Abstract
The purpose of this article is to show how the multivariate structure (the “shape” of the distribution) can be separated from the marginal distributions when generating scenarios. To do this we use the copula. As a result, we can define combined approaches that capture shape with one method and handle margins with another. In some cases the combined approach is exact, in other cases, the result is an approximation. This new approach is particularly useful if the shape is somewhat peculiar, and substantially different from the standard normal elliptic shape. But it can also be used to obtain the shape of the normal but with margins from different distribution families, or normal margins with for example tail dependence in the multivariate structure. We provide an example from portfolio management. Only one-period problems are discussed.
Similar content being viewed by others
References
Adcock CJ (2002) Asset pricing and portfolio selection based on the multivariate skew-student distribution. In: Non-linear Asset Pricing Workshop, Paris
Azzalini A, Capitanio A (2003) Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution. J R Stat Soc Ser B (Stat Methodol) 65: 367–389
Bauwens L, Laurent S (2005) A new class of multivariate skew densities, with application to GARCH models. J Bus Econ Stat 23(3):346–354. Available at SSRN: http://ssrn.com/abstract=691865
Bayraksan G, Morton DP (2006) Assessing solution quality in stochastic programs. Math Program 108(2–3): 495–514
Bouyé E, Durrleman V, Nikeghbali A, Riboulet G, Roncalli T (2000) Copulas for finance: a reading guide and some applications. Working paper, Crédit Lyonnais, Paris. Available at SSRN: http://ssrn.com/abstract=1032533
Chiralaksanakul A, Morton DP (2004) Assessing policy quality in multi-stage stochastic programs. Stochastic Programming E-Print Series. http://www.speps.org
Clemen RT, Reilly T (1999) Correlations and copulas for decision and risk analysis. Manage Sci 45(2): 208–224
Demarta S, McNeil AJ (2005) The t copula and related copulas. Int Stat Rev 73(1): 111–129
Dupačová J, Consigli G, Wallace SW (2000) Scenarios for multistage stochastic programs. Ann Oper Res 100: 25–53 ISSN 0254-5330
Dupačová J, Gröwe-Kuska N, Römisch W (2003) Scenario reduction in stochastic programming. Math Program 95(3): 493–511
Eaton JW (2006) GNU Octave Manual. Free Software Foundation, Inc
Embrechts P, Lindskog F, Mcneil A (2003) Modelling dependence with copulas and applications to risk management. In: Rachev ST (ed) Handbook of heavy tailed distributions in finance, handbooks in finance, chapter 8. Elsevier, pp 329–384
Fleishman AI (1978) A method for simulating nonnormal distributions. Psychometrika 43: 521–532
Heinrich J (2004) A guide to the pearson type IV distribution. Technical Report Memo 6820, The Collider Detector at Fermilab, Fermilab, Batavia. Available at http://www-cdf.fnal.gov/publications/cdf6820_pearson4.pdf
Heitsch H, Römisch W (2003) Scenario reduction algorithms in stochastic programming. Comput Optim Appl 24(2–3): 187–206
Heitsch H, Römisch W (2009) Scenario tree modelling for multistage stochastic programs. Math Program 118(2): 371–406. doi:10.1007/s10107-007-0197-2
Heitsch H, Römisch W, Strugarek C (2006) Stability of multistage stochastic programs. SIAM J Optim 17(2): 511–525
Heitsch H, Römisch W (2009) Scenario tree reduction for multistage stochastic programs. Comput Manage Sci 6(2): 117–133. doi:10.1007/s10287-008-0087-y
Høyland K, Wallace SW (2001) Generating scenario trees for multistage decision problems. Manage Sci 47(2): 295–307
Høyland K, Michal K, Wallace SW (2003) A heuristic for moment-matching scenario generation. Comput Optim Appl 24(2–3): 169–185 ISSN 0926-6003
Hu L (2006) Dependence patterns across financial markets: a mixed copula approach. Appl Finan Econ 16(10): 717–729
Joe H (1997) Multivariate models and dependence concepts. Chapman & Hall, London
Jondeau E, Rockinger M (2003) Conditional volatility, skewness, and kurtosis: existence, persistence, and comovements. J Econ Dynam Control 27(10): 1699–1737. ISSN 0165-1889. doi:10.1016/S0165-1889(02)00079-9
Jones MC (2001) Multivariate t and beta distributions associated with the multivariate f distribution. Metrika 54: 215–231
Kaut M, Wallace SW (2007) Evaluation of scenario-generation methods for stochastic programming. Pac J Optim 3(2): 257–271
Kaut M, Wallace SW, Vladimirou H, Zenios S (2007) Stability analysis of portfolio management with conditional value-at-risk. Quant Finance 7(4): 397–409. doi:10.1080/14697680701483222
Linderoth JT, Shapiro A, Wright SJ (2006) The empirical behavior of sampling methods for stochastic programming. Ann Oper Res 142(1): 215–241
Longin F, Solnik B (2001) Extreme correlation of international equity markets. J Finance 56(2): 649–676
Makhorin A (2006a) GNU Linear Programming Kit—Reference Manual, Version 4.9. Free Software Foundation, Inc
Makhorin A (2006b) GNU Linear Programming Kit—Modeling Language GNU MathProg, Version 4.9. Free Software Foundation, Inc
MathWorks (2006) Statistics Toolbox For Use with MATLAB®—User’s Guide. The MathWorks, Inc., 3 Apple Hill Drive Natick, MA 01760-2098
MSCI Morgan Stanley Capital International Inc (2006). http://www.msci.com/equity/
Nelsen RB (1998) An introduction to copulas. Springer, New York
Patton A (2002) Skewness, asymmetric dependence, and portfolios. In: PhD Thesis, chapter 3. Department of Economics, University of California, San Diego
Patton AJ (2004) On the out-of-sample importance of skewness and asymmetric dependence for asset allocation. J Finan Econom 2(1): 130–168
Pflug GC (2001) Scenario tree generation for multiperiod financial optimization by optimal discretization. Math Program 89(2): 251–271
Rockafellar RT, Uryasev S (2000) Optimization of conditional value-at-risk. J Risk 2(3): 21–41
Romano C (2002) Calibrating and simulating copula functions: an application to the italian stock market. working paper 12. Centro Interdipartimale sul Diritto e l’Economia dei Mercati
Rosenberg JV (2003) Non-parametric pricing of multivariate contingent claims. J Deriv 10(3): 9–26
Sklar A (1959) Fonctions de répartition à n dimensions et leurs marges. Publications de l’Institut de Statistique de l’Université de Paris 8: 229–231
Sklar A (1996) Random variables, distribution functions, and copulas—a personal look backward and forward. In: Rüschendorff L, Schweizer B, Taylor M (eds) Distributions with fixed marginals and related topics. Institute of Mathematical Statistics, Hayward, pp 1–14
Uryasev S (2000) Conditional value-at-risk: optimization algorithms and applications. Finan Eng News 14: 1–5. URL http://fenews.com/
Wallace, SW, Ziemba, WT (eds) (2005) Applications of stochastic programming. MPS-SIAM Series on Optimization, Philadelphia
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kaut, M., Wallace, S.W. Shape-based scenario generation using copulas. Comput Manag Sci 8, 181–199 (2011). https://doi.org/10.1007/s10287-009-0110-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10287-009-0110-y