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Stochastic Discount Factors

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Modeling with Stochastic Programming

Abstract

In this chapter, we discuss ways to use information from financial markets to calibrate models for discounting future risks. This type of information is important for modeling the impact of future uncertainty on present decisions. In some areas of activity, there exist well developed financial markets with hordes of traders using the tools and information available to them to decide the present value of future events. This chapter describes a methodology to use market information.

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References

  1. Laurent El Ghaoui Aharon Ben-Tal and Arkadi Nemirovski. Robust optimization. Princeton Series in Applied Mathematics. Princeton University Press, 2009.

    Google Scholar 

  2. M. Ball, C. Barnhart, G. Nemhauser, and A. Odoni. Air transportation: Irregular operations and control. In C. Barnhart and G. Laporte, editors, Transportation, number 14 in Handbooks in Operations Research and Management Science, chapter 1, pages 1–67. Elsevier, 2007.

    Google Scholar 

  3. Güzin Bayraksan and David P. Morton. Assessing solution quality in stochastic programs. Mathematical Programming, 108(2–3):495–514, sep 2006. doi: 10.1007/s10107-006-0720-x.

    Google Scholar 

  4. Güzin Bayraksan and David P. Morton. A sequential sampling procedure for stochastic programming. Operations Research, 59(4):898–913, 2011. doi: 10.1287/opre.1110.0926.

    Google Scholar 

  5. Dimitris Bertsimas and Melvyn Sim. The price of robustness. Operations Research, 52 (1):35–53, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  6. John R. Birge. Option methods for incorporating risk into linear capacity planning models. Manufacturing & Service Operations Management, 2(1):19–31, 2000.

    Article  Google Scholar 

  7. John R. Birge and François Louveaux. Introduction to Stochastic Programming. Springer, Berlin Heidelberg New York, 1997.

    MATH  Google Scholar 

  8. George B. Dantzig and Gerd Infanger. Large-scale stochastic linear programs—importance sampling and Benders decomposition. In Computational and applied mathematics, I (Dublin, 1991), pages 111–120. North-Holland, Amsterdam, 1992.

    Google Scholar 

  9. Jitka Dupačová and Werner Römisch. Quantitative stability for scenario-based stochastic programs. In Marie Hušková, Petr Lachout, and Jan Ámos Víšek, editors, Prague Stochastics ’98, pp 119–124. JČMF, 1998.

    Google Scholar 

  10. Jitka Dupačová, Nicole Gröwe-Kuska, and Werner Römisch. Scenario reduction in stochastic programming: An approach using probability metrics. Mathematical Programming, 95(3):493–511, 2003. doi: 10.1007/s10107-002-0331-0.

    Google Scholar 

  11. M. Ehrgott and D.M. Ryan. Constructing robust crew schedules with bicriteria optimization. Journal of Multi-Criteria Decision Analysis, 11(3):139–150, 2002.

    Article  MATH  Google Scholar 

  12. Matthias Ehrgott and David M. Ryan. The method of elastic constraints for multiobjective combinatorial optimization and its application in airline crew scheduling. In T. Tanino, T. Tanaka, and M. Inuiguchi, editors, Multi-Objective Programming and Goal Programming – Theory and Applications, pages 117–122. Springer, Berlin Heidelberg New York, 2003.

    Google Scholar 

  13. Y. Ermoliev. Stochastic quasigradient methods and their application to system optimization. Stochastics, 9:1–36, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  14. Olga Fiedler and Werner Römisch. Stability in multistage stochastic programming. Annals of Operations Research, 56(1):79–93, 2005. doi: 10.1007/BF02031701.

    Google Scholar 

  15. K. Froot and J. Stein. Risk management, capital budgeting and capital structure policy for financial institutions: An integrated approach. Journal of Financial Economics, 47:55–82, 1998.

    Article  Google Scholar 

  16. A. Gaivoronski. Stochastic quasigradient methods and their implementation. In Numerical techniques for stochastic optimization, volume 10 of Springer Ser. Comput. Math., pp 313–351. Springer, Berlin Heidelberg New York, 1988.

    Google Scholar 

  17. V. Gaur and S. Seshadri. Hedging inventory risk through market instruments. Manufacturing and Service Operations Management, 7(2):103–120, 2005.

    Article  Google Scholar 

  18. R.C. Grinold. Model building techniques for the correction of end effects in multistage convex programs. Operations Research, 31(4):407–431, 1983.

