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Efficient randomized quasi-Monte Carlo methods for portfolio market risk. (English) Zbl 1395.91502

Summary: We consider the problem of simulating loss probabilities and conditional excesses for linear asset portfolios under the \(t\)-copula model. Although in the literature on market risk management there are papers proposing efficient variance reduction methods for Monte Carlo simulation of portfolio market risk, there is no paper discussing combining the randomized quasi-Monte Carlo method with variance reduction techniques. In this paper, we combine the randomized quasi-Monte Carlo method with importance sampling and stratified importance sampling. Numerical results for realistic portfolio examples suggest that replacing pseudorandom numbers (Monte Carlo) with quasi-random sequences (quasi-Monte Carlo) in the simulations increases the robustness of the estimates once we reduce the effective dimension and the impact of the non-smoothness of the integrands.

MSC:

91G60 Numerical methods (including Monte Carlo methods)
65C05 Monte Carlo methods
91G10 Portfolio theory
62P05 Applications of statistics to actuarial sciences and financial mathematics
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[2] Arouna, B., Adaptative Monte Carlo method, a variance reduction technique, Monte Carlo Methods Appl., 10, 1, 1-24 (2004) · Zbl 1063.65003
[4] Başoğlu, İ.; Hörmann, W.; Sak, H., Optimally stratified importance sampling for portfolio risk with multiple loss thresholds, Optimization, 62, 11, 1451-1471 (2013) · Zbl 1280.91190
[6] Boyle, P.; Broadie, M.; Glasserman, P., Monte Carlo methods for security pricing, J. Econom. Dynam. Control, 21, 89, 1267-1321 (1997), Computational financial modelling · Zbl 0901.90007
[7] Bratley, P.; Fox, B. L., Algorithm 659: Implementing Sobol’s quasirandom sequence generator, ACM Trans. Math. Softw., 14, 88-100 (1988) · Zbl 0642.65003
[8] Broadie, M.; Du, Y.; Moallemi, C. C., Efficient risk estimation via nested sequential simulation, Manage. Sci., 57, 6, 1172-1194 (2011) · Zbl 1218.91170
[9] Caflisch, R. E., Monte Carlo and quasi-Monte Carlo methods, Acta Numer., 1-49 (1998) · Zbl 0949.65003
[10] Cheng, R. C.H.; Davenport, T., Quasi-Monte Carlo integration, Manage. Sci., 35, 11, 1278-1296 (1989) · Zbl 0681.62020
[12] Demarta, S.; McNeil, A., The t copula and related copulas, Int. Statist. Rev., 73, 1, 111-129 (2005) · Zbl 1104.62060
[13] Derflinger, G.; Hörmann, W.; Leydold, J., Random variate generation by numerical inversion when only the density function is known, ACM Trans. Model. Comput. Simul., 20, 4, 18:1-18:25 (2010) · Zbl 1386.65028
[14] Derflinger, G.; Hörmann, W.; Leydold, J.; Sak, H., Efficient numerical inversion for financial simulations, (L’Ecuyer, P.; Owen, A. B., Monte Carlo and Quasi-Monte Carlo Methods 2008 (2009), Springer-Verlag: Springer-Verlag Heidelberg), 297-304 · Zbl 1182.91195
[16] Embrechts, P.; McNeil, A.; Straumann, D., Correlation and dependence in risk management: properties and pitfalls, (Dempster, M. A.H., Risk Management: Value at Risk and Beyond (2002), Cambridge University Press: Cambridge University Press Cambridge), 176-223
[17] Etoré, P.; Jourdain, B., Adaptive optimal allocation in stratified sampling methods, Methodol. Comput. Appl. Probab., 12, 335-360 (2010) · Zbl 1208.65005
[18] Glasserman, P., Monte Carlo Methods in Financial Engineering (2004), Springer-Verlag: Springer-Verlag New York · Zbl 1038.91045
[19] Glasserman, P.; Heidelberger, P.; Shahabuddin, P., Asymptotically optimal importance sampling and stratification for pricing path dependent options, Math. Finance, 9, 2, 117-152 (1999) · Zbl 0980.91034
[20] Glasserman, P.; Heidelberger, P.; Shahabuddin, P., Portfolio value-at-risk with heavy-tailed risk factors, Math. Finance, 12, 3, 236-269 (2002) · Zbl 1147.91325
[21] Imai, J.; Tan, K. S., Pricing derivative securities using integrated quasi-Monte Carlo methods with dimension reduction and discontinuity realignment, SIAM J. Sci. Comput., 36, 5, A2101-A2121 (2014) · Zbl 1307.65003
[22] Jin, X.; Zhang, A. X., Reclaiming quasi-Monte Carlo efficiency in portfolio value-at-risk simulation through Fourier transform, Manage. Sci., 52, 6, 925-938 (2006) · Zbl 1232.91709
[23] Kreinin, A.; Merkoulovitch, L.; Rosen, D.; Zerbs, M., Principal component analysis in quasi Monte Carlo simulation, Algo Research Quarterly, 1, 21-30 (1998)
[24] Lemieux, C., Monte Carlo and Quasi-Monte Carlo Sampling (2009), Princeton University Press, Springer: Princeton University Press, Springer New York · Zbl 1269.65001
[26] Mashal, R.; Naldi, M.; Zeevi, A., Comparing the dependence structure of equity and asset returns, Risk, 16, 82-87 (2003)
[27] Morokoff, W.; Caflisch, R. E., Quasi-Monte Carlo integration, J. Comput. Phys., 122, 218-230 (1995) · Zbl 0863.65005
[29] R Core Team, R: A Language and Environment for Statistical Computing (2015), R Foundation for Statistical Computing: R Foundation for Statistical Computing Vienna, Austria, URL http://www.R-project.org/
[30] Sak, H.; Hörmann, W.; Leydold, J., Efficient risk simulations for linear asset portfolios in the t-copula model, European J. Oper. Res., 202, 802-809 (2010) · Zbl 1176.91150
[31] Sun, W.; Rachev, S. T.; Stoyanov, S. V.; Fabozzi, F. J., Multivariate skewed student’s t copula in the analysis of nonlinear and asymmetric dependence in the German equity market, Stud. Nonlinear Dyn. Econom., 12, 2, 1-37 (2008) · Zbl 1193.91183
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