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Optimal Investment in Incomplete Financial Markets

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Mathematical Finance — Bachelier Congress 2000

Part of the book series: Springer Finance ((FINANCE))

Abstract

We give a review of classical and recent results on maximization of expected utility for an investor who has the possibility of trading in a financial market. Emphasis will be given to the duality theory related to this convex optimization problem.

For expository reasons we first consider the classical case where the underlying probability space Ω is finite. This setting has the advantage that the technical difficulties of the proofs are reduced to a minimum, which allows for a clearer insight into the basic ideas, in particular the crucial role played by the Legendre-transform. In this setting we state and prove an existence and uniqueness theorem for the optimal investment strategy, and its relation to the dual problem; the latter consists in finding an equivalent martingale measure optimal with respect to the conjugate of the utility function. We also discuss economic interpretations of these theorems.

We then pass to the general case of an arbitrage-free financial market modeled by an ℝd-valued semi-martingale. In this case some regularity conditions have to be imposed in order to obtain an existence result for the primal problem of finding the optimal investment, as well as for a proper duality theory. It, turns out, that, one may give a necessary and sufficient condition, namely a mild condition on the asymptotic behavior of the utility function, its so-called reasonable asymptotic elasticity. This property allows for an economic interpretation motivating the term “reasonable”. The remarkable fact, is that, this regularity condition only pertains to the behavior of the utility function, while we do not, have to impose any regularity conditions on the stochastic process modeling the financial market, (to be precise: of course, we have to require the arbitrage-freeness of this process in a proper sense; also we have to assume in one of the cases considered below that, this process is locally bounded; but, otherwise it, may be an arbitrary ℝd-valued semi-martingale).

We state two general existence and duality results pertaining to the setting of optimizing expected utility of terminal consumption. We also survey some of the ramifications of these results allowing for intermediate consumption, state-dependent, utility, random endowment,, non-smooth utility functions and transaction costs.

Support by the Austrian Science Foundation (FWF) under the Wittgenstein-Preis program Z36-MAT and grant SFB#010 and the Austrian National Bank under grant ’Jubiläumsfondprojekt Number 8699’ is gratefully acknowledged.

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References

  1. P. Artzner, (1997), On the numeraire portfolio. Mathematics of Derivative Securities, M. Dempster and S. Pliska, eds., Cambridge University Press, pp. 53–60.

    Google Scholar 

  2. J.P. Ansel, C. Strieker, (1994), Couverture des actifs contingents et prix maximum. Ann. Inst,. Henri Poincaré, Vol. 30, pp. 303–315.

    MATH  Google Scholar 

  3. D. Becherer, (2000), The numeraire portfolio for unbounded semimartingales. preprint,, TU Berlin.

    Google Scholar 

  4. F. Bellini, M. Frittelli, (2000), On the existence of minimax martin gale measures, preprint,.

    Google Scholar 

  5. W. Brannat.il, W. Schachermayer, (1999), A Bipolar Theorem for Subsets of L\(fl, F, P). Séminaire de Probabilités, Vol. XXXIII, pp. 349–354.

    Google Scholar 

  6. I. Csiszar, (1975), I-Divergence Geometry of Probability Distributions and Minimization Problems. Annals of Probability, Vol. 3, No. 1, pp. 146–158.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Cvitanic, (2000), Minimizing expected loss of hedging in incomplete and constrained markets. Preprint, Columbia University, New York.

    Google Scholar 

  8. P. Collin-Dufresne, J.-N. Huggonnier, (2000), Utility-based pricing of contingent claims subject to counterparty credit risk. Working paper GSIA and Department, of Mathematics, Carnegie Mellon University.

    Google Scholar 

  9. J.C. Cox, C.F. Huang, (1989), Optimal consumption and portfolio policies when asset prices follow a diffusion process. J. Economic Theory, Vol. 49, pp. 33–83.

    Article  MathSciNet  MATH  Google Scholar 

  10. J J.C. Cox, C.F. Huang, (1991), A variational problem arising in fi nancial economics. J. Math. Econ., Vol. 20, pp. 465–487.

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Cvitanic, I. Karatzas, (1996), Hedging and portfolio optimization under transaction costs: A martingale approach. Mathematical Finance 6, pp. 133–165.

