Optimal Financial Decision Making Under Uncertainty

  • Giorgio ConsigliEmail author
  • Daniel Kuhn
  • Paolo Brandimarte
Part of the International Series in Operations Research & Management Science book series


We use a fairly general framework to analyze a rich variety of financial optimization models presented in the literature, with emphasis on contributions included in this volume and a related special issue of OR Spectrum. We do not aim at providing readers with an exhaustive survey, rather we focus on a limited but significant set of modeling and methodological issues. The framework is based on a benchmark discrete-time stochastic control optimization framework, and a benchmark financial problem, asset-liability management, whose generality is considered in this chapter. A wide set of financial problems, ranging from asset allocation to financial engineering problems, is outlined, in terms of objectives, risk models, solution methods, and model users. We pay special attention to the interplay between alternative uncertainty representations and solution methods, which have an impact on the kind of solution which is obtained. Finally, we outline relevant directions for further research and optimization paradigms integration.


Stochastic control Dynamic programming Multistage stochastic programming Robust optimization Distributionally robust optimization Decision rules Asset-liability management Pension fund management 


  1. 1.
    H. Aro, T. Pennanen, Liability driven investment in longevity risk management. This volumeGoogle Scholar
  2. 2.
    P. Artzner, F. Delbaen, J.-M. Eber, D. Heath, Coherent measures of risk. Math. Financ. 9, 203–228 (1999)CrossRefGoogle Scholar
  3. 3.
    M. Avriel, H. Pri-Zan, R. Meiri, A. Peretz, Opti-money at Bank Hapoalim: a model-based investmentdecision-supportsystem for individual customers Interfaces 34 (1), 39–50 (2004)Google Scholar
  4. 4.
    C. Bandi, D. Bertsimas, Tractable stochastic analysis in high dimensions via robust optimization. Math. Program. 134 (1), 23–70 (2012)CrossRefGoogle Scholar
  5. 5.
    C. Bandi, D. Bertsimas, Robust option pricing. Eur. J. Oper. Res. 239 (3), 842–853 (2014)CrossRefGoogle Scholar
  6. 6.
    A. Ben-Tal, T. Margalit, A. Nemirovski, Robust modeling of multi-stage portfolio problems, in High Performance Optimization, ed. by H. Frenk, K. Roos, T. Terlaky, S. Zhang (Springer US, New York, 2000), pp. 303–328CrossRefGoogle Scholar
  7. 7.
    A. Ben-Tal, A. Goryashko, E. Guslitzer, A. Nemirovski, Adjustable robust solutions of uncertain linear programs. Math. Program. 99 (2), 351–376 (2004)CrossRefGoogle Scholar
  8. 8.
    A. Ben-Tal, S. Boyd, A. Nemirovski, Extending scope of robust optimization: comprehensive robust counterparts of uncertain problems. Math. Program. 107 (2), 63–89 (2006)CrossRefGoogle Scholar
  9. 9.
    A. Ben-Tal, L. El Ghaoui, A. Nemirovski, Robust Optimization (Princeton University Press, Princeton, 2009)CrossRefGoogle Scholar
  10. 10.
    M. Bertocchi, G. Consigli, M.A.H. Dempster (eds.), Stochastic Optimization Methods in Finance and Energy. New Financial Products and Energy Market Strategies. Fred Hillier International Series in Operations Research and Management Science (Springer US, New York, 2011)Google Scholar
  11. 11.
    D. Bertsimas, D. Pachamanova, Robust multiperiod portfolio management in the presence of transaction costs. Comput. Oper. Res. 35 (1), 3–17 (2008)CrossRefGoogle Scholar
  12. 12.
    D. Bertsimas, M. Sim, The price of robustness. Oper. Res. 52 (1), 35–53 (2004)CrossRefGoogle Scholar
  13. 13.
    D. Bertsimas, C. Darnell, R. Soucy, Portfolio construction through mixed-integer programming at Grantham, Mayo, Van Otterloo and company. Interfaces 29 (1), 49–66 (1999)Google Scholar
  14. 14.
    D. Bertsimas, V. Gupta, N. Kallus, Data-driven robust optimization (2013). Available from arXiv:1401.0212Google Scholar
  15. 15.
