A copula-based scenario tree generation algorithm for multiperiod portfolio selection problems

  • Zhe Yan
  • Zhiping ChenEmail author
  • Giorgio Consigli
  • Jia Liu
  • Ming Jin
S.I.: Stochastic Optimization:Theory&Applications in Memory of M.Bertocchi


Global financial investors have been confronted in recent years with an increasing frequency of market shocks and returns’ outliers, until the unprecedented surge of financial risk observed in 2008. From a statistical viewpoint, those market dynamics have shown not only asymmetric returns and fat tails but also a time-varying tail dependence, stimulating the formulation of portfolio selection models based on such assumptions. The concept of tail dependence on upper or lower tails, roughly speaking, focuses on the risk that tail events may occur jointly in different markets. This notion can be given a rigorous probabilistic definition, and it turns out that a distinction between upper and lower tails is relevant in portfolio management. In this paper, relying on a discrete modeling framework, we present a scenario generation algorithm able to capture this time-varying asymmetric tail dependence, and evaluate resulting optimal investment policies based on 4-stages 1-month planning horizons. The scenario tree aims at approximating a stochastic process combining an ARMA-GARCH model and a dynamic Student-t-Clayton copula. From a methodological viewpoint, scenario trees are generated from this model by stage-wisely sampling and clustering and to improve tail fitting with original data, the scenarios’ nodal probabilities are calibrated on the returns’ lower tails for a set of equity indices. The resulting scenario trees are then applied to solve a multiperiod portfolio selection problem. We present a set of empirical results to validate the adopted statistical approach and the optimal portfolio strategies able to capture asymmetric tail returns.


Copula Scenario tree generation Tail of the distribution Portfolio selection 



The authors are grateful to the editor and three anonymous reviewers for their extremely detailed and insightful comments and suggestions, which have led to a substantial improvement of the paper in both content and style. This research was supported by the National Natural Science Foundation of China (Grant Numbers 11571270 and 71371152 ).