    Article  MATH  Google Scholar 

  19. J. Michael Harrison and Stanley R. Pliska. Martingales and stochastic integrals in the theory of continuous time trading. Stochastic Processes and Their Applications, 11:215–260, 1981.

    Google Scholar 

  20. H. Heitsch and W. Römisch. Scenario reduction algorithms in stochastic programming. Computational Optimization and Applications, 24(2–3):187–206, 2003. doi: 10.1023/A:1021805924152.

    Article  MathSciNet  MATH  Google Scholar 

  21. H. Heitsch, W. Römisch, and C. Strugarek. Stability of multistage stochastic programs. SIAM Journal on Optimization, 17(2):511–525, 2006. doi: 10.1137/050632865.

    Article  MathSciNet  MATH  Google Scholar 

  22. Holger Heitsch and Werner Römisch. A note on scenario reduction for two-stage stochastic programs. Operations Research Letters, 35(6):731–738, 2007. doi: 10.1016/j.orl.2006.12.008.

    Google Scholar 

  23. Holger Heitsch and Werner Römisch. Scenario tree reduction for multistage stochastic programs. Computational Management Science, 6(2):117–133, 2009. doi: 10.1007/s10287-008-0087-y.

    Google Scholar 

  24. J. L. Higle and S. Sen. Stochastic decomposition: An algorithm for two-stage linear programs with recourse. Mathematics of Operations Research, 16:650–669, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  25. J. L. Higle and S. Sen. Statistical verification of optimality conditions for stochastic programs with recourse. Annals of Operations Research, 30:215–240, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  26. J. L. Higle and S. W. Wallace. Sensitivity analysis and uncertainty in linear programming. Interfaces, 33:53–60, 2003.

    Article  Google Scholar 

  27. K. Høyland and S. W. Wallace. Generating scenario trees for multistage decision problems. Management Science, 47(2):295–307, 2001. doi: 10.1287/mnsc.47.2.295.9834.

    Article  Google Scholar 

  28. Kjetil Høyland, Michal Kaut, and Stein W. Wallace. A heuristic for moment-matching scenario generation. Computational Optimization and Applications, 24(2–3):169–185, 2003. ISSN 0926-6003.

    Google Scholar 

  29. Gerd Infanger. Monte Carlo (importance) sampling within a Benders decomposition algorithm for stochastic linear programs. Annals of Operations Research, 39(1–4): 69–95 (1993), 1992. ISSN 0254-5330.

    Google Scholar 

  30. P. Kall and S.W. Wallace. Stochastic Programming. Wiley, Chichester, 1994.

    MATH  Google Scholar 

  31. Michal Kaut and Stein W. Wallace. Evaluation of scenario-generation methods for stochastic programming. Pacific Journal of Optimization, 3(2):257–271, 2007.

    MathSciNet  MATH  Google Scholar 

  32. Michal Kaut and Stein W. Wallace. Shape-based scenario generation using copulas. Computational Management Science, 8(1–2):181–199, 2011. doi: 10.1007/s10287-009-0110-y.

    Google Scholar 

  33. Michal Kaut, Stein W. Wallace, Hercules Vladimirou, and Stavros Zenios. Stability analysis of portfolio management with conditional value-at-risk. Quantitative Finance, 7(4):397–409, 2007. doi: 10.1080/14697680701483222.

    Google Scholar 

  34. A. J. King. Asymmetric risk measures and tracking models for portfolio optimization under uncertainty. Annals of Operations Research, 45:165–177, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  35. Alan J. King. Duality and martingales: A stochastic programming perspective on contingent claims. Mathematical Programming, 91(3):543–562, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  36. Alan J. King, Teemu Pennanen, and Matti Koivu. Calibrated option bounds. International Journal of Theoretical and Applied Finance, 8:141–159, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  37. Alan J. King, Olga Streltchenko, and Yelena Yesha. Private valuation of contingent claims in a discrete time/state model. In John B. Guerard, editor, Handbook of Portfolio Construction: Contemporary Applications, pp 691–710. Springer, Berlin Heidelberg New York, 2010.

    Google Scholar 

  38. Anton J. Kleywegt, Alexander Shapiro, and Tito Homem-de Mello. The sample average approximation method for stochastic discrete optimization. SIAM Journal on Optimization, 12(2):479–502, 2001. doi: 10.1137/S1052623499363220.