    J.R. Birge, F. Louveaux, Introduction to Stochastic Programming, 2nd edn. Springer Series in Operations Research and Financial Engineering (Springer, New York, 2011)Google Scholar
  16. 16.
    R. Bookstaber, A Demon of Our Own Design (Wiley, New York, 2007)Google Scholar
  17. 17.
    R. Bookstaber, J. Langsam, On the optimality of coarse behavior rules. J. Theor. Biol. 116 (2), 161–193 (1985)CrossRefGoogle Scholar
  18. 18.
    P. Brandimarte, Handbook in Monte Carlo Simulation: Applications in Financial Engineering, Risk Management, and Economics (Wiley, New York, 2014)CrossRefGoogle Scholar
  19. 19.
    M.W. Brandt, P. Santa-Clara, R. Valkanov, Parametric portfolio policies: exploiting characteristics in the cross-section of equity returns. Rev. Financ. Stud. 22 (9), 3411–3447 (2009)CrossRefGoogle Scholar
  20. 20.
    M. Broadie, Computing efficient frontiers using estimated parameters. Ann. Oper. Res. 45 (1), 21–58 (1993)CrossRefGoogle Scholar
  21. 21.
    R. Bruni, F. Cesarone, A. Scozzari, F. Tardella, A linear risk-return model for enhanced indexation in portfolio optimization. OR Spectr. 37 (3), 735–759 (2015)CrossRefGoogle Scholar
  22. 22.
    G.C. Calafiore, Multi-period portfolio optimization with linear control policies. Automatica 44 (10), 2463–2473 (2008)CrossRefGoogle Scholar
  23. 23.
    G.C. Calafiore, An affine control method for optimal dynamic asset allocation with transaction costs. SIAM J. Control. Optim. 48 (4), 2254–2274 (2009)CrossRefGoogle Scholar
  24. 24.
    G.C. Calafiore, Random convex programs. SIAM J. Optim. 20 (6), 3427–3464 (2010)CrossRefGoogle Scholar
  25. 25.
    G.C. Calafiore, Scenario optimization methods in portfolio analysis and design. This volumeGoogle Scholar
  26. 26.
    G.C. Calafiore, M.C. Campi, The scenario approach to robust control design. IEEE Trans. Autom. Control 51 (5), 742–753 (2006)CrossRefGoogle Scholar
  27. 27.
    J.Y. Campbell, L.M. Viceira, Strategic Asset Allocation (Oxford University Press, Oxford, 2002)CrossRefGoogle Scholar
  28. 28.
    M.C. Campi, S. Garatti, The exact feasibility of randomized solutions of uncertain convex programs. SIAM J. Optim. 19 (3), 1211–1230 (2008)CrossRefGoogle Scholar
  29. 29.
    D.R. Cariño, T. Kent, D.H. Myers, C. Stacy, M. Sylvanus, A.L. Turner, K. Watanabe, W.T. Ziemba, The Russell-Yasuda Kasai Model: an asset/liability model for a Japanese insurance company using multistage stochastic programming. Interfaces 24 (1), 29–49 (1994)CrossRefGoogle Scholar
  30. 30.
    S. Ceria, R. Stubbs, Incorporating estimation errors into portfolio selection: Robust portfolio construction. J. Asset Manag. 7 (2), 109–127 (2006)CrossRefGoogle Scholar
  31. 31.
    R. Cerqueti, P. Falbo, G. Guastaroba, C. Pelizzari, Approximating multivariate Markov chains for bootstrapping through contiguous partitions. OR Spectr. 37 (3), 803–841 (2015)CrossRefGoogle Scholar
  32. 32.
    E. Çetinkaya, A. Thiele, Data-driven portfolio management with quantile constraints. OR Spectr. 37 (3), 761–786 (2015)CrossRefGoogle Scholar
  33. 33.
    X. Chen, M. Sim, P. Sun, A robust optimization perspective on stochastic programming. Oper. Res. 55 (6), 1058–1071 (2007)CrossRefGoogle Scholar
  34. 34.
    Z. Chen, G. Consigli, J. Liu, G. Li, T. Fu, Q. Hu, Multi-period risk measures and dynamic risk control. This volumeGoogle Scholar
  35. 35.
    V.K. Chopra, W.T. Ziemba, The effect of errors in means, variances and covariances on optimal portfolio choice. J. Portf. Manag. 19 (2), 6–11 (1993)CrossRefGoogle Scholar
  36. 36.
