Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-05T23:29:23.565Z Has data issue: false hasContentIssue false
  • English
  • Français

Efficient simulation of tail probabilities of sums of dependent random variables

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Jose H. Blanchet  and
Leonardo Rojas-Nandayapa
Jose H. Blanchet
Affiliation:
Columbia University, Department of Industrial Engineering and Operations Research, Columbia University, S.W. Mudd Building, 500 West 120th Street, New York, USA. Email address: [email protected]
Leonardo Rojas-Nandayapa
Affiliation:
The University of Queensland, Department of Mathematics, The University of Queensland, Brisbane, Queensland 4072, Australia. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study asymptotically optimal simulation algorithms for approximating the tail probability of P(eX1+⋯+ eXd>u) as u→∞. The first algorithm proposed is based on conditional Monte Carlo and assumes that (X1,…,Xd) has an elliptical distribution with very mild assumptions on the radial component. This algorithm is applicable to a large class of models in finance, as we demonstrate with examples. In addition, we propose an importance sampling algorithm for an arbitrary dependence structure that is shown to be asymptotically optimal under mild assumptions on the marginal distributions and, basically, that we can simulate efficiently (X1,…,Xd|Xj >b) for large b. Extensions that allow us to handle portfolios of financial options are also discussed.

Keywords

Rare-event simulationefficiencydependenceheavy-tailed distributionlog-elliptical distributiontail probabilityvariance reductionimportance samplingconditional Monte Carlo