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Cvitanic, H. Wang, (2000), On optimal terminal wealth under transaction costs. Preprint.

    Google Scholar 

  13. J. Cvitanic, H. Wang, W. Schachermayer, (2000), Utility Maximization in Incomplete Markets with Random Endowment, preprint (12 pages), to appear in Finance and Stochastics.

    Google Scholar 

  14. M. Davis, (1997), Option pricing in incomplete markets. Mathemat ics of Derivative Securities, eds. M.A.H. Dempster and S.R. Pliska, Cambridge University Press, pp. 216–226.

    Google Scholar 

  15. M. Davis, (2000), Optimal hedging with basis risk. Preprint of the TU Vienna.

    Google Scholar 

  16. F. Delbaen, P. Grandits, T. Rheinländer, D. Samperi, M. Schweizer, C. Strieker, (2000), Exponential hedging and entropie penalties. preprint.

    Google Scholar 

  17. G. Deelstra, H. Pham, N. Touzi, (2000), Dual formulation of the utility maximisation problem under transaction costs. Preprint of EN- SAE and CREST.

    Google Scholar 

  18. F. Delbaen, W. Schachermayer, (1994), A General Version of the Fundamental Theorem of Asset Pricing. Math. Annalen, Vol. 300, pp. 463–520.

    Article  MathSciNet  MATH  Google Scholar 

  19. F. Delbaen, W. Schachermayer, (1995), The No-Arbitrage Property under a change of numéraire. Stochastics and Stochastic Reports, Vol. 53, pp. 213–226.

    MathSciNet  MATH  Google Scholar 

  20. F. Delbaen, W. Schachermayer, (1998), A Simple Counter-example to Several Problems in the Theory of Asset Pricing, which arises in many incomplete markets. Mathematical Finance, Vol. 8, pp. 1–12.

    Article  MathSciNet  MATH  Google Scholar 

  21. F. Delbaen, W. Schachermayer, (1998), The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes. Mathematische Annalen, Vol. 312, pp. 215–250.

    Article  MathSciNet  MATH  Google Scholar 

  22. F. Delbaen, W. Schachermayer, (1999), A Compactness Principle for Bounded Sequences of Martingales with Applications. Proceedings of the Seminar of Stochastic Analysis, Random Fields and Applications, Progress in Probability, Vol. 45, pp. 137–173.

    MathSciNet  Google Scholar 

  23. M. Emery, (1980), Compensation de processus à variation finie non localement integrables. Séminaire de Probabilités XIV, Springer Lecture Notes in Mathematics, Vol. 784, pp. 152–160.

    Article  MathSciNet  Google Scholar 

  24. I. Ekeland, R. Temam, (1976), Convex Analysis and Variational Problems. North Holland.

    Google Scholar 

  25. M. Frittelli, (2000), The minimal entropy martingale measure and the valuation problem in incomplete markets. Mathematical Finance, Vol. 10, pp. 39–52.

    Article  MathSciNet  MATH  Google Scholar 

  26. L.P. Foldes, (1990), Conditions for optimality in the infinitehorizon portfolio-cum-savings problem with semimartingale investments. Stochastics and Stochastics Report, Vol. 29, pp. 133–171.

    Article  MathSciNet  MATH  Google Scholar 

  27. H. Föllmer, P. Leukert,, (2000), Efficient Hedging: Cost versus Short fall Risk. Finance and Stochastics, Vol. 4, No. 2, pp. 117–146.

    Article  MathSciNet  MATH  Google Scholar 

  28. H. Föllmer, M. Schweizer, (1991), Hedging of contingent claims under incomplete information. Applied Stochastic Analysis, Stochastic Monographs, M.H.A. Davis and R.J. Elliott, eds., Gordon and Breach, London New York, Vol. 5, pp. 389–414.

    Google Scholar 

  29. T. Göll, J. Kallsen, (2000), Optimal portfolios for logarithmic utility. Stochastic Processes and Their Applications, Vol. 89, pp. 31–48.

    Article  MathSciNet  Google Scholar 

  30. T. Göll, L. Rüschendorf, (2000), Minimax and minimal distance martingale measures and their relationship to portfolio optimization. Preprint, of the Universität, Freiburg, Germany.