    G. Consigli, V. Moriggia, Applying stochastic programming to insurance portfolios stress-testing. Quant. Financ. Lett. 2 (1), 7–13 (2014)CrossRefGoogle Scholar
  37. 37.
    G. Consigli, G. Iaquinta, V. Moriggia, Path-dependent scenario trees for multistage stochastic programs in finance. Quant. Financ. 12 (8), 1265–1281 (2012)CrossRefGoogle Scholar
  38. 38.
    G. Consigli, P. Brandimarte, D. Kuhn, Editorial: financial optimization: optimization paradigms and financial planning under uncertainty. OR Spectr. 37 (3), 553–557 (2015)CrossRefGoogle Scholar
  39. 39.
    M.H.A. Davis, S. Lleo, Jump-diffusion asset-liability management via risk-sensitive control. OR Spectr. 37 (3), 655–675 (2015)CrossRefGoogle Scholar
  40. 40.
    E. Delage, Y. Ye, Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58 (3), 595–612 (2010)CrossRefGoogle Scholar
  41. 41.
    V. DeMiguel, L. Garlappi, R. Uppal, Optimal versus naive diversification: how inefficient is the 1/N portfolio strategy? Rev. Financ. Stud. 22 (5), 1915–1953 (2009)CrossRefGoogle Scholar
  42. 42.
    M.A.H. Dempster, E.A. Medova, Y.S. Yong, Stabilizing implementable decisions in dynamic stochastic programming. This volumeGoogle Scholar
  43. 43.
    M.A.H. Dempster, M. Germano, E.A. Medova, M. Villaverde, Global asset liability management. Br. Actuar. J. 9 (1), 137–195 (2003)CrossRefGoogle Scholar
  44. 44.
    S. Desmettre, R. Korn, P. Ruckdeschel, F.T. Seifried, Robust worst-case optimal investment. OR Spectr. 37 (3), 677–701 (2015)CrossRefGoogle Scholar
  45. 45.
    X. Doan, X. Li, K. Natarajan, Robustness to dependency in portfolio optimization using overlapping marginals. Available on Optimization Online (2013)Google Scholar
  46. 46.
    T. Driouchi, L. Trigeorgis, Y. Gao, Choquet-based European option pricing with stochastic (and fixed) strikes. OR Spectr. 37 (3), 787–802 (2015)CrossRefGoogle Scholar
  47. 47.
    J. Dupačová, The minimax approach to stochastic programming and an illustrative application. Stochastics 20, 73–88 (1987)CrossRefGoogle Scholar
  48. 48.
    J. Dupacova, G. Consigli, S.W. Wallace, Scenarios for multistage stochastic programmes. Ann. Oper. Res. 100, 25–53 (2001)CrossRefGoogle Scholar
  49. 49.
    J. Dupačová, V. Kozmík, Structure of risk-averse multistage stochastic programs. OR Spectr. 37 (3), 559–582 (2015)CrossRefGoogle Scholar
  50. 50.
    L. El Ghaoui, M. Oks, F. Outstry, Worst-case value-at-risk and robust portfolio optimization: a conic programming approach. Oper. Res. 51 (4), 543–556 (2003)CrossRefGoogle Scholar
  51. 51.
    E. Erdoğan, G. Iyengar, Ambiguous chance constrained problems and robust optimization. Math. Program. B 107 (1–2), 37–61 (2006)CrossRefGoogle Scholar
  52. 52.
    D. Fabozzi, P. Kolm, D. Pachamanova, Robust Portfolio Optimization and Management (Wiley, New York, 2007)Google Scholar
  53. 53.
    A. Georghiou, W. Wiesemann, D. Kuhn, Generalized decision rule approximations for stochastic programming via liftings. Math. Program. 152 (1–2), 301–338 (2015)CrossRefGoogle Scholar
  54. 54.
    M. Giandomenico, M. Pinar, Pricing American options: binomial and trinomial non-recombinant trees and two exercise rights. This volumeGoogle Scholar
  55. 55.
    M. Gilli, E. Schumann, Constructing portfolios with heuristics. This volumeGoogle Scholar
  56. 56.
