Finance and Stochastics

, Volume 17, Issue 3, pp 587–613

Dynamic no-good-deal pricing measures and extension theorems for linear operators on L

Article

Abstract

In an L-framework, we present majorant-preserving and sandwich-preserving extension theorems for linear operators. These results are then applied to price systems derived by a reasonable restriction of the class of applicable equivalent martingale measures. Our results prove the existence of a no-good-deal pricing measure for price systems consistent with bounds on the Sharpe ratio. We treat both discrete- and continuous-time market models. Within this study we present definitions of no-good-deal pricing measures that are equivalent to the existing ones and extend them to discrete-time models. We introduce the corresponding version of dynamic no-good-deal pricing measures in the continuous-time setting.

Keywords

Price operator Dynamic risk measure Extension theorem Representation theorem Fundamental theorem Equivalent martingale measure Good deal 

Mathematics Subject Classification (2010)

46E30 91B70 

JEL Classification

G12 G13 

References

  1. 1.
    Albeverio, S., Di Nunno, G., Rozanov, Yu.A.: Price operators analysis in L p-spaces. Acta Appl. Math. 89, 85–108 (2005) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Barrieu, P., El Karoui, N.: Pricing, hedging and optimally designing derivatives via minimization of risk measures. In: Carmona, R. (ed.) Volume on Indifference Pricing, pp. 77–147. Princeton University Press, Princeton (2009) Google Scholar
  3. 3.
    Bernardo, A.E., Ledoit, O.: Gain, loss, and asset pricing. J. Polit. Econ. 108, 144–172 (2000) CrossRefGoogle Scholar
  4. 4.
    Bion-Nadal, J.: Dynamic risk measures: Time consistency and risk measures from BMO martingales. Finance Stoch. 12, 219–244 (2008) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bion-Nadal, J.: Time consistent dynamic risk processes. Stoch. Process. Appl. 119, 633–654 (2009) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bion-Nadal, J.: Bid-ask dynamic pricing in financial markets with transaction costs and liquidity risk. J. Math. Econ. 45, 738–750 (2009) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bion-Nadal, J.: Dynamic pricing models calibrated on both liquid and illiquid assets. Int. J. Econ. Res. 9, 263–283 (2012) Google Scholar
  8. 8.
    Bion-Nadal, J., Kervarec, M.: Dynamic risk measuring under model uncertainty: Taking advantage of the hidden probability measure. Preprint ArXiv 1012.5850 (Dec. 2010)
  9. 9.
    Björk, T., Slinko, I.: Toward a general theory of good deal bounds. Rev. Finance 10, 221–260 (2006) MATHCrossRefGoogle Scholar
  10. 10.
    Černý, A.: Generalised sharpe ratios and asset pricing in incomplete markets. Eur. Finance Rev. 7, 191–233 (2003) MATHCrossRefGoogle Scholar
  11. 11.
    Cheridito, P., Delbaen, F., Kupper, M.: Dynamic monetary risk measures for bounded discrete time processes. Electron. J. Probab. 11, 57–106 (2006) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cochrane, J.H., Saa-Requejo, J.: Beyond arbitrage: Good deal asset price bounds in incomplete markets. J. Polit. Econ. 108, 1–22 (2000) CrossRefGoogle Scholar
  13. 13.
    Delbaen, F., Schachermayer, W.: The Mathematics of Arbitrage. Springer, Berlin (2006) MATHGoogle Scholar
  14. 14.
    Delbaen, F., Peng, S., Rosazza Gianin, E.: Representation of the penalty term of dynamic concave utilities. Finance Stoch. 14, 449–472 (2010) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Di Nunno, G., Eide, I.B.: Lower and upper bounds of martingale measure densities in continuous-time markets. Math. Finance 21, 475–492 (2011) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, 2nd edn. de Gruyter Studies in Mathematics, vol. 27 (2004) CrossRefGoogle Scholar
  17. 17.
    Fuchssteiner, B., Lusky, W.: Convex Cones. North Holland, Amsterdam (1981) MATHGoogle Scholar
  18. 18.
    Jaschke, S., Küchler, U.: Coherent risk measures and good deal bounds. Finance Stoch. 5, 181–200 (2001) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Klöppel, S., Schweizer, M.: Dynamic utility-based good deal bounds. Stat. Decis. 25, 285–309 (2007) MATHCrossRefGoogle Scholar
  20. 20.
    Klöppel, S., Schweizer, M.: Dynamic utility indifference valuation via convex risk measures. Math. Finance 17, 599–627 (2007) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Neveu, J.: Discrete Parameter Martingales. North Holland, Amsterdam (1974) Google Scholar
  22. 22.
    Peng, S.: Nonlinear expectations, nonlinear evaluations and risk measures. In: Back, K., Bielecki, T.R., Hipp, C., Peng, S., Schachermayer, W., Frittelli, M., Runggaldier, W.J. (eds.) Stochastic Methods in Finance. Lecture Notes in Mathematics, vol. 1856, pp. 165–253. Springer, Berlin (2004) CrossRefGoogle Scholar
  23. 23.
    Staum, J.: Fundamental theorems of asset pricing for good deal bounds. Math. Finance 14, 141–161 (2004) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.UMR 7641 CNRS-Ecole PolytechniqueEcole PolytechniquePalaiseau CedexFrance
  2. 2.Centre of Mathematics for Applications (CMA), Department of MathematicsUniversity of OsloOsloNorway
  3. 3.Norwegian School of Economics and Business Administration (NHH)BergenNorway

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