    Google Scholar 

  39. A. G. Kök, M.L. Fisher, and R. Vaidyanathan. Assortment planning: Review of literature and industry practice. In N. Agrawal and S.A. Smith, editors, Retail Supply Chain Management, pp 99–154. Springer, Berlin Heidelberg New York, 2008.

    Chapter  Google Scholar 

  40. A.-G. Lium, T. G. Crainic, and S. W. Wallace. A study of demand stochasticity in stochastic network design. Transportation Science, 43(2):144–157, 2009. doi: 10.1287/trsc.1090.0265.

    Article  Google Scholar 

  41. Leonard C. MacLean, Edward O. Thorp, and William T. Ziemba. The Kelly capital growth investment criterion: Theory and practice. Handbook in Financial Economics. World Scientific, Singapore, 2011.

    Google Scholar 

  42. S. Mahajan and G. van Ryzin. Retail inventories and consumer choice. In S. Tayur, R. Ganesham, and M. Magasine, editors, Quantitative methods in Supply Chain Management. Kluwer, Amsterdam, 1998.

    Google Scholar 

  43. W.K. Mak, D.P. Morton, and R.K. Wood. Monte carlo bounding techniques for determining solution quality in stochastic programs. Operations Research Letters, 24: 47–56, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  44. H.M. Markowitz. Portfolio selection: Efficient diversification of investment. Yale University Press, New Haven, CT, 1959.

    Google Scholar 

  45. G. C. Pflug. Scenario tree generation for multiperiod financial optimization by optimal discretization. Mathematical Programming, 89(2):251–271, 2001. doi: 10.1007/PL00011398.

    Article  MathSciNet  MATH  Google Scholar 

  46. András Prékopa. Stochastic programming, vol 324. Mathematics and Its Applications. Kluwer, Dordrecht, 1995. ISBN 0-7923-3482-5.

    Google Scholar 

  47. J. M. Rosenberger, E. L. Johnson, and G. L. Nemhauser. A robust fleet-assignment model with hub isolation and short cycles. Transportation Science, 38(3):357–368, 2004.

    Article  Google Scholar 

  48. Andrej Ruszczynski and Alexanger Shapiro. Stochastic Programming. Handbooks in Operations Research and Management Science. Elsevier, Amsterdam, 2002.

    Google Scholar 

  49. Alexander Shapiro. Monte carlo sampling approach to stochastic programming. ESAIM: Proceedings, 13: 65–73, 2003. doi: 10.1051/proc:2003003. Proceedings of 2003 MODE-SMAI Conference.

    Google Scholar 

  50. Alexander Shapiro. Monte Carlo sampling methods. In A. Ruszczyński and A. Shapiro, editors, Stochastic Programming, volume 10 of Handbooks in Operations Research and Management Science, chapter 6, pp 353–425. Elsevier, Amsterdam, 2003. doi: 10.1016/S0927-0507(03)10006-0.

    Google Scholar 

  51. Gordon Sick. Real options. In Finance, vol 9. Handbooks in Operations Research and Management Science, chap 21, pp 631–691. Elsevier, Amsterdam, 1995.

    Google Scholar 

  52. J.W. Suurballe and Robert E. Tarjan. A quick method for finding shortest pairs of paths. Networks, 14:325–336, 1984.

    Google Scholar 

  53. Hajnalka Vaagen and Stein W. Wallace. Product variety arising from hedging in the fashion supply chains. International Journal of Production Economics, 114(2):431–455, 2008. doi: 10.1016/j.ijpe.2007.11.013.

    Google Scholar 

  54. J. von Neumann and O. Morgenstern. Theory of Games and Economic Behavior, 2nd edn. Princeton University Press, Princeton, NJ, 1947.

    MATH  Google Scholar 

  55. S.W. Wallace. Decision making under uncertainty: Is sensitivity analysis of any use? Operations Research, 48:20–25, 2000.

    Article  Google Scholar 

  56. S.W. Wallace and W.T. Ziemba, editors. Applications of Stochastic Programming. MPS-SIAM Series on Optimization, Philadelphia, 2005.

    MATH  Google Scholar 

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King, A.J., Wallace, S.W. (2012). Stochastic Discount Factors. In: Modeling with Stochastic Programming. Springer Series in Operations Research and Financial Engineering. Springer, New York, NY. https://doi.org/10.1007/978-0-387-87817-1_7

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