar, R.T., Uryasev, S. (2001) “Conditional Value-at-Risk for General Loss Distributions”. Research report 2001-5, ISE Dept., University of Florida.
Rutz, R. D. (1988). “Clearance, Payment, and Settlement System in the Futures, Options and Stock Markets.” Review of Futures Markets,7.346-370.
Smith, R.L (1986). “Estimating Tails of Probability Distributions” , Ann. Statist.15,1174-1207.
Yamai, Y., and T. Yoshiba., (2002) “On the Validity of Value-at-Risk: Comparative Analyses with Expected Shortfall,” Monetary and Economic Studies, 20 .
Yu M.T., Chou P.H & Lin M.C., (2000) "Price Limits, Default Risks, and Margin Requirements," Journal of Futures Markets 20,573-602.
Warshawsky, M. J. (1989). ”The adequacy and consistency of margin requirements: The cash, and options segments of the equity markets”. Review of futures Markets,8, 420-437.