    Google Scholar 

  31. C.-F. Huang, R.H. Litzenberger, (1988), Foundations for Fiancial Economics. North-Holland Publishing Co. New York.

    Google Scholar 

  32. S.D. Hodges, A. Neuberger, (1989), Optimal replication of contingent claims under transaction costs. Review of Futures Markets, Vol. 8, pp. 222–239.

    Google Scholar 

  33. J.M. Harrison, S.R. Pliska, (1981), Martingales and Stochastic intefrals in the theory of continuous trading. Stoch. Proc. and Appl., Vol. 11, pp. 215–260.

    Article  MathSciNet  MATH  Google Scholar 

  34. H. He, N.D. Pearson, (1991), Consumption and Portfolio Policies with Incomplete Markets and Short-Sale Constraints: The Finite- Dimensional Case. Mathematical Finance, Vol. 1, pp. 1–10.

    Article  MathSciNet  Google Scholar 

  35. HP91aJ H. He, N.D. Pearson, (1991), Consumption and Portfolio Policies with Incomplete Markets and Short-Sale Constraints: The Infinite- Dimensional Case. Journal of Economic Theory, Vol. 54, pp. 239–250.

    Google Scholar 

  36. S.D. Jacka, (1992), A martingale representation result and an appli cation to incomplete financial markets. Mathematical Finance, Vol. 2, pp. 239–250.

    Article  MATH  Google Scholar 

  37. B.E. Johnson, (1996), The pricing property of the optimal growth portfolio: extensions and applications, preprint,, Department, of Engineering-Economic-Systems, Stanford University.

    Google Scholar 

  38. M. Jonsson, K.R. Sircar, (2000), Partial hedging in a stochastic volatility environment. Preprint, of the Princeton University, Dept.. of Operation Research and Financial Engineering.

    Google Scholar 

  39. J. Kallsen, (2000), Optimal portfolios for exponential Levy pro cesses Mathematical Methods of Operation Research, Vol. 51, No. 3, pp. 357–374.

    MathSciNet  Google Scholar 

  40. Yu.M. Kabanov, C. Strieker, (2000), On the optimal portfolio for the exponential utility maximization: remarks to the six-author paper. Preprint,.

    Google Scholar 

  41. N. El Karoui, M. Jeanblanc, (1998), Optimization of consumptions with labor income. Finance and Stochastics, Vol. 4, pp. 409–440.

    Article  Google Scholar 

  42. I. Karatzas, J.P. Lehoczky, S.E. Shreve, (1987), Optimal portfolio and consumption decisions for a “small investo” on a finite horizon. SIAM Journal of Control and Optimization, Vol. 25, pp. 1557–1586.

    Article  MathSciNet  MATH  Google Scholar 

  43. I. Karatzas, J.P. Lehoczky, S.E. Shreve, G.L. Xu, (1991), Martingale and duality methods for utility maximization in an incomplete market. SIAM Journal of Control and Optimization, Vol. 29, pp. 702–730.

    Article  MathSciNet  MATH  Google Scholar 

  44. N. El Karoui, M.-C. Quenez, (1995), Dynamic programming and pric ing of contingent claims in an incomplete market. SIAM J. Control Optim., Vol. 33, pp. 29–66.

    Article  MATH  Google Scholar 

  45. N. Karoui, R. Rouge, (2000), Pricing via utility maximization and entropy. Preprint, to appear in Mathematical Finance.

    Google Scholar 

  46. J. Komlos, (1967), A generalization of a problem of Steinhaus. Acta Math. Sci. Hung., Vol*. 18, pp. 217–229.

    Article  MathSciNet  MATH  Google Scholar 

  47. D. Kramkov, W. Schachermayer, (1999), The Asymptotic Elasticity of Utility Functions and Optimal Investment in Incomplete Markets. Annals of Applied Probability, Vol. 9, No. 3, pp. 904–950.

    Article  MathSciNet  MATH  Google Scholar 

  48. P. Lakner, (2000), Portfolio Optimization with an Insurance Con straint. Preprint of the NYU, Dept. of Statistics and Operation Research.