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60G70: Extreme value theory; extremal processes
Type
Part 4. Simulation
Information
Journal of Applied Probability , Volume 48 , Issue A: New Frontiers in Applied Probability (Journal of Applied Probability Special Volume 48A) , August 2011 , pp. 147 - 164
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Abramowitz, M. and Stegun, I. A., (1965). Handbook of Mathematical Functions. Dover, New York.Google Scholar
[2] Adler, R. J., Blanchet, J. H. and Liu, J., (2010). Efficient Monte Carlo for high excursions of Gaussian random fields. Preprint. Available at http://arxiv.org/abs/1005.0812v3.Google Scholar
[3] Albrecher, H., Asmussen, S. and Kortschak, D., (2006). Tail asymptotics for the sum of two heavy-tailed dependent risks. Extremes 9, 107–130.CrossRefGoogle Scholar
[4] Asmussen, S., (2000). Ruin Probabilities. World Scientific, River Edge, NJ.CrossRefGoogle Scholar
[5] Asmussen, S., (2003). Applied Probability and Queues. Springer, New York.Google Scholar
[6] Asmussen, S. and Binswanger, K., (1997). Simulation of ruin probabilities for subexponential claims. ASTIN Bull. 27, 297–318.Google Scholar
[7] Asmussen, S. and Glynn, P. W., (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York.CrossRefGoogle Scholar
[8] Asmussen, S. and Kroese, D. P., (2006). Improved algorithms for rare event simulation with heavy tails. Adv. Appl. Prob. 38, 545–558.Google Scholar
[9] Asmussen, S. and Rojas-Nandayapa, L., (2008). Asymptotics of sums of lognormal random variables with Gaussian copula. Statist. Prob. Lett. 78, 2709–2714.CrossRefGoogle Scholar
[10] Asmussen, S., Binswanger, K. and Höjgaard, B., (2000). Rare events simulation for heavy-tailed distributions. Bernoulli 6, 303–322.Google Scholar
[11] Asmussen, S., Blanchet, J., Juneja, S. and Rojas-Nandayapa, L., (2009). Efficient simulation of tail probabilities of sums of correlated lognormals. Ann. Operat. Res., 19 pp.Google Scholar
[12] Barndorff-{Nielsen}, O. E., (1977). Exponentially decreasing distributions for the logarithm of particle size. Proc. R. Soc. London A 353, 401–419.Google Scholar
[13] Blanchet, J. and Glynn, P. W., (2008). Efficient rare-event simulation for the maximum of a heavy-tailed random walk. Ann. Appl. Prob. 18, 1351–1378.CrossRefGoogle Scholar
[14] Blanchet, J. and Li, C., (2011). Efficient rare-event simulation for heavy-tailed compound sums. To appear in ACM Trans. Model. Comput. Simul. Google Scholar
[15] Blanchet, J., Juneja, S. and Rojas-Nandayapa, L., (2008). Efficient tail estimation for sums of correlated lognormals. In Proc. 40th Winter Conf. Simulation (Miami, FL, 2008), pp. 607–614.Google Scholar
[16] Boots, N. K. and Shahabuddin, P., (2001). Simulating ruin probabilities in insurance risk processes with subexponential claims. In Proc. 33rd Winter Conf. Simulation (Arlington, VA, 2001), pp. 468–476.CrossRefGoogle Scholar
[17] Chan, J. C. C. and Kroese, D. P., (2009). Rare-event probability estimation with conditional Monte Carlo. Ann. Operat. Res., 19pp.Google Scholar
[18] Chan, J. C. C. and Kroese, D. P., (2010). Efficient estimation of large portfolio loss probabilities in t-copula models. Europ. J. Operat. Res. 205, 361–367.Google Scholar
[19] Cont, R. and Tankov, P., (2003). Financial Modelling with Jump Processes. Chapman and Hall, New York.Google Scholar
[20] Duffie, D., (2001). Dynamic Asset Pricing Theory. Princeton University Press.Google Scholar
[21] Dupuis, P., Leder, K. and Wang, H., (2006). Importance sampling for sums of random variables with regularly varying tails. ACM Trans. Model. Comput. Simul. 17, 1–21.Google Scholar
[22] Embrechts, P., Klüppelberg, C. and Mikosch, T., (1997). Modelling Extremal Events. Springer, Berlin.Google Scholar
[23] Foss, S. and Richards, A., (2010). On sums of conditionally independent subexponential random variables. Math. Operat. Res. 35, 102–119.Google Scholar
[24] Hartinger, J. and Kortschak, D., (2009). On the efficiency of the Asmussen-Kroese-estimator and its applications to stop-loss transforms. Blätter DGVFM 30, 363–377.CrossRefGoogle Scholar
[25] Huang, Z. and Kou, S., (2006). First passage times and analytical solutions for options on two assets with jump risks. Tech. Rep., Columbia University.Google Scholar
[26] {Jörgensen}, B., (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution (Lecture Notes Statist. 9). Springer, New York.CrossRefGoogle Scholar
[27] Juneja, S., (2007). Estimating tail probabilities of heavy-tailed distributions with asymptotically zero relative error. Queueing Systems 57, 115–127.Google Scholar
[28] Juneja, S. and Shahabuddin, P., (2002). Simulating heavy-tailed processes using delayed hazard rate twisting. ACM Trans. Model. Comput. Simul. 12, 94–118.CrossRefGoogle Scholar
[29] Klöppel, S., Reda, R. and Scharchermayer, W., (2010). A rotationally invariant technique for rare event simulation. Risk Mag. 22, 90–94.Google Scholar
[30] Kortschak, D. and Albrecher, H., (2008). Asymptotic results for the sum of dependent non-identically distributed random variables. Methodology Comput. Appl. Prob. 11, 279–306.Google Scholar
[31] Kou, S., (2002). A jump diffusion model for option pricing. Manag. Sci. 48, 1086–1101.Google Scholar
[32] McNeil, A., Frey, R. and Embrechts, P., (2005). Quantitative Risk Management. Princeton University Press.Google Scholar
[33] Mikosch, T., (2009). Non-Life Insurance Mathematics. Springer, Berlin.Google Scholar
[34] Mitra, A. and Resnick, S. I., (2009). Aggregation of rapidly varying risks and asymptotic independence. Adv. Appl. Prob. 41, 797–828.Google Scholar