Skip to main content
SpringerLink
Log in

Bid and ask prices as non-linear continuous time G-expectations based on distortions

  • Published:
Mathematics and Financial Economics Aims and scope Submit manuscript

Abstract

Probability distortions for constructing nonlinear G-expectations for the bid and ask or lower and upper prices in continuous time are here extended to the direct use of measure distortions. Fairly generally measure distortions can be constructed as probability distortions applied to an exponential distribution function on the half line. The valuation methodologies are extended beyond contract valuation to the valuation of economic activities with infinite lives. Explicit computations illustrate the procedures for stock indices and insurance loss processes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Artzner, P., Delbaen, F., Eber, M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203–228 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bielecki, T., Cialenco, I., Zhang, Y.: Dynamic coherent acceptability indices and their applications to finance. Math. Finance (2011). doi:10.1111/j.1467-9965.2012.00524x

  3. Carr, P., Madan, D.B., Vicente Alvarez, J.J.: Markets profits, capital, leverage and returns. J. Risk 14, 95–122 (2011)

    Google Scholar 

  4. Cherny, A., Madan, D.: New measures for performance evaluation. Rev. Financ. Stud. 22, 2571–2606 (2009)

    Article  Google Scholar 

  5. Cherny, A., Madan, D.B.: Markets as a counterparty: an introduction to conic finance. Int. J. Theor. Appl. Finance 13, 1149–1177 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cohen, S., Elliott, R.J.: A general theory of finite state backward stochastic difference equations. Stoch. Process. Appl. 120, 442–466 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Eberlein, E., Madan, D.B.: Unlimited liabilities reserve capital requirements and the taxpayer put option. Quant. Finance 12, 709–724 (2012)

    Google Scholar 

  8. Eberlein, E., Madan, D.B., Pistorius, M., Schoutens, W., Yor, M.: Two price economies in continuous time. Ann. Finance (2012). doi:10.1007/s10436-013-0228-3

  9. Kusuoka, S.: On law invariant coherent risk measures. Adv. Math. Econ. 3, 83–95 (2001)

    Article  MathSciNet  Google Scholar 

  10. Madan, D.B.: Conserving Capital by adjusting Deltas for Gammas in the Presence of Skewness, J. Risk Financial Manag. 3, 1–25 (2010)

    Google Scholar 

  11. Madan, D.B.: A two price theory of financial equilibrium with risk management implications. Ann. Finance 8(4), 489–505 (2012)

    Article  MathSciNet  Google Scholar 

  12. Madan, D.B., Schoutens, W.: Conic finance and the corporate balance sheet. Int. J. Theor. Appl. Finance 14, 587–610 (2011)

    Google Scholar 

  13. Madan, D.B., Schoutens, W.: Conic coconuts: the pricing of contingent capital notes using conic finance. Math. Financ. Econ. 4, 87–106 (2011)

    Google Scholar 

  14. Madan, D.B., Schoutens, W.: Structured products equilibria in conic two price markets. Math. Financ. Econ. 6, 37–57 (2012)

    Google Scholar 

  15. Madan, D.B., Schoutens, W.: Tenor specific pricing. Int. J. Theor. Appl. Finance 15, 6 (2012). doi:10.1142/S0219024912500434

  16. Madan, D., Seneta, E.: The variance gamma (V.G.) model for share market returns. J. Bus. 63, 511–524 (1990)

    Article  Google Scholar 

  17. Madan, D.B., Yor, M.: On valuing stochastic perpetuities using new long horizon stock price models distinguishing booms, busts and balanced markets. Working Paper, Robert H. Smith School of Business (2012)

  18. Madan, D., Carr, P., Chang, E.: The variance gamma process and option pricing. Eur. Finance Rev. 2, 79–105 (1998)

    Article  MATH  Google Scholar 

  19. Madan, D.B., Wang, S., Heckman P.: A theory of risk for two price market equilibria. www.casact.org/liquidity/LRP9.1.pdf (2011)

  20. Madan, D.B., Pistorius, M., Stadje M.: On consistent valuation based on distortions: from multinomial random walks to Lévy processes, Imperial College (2012). http://arxiv.org/abs/1301.3531

  21. Peng, S.: Dynamically consistent nonlinear evaluations and expectations, preprint no. 2004-1. Institute of Mathematics, Shandong University (2004). http://arxiv.org/abs/math/0501415

  22. Wang, S., Young, Y.R., Panjer, H.H.: Axiomatic characterization of insurance prices. Insur. Math. Econ. 21, 173–183 (1997)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Freiburg, Freiburg, Germany

    Ernst Eberlein

  2. Robert H. Smith School of Business, College Park, MD, USA

    Dilip B. Madan

  3. Imperial College, London, UK

    Martijn Pistorius

  4. Université Pierre et Marie Curie, Paris, France

    Marc Yor

Authors
  1. Ernst Eberlein
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dilip B. Madan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Martijn Pistorius
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Marc Yor
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dilip B. Madan.

Additional information

Marc Yor passed away January 9 2014.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Eberlein, E., Madan, D.B., Pistorius, M. et al. Bid and ask prices as non-linear continuous time G-expectations based on distortions. Math Finan Econ 8, 265–289 (2014). https://doi.org/10.1007/s11579-014-0117-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11579-014-0117-1

Keywords

Search

Navigation