    Google Scholar 

  49. J.B. Long, (1990), The numeraire portfolio. Journal of Financial Eco nomics, Vol. 26, pp. 29–69.

    Article  Google Scholar 

  50. R.C. Merton, (1969), Lifetime portfolio selection under uncertainty: the continuous-time model. Rev. Econom. Statist,., Vol. 51, pp. 2472. 57.

    Google Scholar 

  51. R.C. Merton, (1971), Optimum consumption and portfolio rules in a continuous-time model. J. Econom. Theory, Vol. 3, pp. 373–413.

    Article  MathSciNet  MATH  Google Scholar 

  52. R.C. Merton, (1990), Continuous-Time Finance. Basil Blackwell, Oxford.

    Google Scholar 

  53. S.R. Pliska, (1986), A stochastic calculus model of continuous trading: optimal portfolios. Math. Oper. Res., Vol. 11, pp. 371–382.

    Article  MathSciNet  MATH  Google Scholar 

  54. R.T. Rockafellar, (1970), Convex Analysis. Princeton University Press, Princeton, New Jersey.

    Google Scholar 

  55. L. Riischendorf, (1984), On the minimum discrimination information theorem. Statistics and Decisions Supplement, Issue, Vol. 1, pp. 263–283.

    Google Scholar 

  56. P.A. Samuelson, (1969), Lifetime portfolio selection by dynamic stochastic programming. Rev. Econom. Statist,., Vol. 51, pp. 239–246.

    Article  Google Scholar 

  57. W. Schachermayer, (1999), Optimal Investment in Incomplete Markets when Wealth may Become Negative. Preprint, (45 pages), to appear in Annals of Applied Probability.

    Google Scholar 

  58. W. Schachermayer, (2000), Introduction to the Mathematics of Financial Markets. Notes on the St.. Flour summer school 2000, preprint,, to appear in Springer Lecture Notes.

    Google Scholar 

  59. W. Schachermayer, (2000), How Potential Investments may Change the Optimal Portfolio for the Exponential Utility, preprint,.

    Google Scholar 

  60. H.H. Schafer, (1966), Topological Vector Spaces. Graduate Texts in Mathematics.

    Google Scholar 

  61. H. Strasser, (1985), Mathematical theory of statistics: statistical experiments and asymptotic decision theory. De Gruyter studies in mathematics, Vol. 7.

    Google Scholar 

  62. J. Xia, J. Yan, (2000), Martingale measure method for expected utility maximisation and valuation in incomplete markets. Preprint,.

    Google Scholar 

  63. G. Zitkovic, (2000), Maximization of utility of consumption in incomplete semimartingale markets. In preparation.

    Google Scholar 

  64. G. Zitkovic, (2000), A filtered version of the Bipolar Theorem of Brannath and Schachermayer. Preprint, of the Dept. of Statistics, Columbia University, NY.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Financial and Actuarial Mathematics, Vienna University of Technology, Wiedner Hauptstrasse 8-10 / 105, 1040, Vienna, Austria

    Walter Schachermayer

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  1. Walter Schachermayer
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Editors and Affiliations

  1. CEREMADE, UMR CNRS 7534, Université Paris IX-Dauphine, Place du Maréchal De Lattre De Tassigny, 75775, Paris Cedex 16, France

    Hélyette Geman

  2. Groupe ESSEC, Av. Bernhard-Hirsch, B.P. 50105, 95021, Cergy-Pontoise Cedex, France

    Hélyette Geman

  3. Robert H. Smith School of Business, University of Maryland, 4427 Van Munching Hall, 20742, College Park, MD, USA

    Dilip Madan

  4. Department of Finance, University of Illinois, 242 University Hall, 601 South Morgan Street, 60607-7, Chicago, IL, USA

    Stanley R. Pliska

  5. Erasmus Universiteit Rotterdam, Postbus 1738, 3000 DR, Rotterdam, The Netherlands

    Ton Vorst

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Schachermayer, W. (2002). Optimal Investment in Incomplete Financial Markets. In: Geman, H., Madan, D., Pliska, S.R., Vorst, T. (eds) Mathematical Finance — Bachelier Congress 2000. Springer Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-12429-1_20

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  • DOI: https://doi.org/10.1007/978-3-662-12429-1_20

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