Skip to main content
SpringerLink
Log in

Option on a unit-type closed-end investment fund

  • Chapter
Advances in Mathematical Economics

Part of the book series: Advances in Mathematical Economics ((MATHECON,volume 9))

  • 591 Accesses

Abstract

In this paper we study options on a unit-type closed-end investment fund. These options are included among the exotic options, because the underlying asset of the options is the value process of the investment fund and therefore depends on a fund manager (= an option writer)’s action. We prove that a fair price of such option is represented as the value function of the associated stochastic exit time control problem. Using Hajek’s mean comparison theorem, we find an explicit form of the fair option premium in the case of a constant volatility. We also characterize the fair option premium as a limit of a sequence of classical solutions to the associated Hamilton-Jacobi-Bellman equations with a classical Dirichlet boundary condition in the case of a diffusion market model.

I would like to express my gratitude to the anonymous referee for valuable comments and suggestions on this paper.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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. Adachi, T.: On value of American version of Asian option, generalization of passport option and consumption/investment problems. Doctoral dissertation, Department of Computer and Mathematical Sciences, Tohoku University 2001

    Google Scholar 

  2. Ahn, H., Penaud, A., Wilmott, P.: Various passport options and their valuation. Preprint, OCIAM Oxford University (1999)

    Google Scholar 

  3. Barles, G., Burdeau, J.: The Dirichlet problem for semilinear second-order degenerate elliptic stochastic exit time control problems. Comm. in PDEs 20, 129–178 (1995)

    MATH  MathSciNet  Google Scholar 

  4. Barles, G., Rouy, E.: A strong comparison result for the Bellman equation arising stochastic exit time control problems and its applications. Comm. in PDEs 23, 1995–2033 (1998)

    MATH  MathSciNet  Google Scholar 

  5. Borkar, V.S.: Optimal Control of Diffusion Processes. Longman Sci. and Tech., Harlow, UK 1989

    Google Scholar 

  6. Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27, 1–67 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  7. Delbaen, F., Yor, M.: Passport options. Preprint, ETH. (1999)

    Google Scholar 

  8. Duffle, D.: Dynamic Asset Pricing Theory. 3rd ed. Princeton Univ. Press 2001

    Google Scholar 

  9. Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Springer-Verlag, NY 1993

    MATH  Google Scholar 

  10. Hajek, B.: Mean stochastic comparison of diffusions. Z. Wahrscheinlichkeitstheorie verw. Gebiete 68, 315–329 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  11. Henderson, V: Price comparison results and super-replication: An application to passport options. Appl. Stoch. models in Business and Industry 16, 297–310 (2000)

    Article  MATH  Google Scholar 

  12. Henderson, V, Hobson, D.: Local times, coupling and the passport option. Finance & Stochastics 4, 69–80 (1999)

    Article  MathSciNet  Google Scholar 

  13. Hyer, T., Lipton-Lifschitz, A., Pugachevsky, D.: Passport to success. Risk 10, 127–131 (1997)

    Google Scholar 

  14. Ishii, H.: A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 16, 105–135 (1989)

    MATH  MathSciNet  Google Scholar 

  15. Karatzas, I.: Lectures on the Mathematics of Finance. Providence, RI 1997

    Google Scholar 

  16. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. 2nd ed. Springer-Verlag, NY 1991

    MATH  Google Scholar 

  17. Karatzas, I., Shreve, S.E.: Methods of Mathematical Finance. Springer-Verlag, NY 1998

    MATH  Google Scholar 

  18. Krylov, N.V.: Controlled Diffusion Processes. Springer-Verlag, NY 1980

    MATH  Google Scholar 

  19. Lions, P.-L.: Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part 1: The dynamic programming principle and application. Comm. in PDEs 8, 1101–1174(1983)

    Article  MATH  Google Scholar 

  20. Nagai, H.: Stochastic Differential Equations. In Japanese. Kyoritu-Shuppan 1999

    Google Scholar 

  21. Nagayama, I.: Pricing of passport option. J. Math. Sci. Univ. Tokyo 5, 747–785 (1999)

    MathSciNet  Google Scholar 

  22. Shreve, S.E., Večeř, J.: Options on traded account: Vacation calls, vacation puts and passport options. Finance & Stochastics 4, 255–274 (2000)

    Article  MATH  Google Scholar 

  23. Yong, J., Zhou, X.Y.: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer-Verlag, NY 1999

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fukushima Medical University, Fukushima City, Fukushima, 960-1295, Japan

    Takashi Adachi

Authors
  1. Takashi Adachi
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-0041, Japan

    Shigeo Kusuoka (Professor) (Professor)

  2. Department of Economics, Meisei University, Hino Tokyo, 191-8506, Japan

    Akira Yamazaki (Professor) (Professor)

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Tokyo

About this chapter

Cite this chapter

Adachi, T. (2006). Option on a unit-type closed-end investment fund. In: Kusuoka, S., Yamazaki, A. (eds) Advances in Mathematical Economics. Advances in Mathematical Economics, vol 9. Springer, Tokyo. https://doi.org/10.1007/4-431-34342-3_1

Download citation

Publish with us

Policies and ethics

Search

Navigation