×

zbMATH — the first resource for mathematics

Efficient risk simulations for linear asset portfolios in the \(t\)-copula model. (English) Zbl 1176.91150
Summary: We consider the problem of calculating tail probabilities of the returns of linear asset portfolios. As a flexible and accurate model for the logarithmic returns we use the \(t\)-copula dependence structure and marginals following the generalized hyperbolic distribution. Exact calculation of the tail-loss probabilities is not possible and even simulation leads to challenging numerical problems. Applying a new numerical inversion method for the generation of the marginals and importance sampling with carefully selected mean shift we develop an efficient simulation algorithm. Numerical results for a variety of realistic portfolio examples show an impressive performance gain.

MSC:
91G10 Portfolio theory
Software:
ghyp; LBFGS-B; R; Runuran
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aas, K.; Haff, I.H., The generalized hyperbolic skew student’s t-distribution, Journal of financial econometrics, 4, 2, 275-309, (2006)
[2] Barndorff-Nielsen, O.; Blæsild, P., Hyperbolic distributions, (), 700-707 · Zbl 0489.62020
[3] Bassamboo, A.; Juneja, S.; Zeevi, A., Portfolio credit risk with extremal dependence: asymptotic analysis and efficient simulation, Operations research, 56, 3, 593-606, (2008) · Zbl 1167.91362
[4] Behr, A.; Pötter, U., Alternatives to the normal model of stock returns: Gaussian mixture, generalised logf and generalised hyperbolic models, Annals of finance, 5, 1, 49-68, (2009)
[5] Breymann, W., Luethi, D., 2008. ghyp: A package on the generalized hyperbolic distribution and its special cases. R package version 1.3.0, <http://www.idp.zhaw.ch>.
[6] Byrd, R.H.; Lu, P.; Nocedal, J.; Zhu, C., A limited memory algorithm for bound constrained optimization, SIAM journal of scientific computing, 16, 1190-1208, (1995) · Zbl 0836.65080
[7] Derflinger, G., Hörmann, W., Leydold, J., 2008. Random variate generation by numerical inversion when only the density function is known. Research Report Series of the Department of Statistics and Mathematics 78, Wirtschaftsuniversität Wien, Augasse 2-6, A-1090 Wien, Austria, <http://epub.wu-wien.ac.at/english/..
[8] Embrechts, P.; McNeil, A.; Straumann, D., A limited memory algorithm for bound constrained optimization, SIAM journal of scientific computing, 16, 1190-1208, (1995) · Zbl 0836.65080
[9] Frey, R., McNeil, A., 2001. Modelling dependent defaults. Tech. rep., ETH, Eidgenössische Technische Hochschule Zürich, Department of Mathematics, <http://e-collection.ethbib.ethz.ch/view/eth:26296>.
[10] Glasserman, P.; Heidelberger, P.; Shahabuddin, P., Asymptotically optimal importance sampling and stratification for pricing path dependent options, Mathematical finance, 9, 2, 117-152, (1999) · Zbl 0980.91034
[11] Glasserman, P.; Heidelberger, P.; Shahabuddin, P., Portfolio value-at-risk with heavy-tailed risk factors, Mathematical finance, 12, 3, 239–269, (2002) · Zbl 1147.91325
[12] Hörmann, W.; Leydold, J., Continuous random variate generation by fast numerical inversion, ACM transactions on modeling and computer simulation, 13, 4, 347-362, (2003) · Zbl 1390.65010
[13] Kang, W., Shahabuddin, P., 2005. Fast simulation for multifactor portfolio credit risk in the t-copula model. In: Kuhl, M.E., Steiger, N.M., Armstrong, F.B., Joines J.A. (Eds.), WSC’05: Proceedings of the 37th Conference on Winter Simulation, Winter Simulation Conference, pp. 1859-1868.
[14] Karadağ, D.T., 2008. Portfolio risk calculation and stochastic portfolio optimization by a copula based approach. Master’s Thesis, Boğaziçi University Istanbul.
[15] Kole, E.; Koedijk, K.; Verbeek, M., Selecting copulas for risk management, Journal of banking and finance, 31, 8, 2405-2423, (2007)
[16] Leydold, J., Hörmann, W., 2008. Runuran - R interface to the UNU.RAN random variate generators, Version 0.8. Department of Statistics and Mathematics, WU Wien, A-1090 Wien, Austria, <http://cran.r-project.org/>.
[17] Mashal, R.; Naldi, M.; Zeevi, A., Comparing the dependence structure of equity and asset returns, Risk, 16, 82-87, (2003)
[18] Prause, K., 1997. Modelling financial data using generalized hyperbolic distributions. Tech. rep., FDM preprint 48, University of Freiburg.
[19] Prause, K., 1999. The generalized hyperbolic model: Estimation, financial derivatives, and risk measures. Ph.D. Thesis, University of Freiburg. · Zbl 0944.91026
[20] R Development Core Team, 2008. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, <http://www.R-project.org>.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.