Skip to main content
SpringerLink
Log in

Exponential Utility Indifference Valuation in a General Semimartingale Model

  • Chapter
  • First Online:
Optimality and Risk - Modern Trends in Mathematical Finance

Abstract

We study the exponential utility indifference valuation of a contingent claim H when asset prices are given by a general semimartingale S. Under mild assumptions on H and S, we prove that a no-arbitrage type condition is fulfilled if and only if H has a certain representation. In this case, the indifference value can be written in terms of processes from that representation, which is useful in two ways. Firstly, it yields an interpolation expression for the indifference value which generalizes the explicit formulas known for Brownian models. Secondly, we show that the indifference value process is the first component of the unique solution (in a suitable class of processes) of a backward stochastic differential equation. Under additional assumptions, the other components of this solution are BMO-martingales for the minimal entropy martingale measure. This generalizes recent results by Becherer (Ann. Appl. Probab. 16:2027–2054, 2006) and Mania and Schweizer (Ann. Appl. Probab. 15:2113–2143, 2005).

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Becherer, D.: Rational hedging and valuation of integrated risks under constant absolute risk aversion. Insur. Math. Econ. 33, 1–28 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Becherer, D.: Bounded solutions to backward SDEs with jumps for utility optimization and indifference hedging. Ann. Appl. Probab. 16, 2027–2054 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Biagini, S., Frittelli, M.: Utility maximization in incomplete markets for unbounded processes. Finance Stoch. 9, 493–517 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Biagini, S., Frittelli, M.: The supermartingale property of the optimal wealth process for general semimartingales. Finance Stoch. 11, 253–266 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bobrovnytska, O., Schweizer, M.: Mean-variance hedging and stochastic control: Beyond the Brownian setting. IEEE Trans. Automat. Contr. 49, 396–408 (2004)

    Article  MathSciNet  Google Scholar 

  6. Delbaen, F., Grandits, P., Rheinländer, T., Samperi, D., Schweizer, M., Stricker, C.: Exponential hedging and entropic penalties. Math. Financ. 12, 99–123 (2002)

    Article  MATH  Google Scholar 

  7. Doléans-Dade, C., Meyer, P.A.: Inégalités de normes avec poids. In: Séminaire de Probabilités XIII. Lecture Notes in Mathematics, vol. 721, pp. 313–331. Springer, Berlin (1979)

    Chapter  Google Scholar 

  8. Emery, M., Stricker, C., Yan, J.A.: Valeurs prises par les martingales locales continues à un instant donné. Ann. Probab. 11, 635–641 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  9. Föllmer, H., Schweizer, M.: Hedging of contingent claims under incomplete information. In: Davis, M., Elliott, R. (eds.) Applied Stochastic Analysis. Stochastics Monographs, vol. 5, pp. 389–414. Gordon and Breach, New York (1991)

    Google Scholar 

  10. Frei, C., Schweizer, M.: Exponential utility indifference valuation in two Brownian settings with stochastic correlation. Adv. Appl. Probab. 40, 401–423 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Frittelli, M.: The minimal entropy martingale measure and the valuation problem in incomplete markets. Math. Financ. 10, 39–52 (2000)

    Article  MathSciNet  Google Scholar 

  12. Grandits, P., Rheinländer, T.: On the minimal entropy martingale measure. Ann. Probab. 30, 1003–1038 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Henderson, V.: Valuation of claims on nontraded assets using utility maximization. Math. Financ. 12, 351–373 (2002)

    Article  MATH  Google Scholar 

  14. Henderson, V., Hobson, D.: Real options with constant relative risk aversion. J. Econ. Dyn. Control. 27, 329–355 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Henderson, V., Hobson, D.: Substitute hedging. RISK 15, 71–75 (2002)

    Google Scholar 

  16. Kabanov, Yu., Stricker, C.: On the optimal portfolio for the exponential utility maximization: Remarks to the six-author paper. Math. Financ. 12, 125–134 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Karatzas, I., Shreve, S.: Methods of Mathematical Finance. Applications of Mathematics, vol. 39. Springer, New York (1998)

    MATH  Google Scholar 

  18. Kazamaki, N.: Continuous Exponential Martingales and BMO. Lecture Notes in Mathematics, vol. 1579. Springer, New York (1994)

    MATH  Google Scholar 

  19. Mania, M., Schweizer, M.: Dynamic exponential utility indifference valuation. Ann. Appl. Probab. 15, 2113–2143 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Musiela, M., Zariphopoulou, T.: An example of indifference prices under exponential preferences. Finance Stoch. 8, 229–239 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Protter, P.: Stochastic Integration and Differential Equations. 2nd edn., version 2.1. Stochastic Modelling and Applied Probability, vol. 21. Springer, New York (2005)

    Google Scholar 

  22. Rheinländer, T.: An entropy approach to the Stein and Stein model with correlation. Finance Stoch. 9, 399–413 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Schweizer, M.: On the minimal martingale measure and the Föllmer-Schweizer decomposition. Ann. Probab. 24, 573–599 (1995)

    Google Scholar 

  24. Schweizer, M., From actuarial to financial valuation principles. Insur. Math. Econ. 28, 31–47 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  25. Tehranchi, M.: Explicit solutions of some utility maximization problems in incomplete markets. Stoch. Process. Appl. 114, 109–125 (2004)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, ETH Zurich, Rämistr. 101, 8092, Zurich, Switzerland

    Christoph Frei & Martin Schweizer

Authors
  1. Christoph Frei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Schweizer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Schweizer .

Additional information

This paper is dedicated to Yuri Kabanov on the occasion of his 60th birthday. We hope he likes it even if it is not short…

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Frei, C., Schweizer, M. (2009). Exponential Utility Indifference Valuation in a General Semimartingale Model. In: Optimality and Risk - Modern Trends in Mathematical Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02608-9_4

Download citation

Publish with us

Policies and ethics

Search

Navigation