  1. Aepli, M. D. (2015). Portfolio risk forecasting-on the predictive power of multivariate dynamic copula models. Doctoral dissertation, University of St. Gallen.Google Scholar
  2. Bertocchi, M., Consigli, G., & Dempster, M. A. (Eds.). (2011). Stochastic optimization methods in finance and energy: New financial products and energy market strategies. New York: Springer.Google Scholar
  3. Birge, J. R., & Louveaux, F. (2011). Introduction to stochastic programming. New York: Springer.CrossRefGoogle Scholar
  4. Braun, V., & Grizska, M. (2011). Modeling asymmetric dependence of financial returns with multivariate dynamic copulas. Ssrn Electronic Journal.Google Scholar
  5. Calfa, B. A., Agarwal, A., Grossmann, I. E., & Wassick, J. M. (2014). Data-driven multi-stage scenario tree generation via statistical property and distribution matching. Computers & Chemical Engineering, 68, 7–23.CrossRefGoogle Scholar
  6. Chen, Z., Liu, J., & Hui, Y. (2017). Recursive risk measures under regime switching applied to portfolio selection. Quantitative Finance, 1–20.Google Scholar
  7. Chen, Z., & Xu, D. (2014). Knowledge-based scenario tree generation methods and application in multiperiod portfolio selection problem. Applied Stochastic Models in Business and Industry, 30(3), 240–257.CrossRefGoogle Scholar
  8. Consiglio, A., Tumminello, M., & Zenios, S. A. (2015). Designing and pricing guarantee options in defined contribution pension plans. Insurance Mathematics & Economics, 65(65), 267–279.CrossRefGoogle Scholar
  9. Consigli, G. (2002). Tail estimation and mean-VaR portfolio selection in markets subject to financial instability. Journal of Banking & Finance, 26(7), 1355–1382.CrossRefGoogle Scholar
  10. Consigli, G. (2004). Estimation of tail risk and portfolio optimisation with respect to extreme measures. In Risk measures for the 21st century, pp. 365–401. New York, NY: Wiley.Google Scholar
  11. Consigli, G., Iaquinta, G., & Moriggia, V. (2012). Path-dependent scenario trees for multistage stochastic programmes in finance. Quantitative Finance, 12(8), 1265–1281.CrossRefGoogle Scholar
  12. Consiglio, A., Carollo, A., & Zenios, S. A. (2016). A parsimonious model for generating arbitrage-free scenario trees. Quantitative Finance, 16(2), 201–212.CrossRefGoogle Scholar
  13. Dempster, M. A., Medova, E. A., & Yong, Y. S. (2011). Comparison of sampling methods for dynamic stochastic programming. Stochastic optimization methods in finance and energy (pp. 389–425). New York, NY: Springer.CrossRefGoogle Scholar
  14. Embrechts, P., Lindskog, F., & McNeil, A. (2001). Modelling dependence with copulas. Departement de mathematiques, Institut Federal de Technologie de Zurich, Zurich: Rapport technique.Google Scholar
  15. Engle, R. (2002). Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. Journal of Business & Economic Statistics, 20(3), 339–350.CrossRefGoogle Scholar
  16. Engle, R. F., & Sheppard, K. (2001). Theoretical and empirical properties of dynamic conditional correlation multivariate GARCH (No. w8554). National Bureau of Economic Research.Google Scholar
  17. Enthoven, A. C., & Arrow, K. J. (1956). A theorem on expectations and the stability of equilibrium. Econometrica: Journal of the Econometric Society, 288–293.Google Scholar
  18. Hamilton, J. D. (2010). Regime switching models. In Macroeconometrics and time series analysis (pp. 202–209). Palgrave Macmillan UK.Google Scholar
  19. Hansen, B. E. (1994). Autoregressive conditional density estimation. International Economic Review, 705–730.Google Scholar
  20. Heitsch, H., & Römisch, W. (2009). Scenario tree modeling for multistage stochastic programs. Mathematical Programming, 118(2), 371–406.CrossRefGoogle Scholar
  21. Hochreiter, R., & Pflug, G. C. (2007). Financial scenario generation for stochastic multi-stage decision processes as facility location problems. Annals of Operations Research, 152(1), 257–272.CrossRefGoogle Scholar
  22. Høyland, K., & Wallace, S. W. (2001). Generating scenario trees for multistage decision problems. Management Science, 47(2), 295–307.CrossRefGoogle Scholar
  23. Hsieh, C. H., & Huang, S. C. (2012). Time-varying dependency and structural changes in currency markets. Emerging Markets Finance and Trade, 48(2), 94–127.CrossRefGoogle Scholar
  24. Hu, L. (2006). Dependence patterns across financial markets: A mixed copula approach. Applied Financial Economics, 16(10), 717–729.CrossRefGoogle Scholar
  25. Ji, X., Zhu, S., Wang, S., & Zhang, S. (2005). A stochastic linear goal programming approach to multistage portfolio management based on scenario generation via linear programming. IIE Transactions, 37(10), 957–969.CrossRefGoogle Scholar
  26. Jin, X. (2009). Large portfolio risk management with dynamic copulas. McGill University: Unpublished paper.CrossRefGoogle Scholar
  27. Kaut, M. (2014). A copula-based heuristic for scenario generation. Computational Management Science, 11(4), 503–516.CrossRefGoogle Scholar
  28. Kaut, M., & Wallace, S. W. (2011). Shape-based scenario generation using copulas. Computational Management Science, 8(1), 181–199.CrossRefGoogle Scholar
  29. Longin, F., & Solnik, B. (2001). Extreme correlation of international equity markets. The Journal of Finance, 56(2), 649–676.CrossRefGoogle Scholar
  30. MacQueen, J. (1967). Some methods for classification and analysis of multivariate observations. Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, 14(1), 281–297.Google Scholar
  31. McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative risk management: Concepts, techniques and tools (Revised ed.). Princeton: Princeton University Press.Google Scholar
  32. Mehrotra, S., & Papp, D. (2013). Generating moment matching scenarios using optimization techniques. SIAM Journal on Optimization, 23(2), 963–999.CrossRefGoogle Scholar
  33. Nelsen, R. B. (2006). An introduction to copulas, ser., Lecture Notes in Statistics New York, NY: Springer.Google Scholar
  34. Ng, W. L. (2008). Modeling duration clusters with dynamic copulas. Finance Research Letters, 5(2), 96–103.CrossRefGoogle Scholar
  35. Patton, A. J. (2004). On the out-of-sample importance of skewness and asymmetric dependence for asset allocation. Journal of Financial Econometrics, 2(1), 130–168.CrossRefGoogle Scholar
  36. Patton, A. J. (2006). Modelling asymmetric exchange rate dependence. International Economic Review, 47(2), 527–556.CrossRefGoogle Scholar
  37. Pflug, G. C. (2001). Scenario tree generation for multiperiod financial optimization by optimal discretization. Mathematical Programming, 89(2), 251–271.CrossRefGoogle Scholar
  38. Pflug, G. C., & Pichler, A. (2014). Multistage stochastic optimization. Cham: Springer International Publishing.CrossRefGoogle Scholar
  39. Pflug, G. C., & Pichler, A. (2015). Dynamic generation of scenario trees. Computational Optimization and Applications, 62(3), 641–668.CrossRefGoogle Scholar
  40. Pranevicius, H., & Sutiene, K. (2007). Scenario tree generation by clustering the simulated data paths. In Proceedings 21st European conference on modelling and simulation (pp. 203–208).Google Scholar
  41. Rubasheuski, U., Oppen, J., & Woodruff, D. L. (2014). Multi-stage scenario generation by the combined moment matching and scenario reduction method. Operations Research Letters, 42(5), 374–377.CrossRefGoogle Scholar
  42. Römisch, W., & Schultz, R. (1991). Stability analysis for stochastic programs. Annals of Operations Research, 30(1), 241–266.CrossRefGoogle Scholar
  43. Shapiro, A., Dentcheva, D., & Ruszczyski, A. (2009). Lectures on stochastic programming: modeling and theory. Society for Industrial and Applied Mathematics.Google Scholar
  44. Sutiene, K., & Pranevicius, H. (2007). Scenario generation employing copulas. In World congress on engineering (pp. 777–784).Google Scholar
  45. Xu, D., Chen, Z., & Yang, L. (2012). Scenario tree generation approaches using \(K\)-means and LP moment matching methods. Journal of Computational & Applied Mathematics, 236(17), 4561–4579.CrossRefGoogle Scholar
  46. Ziemba, W. T., & Vickson, R. G. (2006). Stochastic optimization models in finance. Singapore: World Scientific.CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Computing ScienceSchool of Mathematics and Statistics, Xi’an Jiaotong UniversityXi’anPeople’s Republic of China
  2. 2.Department of Management, Economics and Quantitative MethodsUniversity of BergamoBergamoItaly

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