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On the game interpretation of a shadow price process in utility maximization problems under transaction costs

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Abstract

To any utility maximization problem under transaction costs one can assign a frictionless model with a price process S , lying in the bid/ask price interval \([\underline{S}, \overline{S}]\). Such a process S is called a shadow price if it provides the same optimal utility value as in the original model with bid-ask spread.

We call S a generalized shadow price if the above property is true for the relaxed utility function in the frictionless model. This relaxation is defined as the lower semicontinuous envelope of the original utility, considered as a function on the set \([\underline{S}, \overline{S}]\), equipped with some natural weak topology. We prove the existence of a generalized shadow price under rather weak assumptions and mark its relation to a saddle point of the trader/market zero-sum game, determined by the relaxed utility function. The relation of the notion of a shadow price to its generalization is illustrated by several examples. Also, we briefly discuss the interpretation of shadow prices via Lagrange duality.

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References

  1. Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis. A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  2. Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces. Applications to PDEs and Optimization. SIAM, Philadelphia (2006)

    Book  MATH  Google Scholar 

  3. Biagini, S., Guasoni, P.: Relaxed utility maximization in complete markets. Math. Finance 21, 703–722 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Benedetti, G., Campi, L., Kallsen, J., Muhle-Karbe, J.: On the existence of shadow prices. Finance Stoch. (2013, to appear). doi:10.1007/s00780-012-0201-4

  5. Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., Montrucchio, L.: Risk measures: rationality and diversification. Math. Finance 21, 743–774 (2011)

    MathSciNet  MATH  Google Scholar 

  6. Choi, J.H., Sîrbu, M., Žitkovič, G.: Shadow prices and well-posedness in the problem of optimal investment and consumption with transaction costs. Preprint (2012). arXiv:1204.0305 [q-fin.PM]

  7. Cvitanić, J.: Karatzas I. Hedging and portfolio optimization under transaction costs: a martingale approach. Math. Finance 6, 133–165 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cvitanić, J., Wang, H.: On optimal terminal wealth under transaction costs. J. Math. Econ. 35, 223–231 (2001)

    Article  MATH  Google Scholar 

  9. Czichowsky, C., Muhle-Karbe, J., Schachermayer, W.: Transaction costs, shadow prices, and connections to duality. Preprint (2012). arXiv:1205.4643 [q-fin.PM]

  10. Dal Maso, G.: An Introduction to Γ-Convergence. Birkhäuser, Basel (1993)

    Book  Google Scholar 

  11. Dunford, N., Schwartz, J.T.: Linear Operators. Part I: General Theory. Interscience, New York (1958)

    Google Scholar 

  12. Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. SIAM, Philadelphia (1999)

    Book  MATH  Google Scholar 

  13. Fiacco, A.V.: Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Academic Press, New York (1983)

    MATH  Google Scholar 

  14. Frittelli, M., Rosazza Gianin, E.: On the penalty function and on continuity properties of risk measures. Int. J. Theor. Appl. Finance 14, 163–185 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gerhold, S., Guasoni, P., Muhle-Karbe, J., Schachermayer, W.: Transaction costs, trading volume, and the liquidity premium. Finance Stoch. 14, 227–272 (2013)

    Google Scholar 

  16. Gerhold, S., Muhle-Karbe, J., Schachermayer, W.: The dual optimizer for the growth-optimal portfolio under transaction costs. Finance Stoch. 17, 325–354 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gerhold, S., Muhle-Karbe, J., Schachermayer, W.: Asymptotics and duality for the Davis and Norman problem. Stochastics 84, 625–641 (2012)

    MathSciNet  MATH  Google Scholar 

  18. Guasoni, P., Muhle-Karbe, J.: Portfolio choice with transaction costs: a user’s guide. Preprint (2012). arXiv:1207.7330 [q-fin.PM]

  19. Ha, C.-W.: Minimax and fixed point theorems. Math. Ann. 248, 73–77 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kallsen, J., Muhle-Karbe, J.: On using shadow prices in portfolio optimization with transaction costs. Ann. Appl. Probab. 20, 1341–1358 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kallsen, J., Muhle-Karbe, J.: Existence of shadow prices in finite probability spaces. Math. Methods Oper. Res. 73, 251–262 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Koopmans, T.C.: Concepts of optimality and their uses. Math. Program. 11, 212–228 (1976)

    Article  MathSciNet  Google Scholar 

  23. Loewenstein, M.: On optimal portfolio trading strategies for an investor facing transaction costs in a continuous trading market. J. Math. Econ. 33, 209–228 (2002)

    Article  MathSciNet  Google Scholar 

  24. Papageorgiou, N.S., Th, K.-Y.S.: Handbook of Applied Analysis. Springer, Dordrecht (2009)

    MATH  Google Scholar 

  25. Pennanen, T.: Convex duality in stochastic programming and mathematical finance. Math. Oper. Res. 36, 340–362 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pennanen, T., Perkkio, A.-P.: Stochastic programs without duality gaps. Math. Program. 136, 91–110 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  28. Rockafellar, R.T.: Conjugate Duality and Optimization. SIAM, Philadelphia (1974)

    Book  MATH  Google Scholar 

  29. Rockafellar, R.T.: Integral functionals, normal integrands and measurable selections. In: Gossez, J.P., et al. (eds.) Nonlinear Operators and the Calculus of Variations, Summer Sch. Bruxelles, 1975. Lect. Notes Math., vol. 543, pp. 157–207. Springer, Berlin (1976)

    Chapter  Google Scholar 

  30. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  31. Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991)

    MATH  Google Scholar 

  32. Simons, S.: Minimax theorems and their proofs. In: Du, D.-Z., Pardalos, P.M. (eds.) Minimax and Applications, pp. 1–23. Kluwer Academic, Dordrecht (1995)

    Chapter  Google Scholar 

  33. Tawarmalani, M., Richard, J.-P.P., Xiong, C.: Explicit convex and concave envelopes through polyhedral subdivisions. Math. Program. 138, 531–577 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Dmitry B. Rokhlin.

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Rokhlin, D.B. On the game interpretation of a shadow price process in utility maximization problems under transaction costs. Finance Stoch 17, 819–838 (2013). https://doi.org/10.1007/s00780-013-0206-7

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