    J. Goh, M. Sim, Distributionally robust optimization and its tractable approximations. Oper. Res. 58 (4), 902–917 (2010)CrossRefGoogle Scholar
  57. 57.
    D. Goldfarb, G. Iyengar, Robust portfolio selection problems. Math. Oper. Res. 28 (1), 1–38 (2003)CrossRefGoogle Scholar
  58. 58.
    L. Györfi, G. Ottucsak, H. Walk, The growth optimal strategy is secure too. This volumeGoogle Scholar
  59. 59.
    G.A. Hanasusanto, D. Kuhn, Robust data-driven dynamic programming, in Advances in Neural Information Processing Systems, pp. 827–835 (2013)Google Scholar
  60. 60.
    Z. Hu, L.J. Hong, Kullback-Leibler divergence constrained distributionally robust optimization. Available from Optimization Online (2013)Google Scholar
  61. 61.
    G. James, D. Witten, T. Hastie, R. Tibshirani, An Introduction to Statistical Learning: with Applications in R (Springer, Berlin, 2013)Google Scholar
  62. 62.
    B. Kawas, A. Thiele, A log-robust optimization approach to portfolio management. OR Spectr. 33 (1), 207–233 (2011)CrossRefGoogle Scholar
  63. 63.
    J.M. Keynes, A Treatise on Probability (MacMillan, London, 1921)Google Scholar
  64. 64.
    A.J. King, Duality and martingales: a stochastic programming perspective on contingent claims. Math. Program. B 91, 543–562 (2002)CrossRefGoogle Scholar
  65. 65.
    F.H. Knight, Risk, Uncertainty and Profit (Hart, Schaffner and Marx, Boston, 1921)Google Scholar
  66. 66.
    A.K. Konicz, D. Pisinger, K.M. Rasmussen, A combined stochastic programming and optimal control approach to personal finance and pensions. OR Spectr. 37 (3), 583–616 (2015)CrossRefGoogle Scholar
  67. 67.
    M. Kopa, T. Post, A general test for SSD portfolio efficiency. OR Spectr. 37 (3), 703–734 (2015)CrossRefGoogle Scholar
  68. 68.
    D. Kuhn, W. Wiesemann, A. Georghiou, Primal and dual linear decision rules in stochastic and robust optimization. Math. Program. 130 (1), 177–209 (2011)CrossRefGoogle Scholar
  69. 69.
    D.G. Luenberger, Investment Science (Oxford University Press, New York, 1998)Google Scholar
  70. 70.
    L.C. MacLean, Y. Zhao, Asset price dynamics: shocks and regimes. This volumeGoogle Scholar
  71. 71.
    H.M. Markowitz, Portfolio Selection: Efficient Diversification of Investments, 2nd edn. (Wiley, New York, 1991)Google Scholar
  72. 72.
    R.C. Merton, On estimating the expected return on the market: an exploratory investigation. J. Financ. Econ. 8 (4), 323–361 (1980)CrossRefGoogle Scholar
  73. 73.
    R.O. Michaud, Efficient Asset Management: A Practical Guide to Stock Portfolio Management and Asset Allocation (Oxford University Press, Oxford, 2001)Google Scholar
  74. 74.
    C.C. Moallemi, M. Saglam, Dynamic portfolio choice with linear rebalancing rules (2015). Available at SSRN:
  75. 75.
    P. Mohajerin Esfahani, D. Kuhn, Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations. Available from Optimization Online (2015)Google Scholar
  76. 76.
    M. Morini, Understanding and Manageing Model RIsk: A Practical Guide for Quants, Traders and Validators (Wiley, New York, 2011)Google Scholar
  77. 77.
    J.M. Mulvey, G. Gould, C. Morgan, An asset and liability management system for towers Perrin-Tillinghast. Interfaces 30 (1), 96–114 (2000)Google Scholar
  78. 78.
    J.M. Mulvey, W.C. Kim, C. Lin, Optimizing a portfolio of liquid and illiquid assets. This volumeGoogle Scholar
  79. 79.
    K. Natarajan, D. Pachamanova, M. Sim, Incorporating asymmetric distributional information in robust value-at-risk optimization. Manag. Sci. 54 (3), 573–585 (2008)CrossRefGoogle Scholar
  80. 80.
    K. Natarajan, M. Sim, J. Uichanco, Tractable robust expected utility and risk models for portfolio optimisation. Math. Financ. 20 (4), 695–731 (2010)CrossRefGoogle Scholar
  81. 81.
    D. Pachamanova, N. Gulpinar, E. Canakoglu, Robust data-driven approaches to pension fund asset-liability management under uncertainty. This volumeGoogle Scholar
  82. 82.
    G.C. Pflug, W. Römisch, Modeling, Measuring and Managing Risk (World Scientific Publishing, Singapore, 2007)CrossRefGoogle Scholar
  83. 83.
    G.C. Pflug, D. Wozabal, Ambiguity in portfolio selection. Quant. Finan. 7 (4), 435–442 (2007)CrossRefGoogle Scholar
  84. 84.
    G.C. Pflug, A. Pichler, D. Wozabal, The 1/n investment strategy is optimal under high model ambiguity. J. Bank. Financ. 36 (2), 410–417 (2012)CrossRefGoogle Scholar
  85. 85.
    S.T. Rachev, J.S.J. Hsu, B.S. Bagasheva, F.J. Fabozzi, Bayesian Methods in Finance (Wiley, New York, 2008)Google Scholar
  86. 86.
    N. Rujeerapaiboon, D. Kuhn, W. Wiesemann, Robust growth-optimal portfolios. Manag. Sci. 62 (7), 2090–2109 (2016)CrossRefGoogle Scholar
  87. 87.
    S. Satchell, A. Scowcroft, A demystification of the Black–Litterman model: managing quantitative and traditional portfolio construction. J. Asset Manag. 1, 138–150 (2000)CrossRefGoogle Scholar
  88. 88.
    M. Sekerke, Bayesian Risk Management: A Guide to Model Risk and Sequential Learning in Financial Markets (Wiley, New York, 2015)CrossRefGoogle Scholar
  89. 89.
    A. Shapiro, D. Dentcheva, A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory (SIAM, Philadelphia, 2009)CrossRefGoogle Scholar
  90. 90.
    A. Shapiro, A. Nemirovski, On complexity of stochastic programming problems, in Continuous Optimization: Current Trends and Applications, ed. by V. Jeyakumar, A.M. Rubinov (Springer, Berlin, 2005), pp. 111–144CrossRefGoogle Scholar
  91. 91.
    G. Szego (ed.), Risk Measures for the 21st Century (Wiley, New York, 2004)Google Scholar
  92. 92.
    R.H. Tütüncü, M. Koenig, Robust asset allocation. Ann. Oper. Res. 132 (1–4), 157–187 (2004)CrossRefGoogle Scholar
  93. 93.
    S. Pagliarani, T. Vargiolu, Portfolio optimization in a defaultable Lévy-driven market model. OR Spectr. 37 (3), 617–654 (2015)CrossRefGoogle Scholar
  94. 94.
    P. Vayanos, D. Kuhn, B. Rustem, A constraint sampling approach for multi-stage robust optimization. Automatica 48 (3), 459–471 (2012)CrossRefGoogle Scholar
  95. 95.
    W. Wiesemann, D. Kuhn, M. Sim, Distributionally robust convex optimization. Oper. Res. 62 (6), 1358–1376 (2014)CrossRefGoogle Scholar
  96. 96.
    H. Xu, D. Zhang, Monte Carlo methods for mean-risk optimization and portfolio selection. Comput. Manag. Sci. 9 (1), 3–29 (2012)CrossRefGoogle Scholar
  97. 97.
    S. Zymler, D. Kuhn, B. Rustem, Worst-case value at risk of nonlinear portfolios. Manag. Sci. 59 (1), 172–188 (2013)CrossRefGoogle Scholar
  98. 98.
    S. Zymler, B. Rustem, D. Kuhn, Robust portfolio optimization with derivative insurance guarantees. Eur. J. Oper. Res. 210 (2), 410–424 (2011)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  • Giorgio Consigli
    • 1
    Email author
  • Daniel Kuhn
    • 2
  • Paolo Brandimarte
    • 3
  1. 1.Department of Management Economics and Quantitative MethodsUniversity of BergamoBergamoItaly
  2. 2.Risk Analytics and Optimization ChairÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland
  3. 3.Dipartimento di Scienze MatematichePolitecnico di TorinoTorinoItaly

Personalised recommendations