Skip to main content
SpringerLink
Log in

Option pricing and Esscher transform under regime switching

  • Published:
Annals of Finance Aims and scope Submit manuscript

Summary

We consider the option pricing problem when the risky underlying assets are driven by Markov-modulated Geometric Brownian Motion (GBM). That is, the market parameters, for instance, the market interest rate, the appreciation rate and the volatility of the underlying risky asset, depend on unobservable states of the economy which are modelled by a continuous-time Hidden Markov process. The market described by the Markov-modulated GBM model is incomplete in general and, hence, the martingale measure is not unique. We adopt a regime switching random Esscher transform to determine an equivalent martingale pricing measure. As in Miyahara [33], we can justify our pricing result by the minimal entropy martingale measure (MEMM).

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

Similar content being viewed by others

References

  1. Avellaneda, M.: Minimum relative-entropy calibration of asset-pricing models. International Journal of Theoretical and Applied Finance 1, 447–472 (1998)

    Google Scholar 

  2. Black, F., Scholes, M.: The pricing of options and corporate liabilities. Journal of Political Economy 81, 637–659 (1973)

    Google Scholar 

  3. Buchen, P. W., Kelly, M.: The maximum entropy distribution of an asset inferred from option prices. Journal of Financial and Quantitative Analysis 31, 143–159 (1996)

    Google Scholar 

  4. Buffington, J., Elliott, R. J.: Regime switching and European options. In: Lawrence, K.S. (ed.) Stochastic theory and control. Proceedings of a Workshop, pp.73–81. Berlin Heidelberg New York: Springer 2002

  5. Buffington, J., Elliott, R. J.: American options with regime switching. International Journal of Theoretical and Applied Finance 5, 497–514 (2002)

    Google Scholar 

  6. Bühlmann, H., Delbaen, F., Embrechts, P., Shiryaev, A. N.: No-arbitrage, change of measure and conditional Esscher transforms, CWI Quarterly 9(4), 291–317 (1996)

    Google Scholar 

  7. Bühlmann, H., Delbaen, F., Embrechts, P., Shiryaev, A. N.: On Esscher transforms in discrete finance models. Astin Bulletin 28(2), 171–186 (1998)

    Google Scholar 

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

    Google Scholar 

  9. Delbaen, F., Schachermayer, W.: What is a free lunch? Notices of the American Mathematical Society 51(5), 526–528 (2004)

    Google Scholar 

  10. Davis, Mark H. A.: Option pricing in incomplete markets. In: Dempster, M.A.H., Pliska, S.R. (eds.) Mathematics of derivative securities, pp. 216–226. Cambridge: Cambridge University Press 1997

  11. Elliott, R. J., Aggoun, L., Moore, J. B.: Hidden Markov models: estimation and control. Berlin Heidelberg New York: Springer 1994

  12. Elliott, R. J., van der Hoek, J.: An application of hidden Markov models to asset allocation problems. Finance and Stochastics 3, 229–238 (1997)

    Google Scholar 

  13. Elliott, R. J., Kopp, P. E.: Mathematics of financial markets. Berlin Heidelberg New York: Springer 1999

  14. Elliott, R. J., Hunter, W. C., Jamieson, B. M.: Financial signal processing. International Journal of Theoretical and Applied Finance 4, 567–584 (2001)

    Google Scholar 

  15. Elliott, R. J., Hinz, J.: Portfolio analysis, hidden Markov models and chart analysis by PF-Diagrams. International Journal of Theoretical and Applied Finance 5 , 385–399 (2002)

    Google Scholar 

  16. Elliott, R. J., Malcolm, W. P., Tsoi, A. H.: Robust parameter estimation for asset price models with Markov modulated volatilities. Journal of Economics Dynamics and Control 27(8), 1391–1409 (2003)

    Google Scholar 

  17. Embrechts, P.: Actuarial versus financial pricing of insurance. Risk Finance 1(4), 17–26 (2000)

    Google Scholar 

  18. Esscher, F.: On the probability function in the collective theory of risk. Skandinavisk Aktuarietidskrift 15, 175–195 (1932)

    Google Scholar 

  19. Föllmer, H., Sondermann, D.: Hedging of contingent claims under incomplete information. In: Hildenbrand, W., Mas-Colell, A. (eds.) Contributions to mathematical economics, pp. 205–223. Amsterdam: North Holland 1986

  20. Föllmer, H., Schweizer, M.: Hedging of contingent claims under incomplete information. In: Davis, M. H. A., Elliott, R. J. (eds.) Applied stochastic analysis, pp. 389–414. London: Gordon and Breach 1991

  21. Föllmer, H., Leukert, P.: Quantile hedging. Finance and Stochastics 3(3), 251–273 (1999)

  22. Frittelli, M.: The minimal entropy martingale measure and the valuation problem in incomplete markets. Mathematical Finance 10(1), 39–52 (2000)

    Google Scholar 

  23. Gerber, H. U., Shiu, Elias S. W.: Option pricing by Esscher transforms (with discussions). Transactions of the Society of Actuaries 46, 99–191 (1994)

    Google Scholar 

  24. Gerber, H. U., Shiu, Elias S. W.: Actuarial bridges to dynamic hedging and option pricing. Insurance: Mathematics and Economics 18, 183–218 (1996)

    Google Scholar 

  25. Guo, X.: Information and option pricings. Quantitative Finance 1, 38–44 (2001)

    Google Scholar 

  26. Harrison, J. M., Kreps, D. M.: Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory 20, 381–408 (1979)

    Google Scholar 

  27. Harrison, J. M., Pliska, S. R.: Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and Their Applications 11, 215–280 (1981)

    Google Scholar 

  28. Harrison, J. M., Pliska, S. R.: A stochastic calculus model of continuous trading: complete markets. Stochastic Processes and Their Applications 15, 313–316 (1983)

    Google Scholar 

  29. Hardy, M.: A regime switching model of long term stock returns. North American Actuarial Journal 5(2), 41–53 (2001)

    Google Scholar 

  30. Kitamura, Y., Stutzer, M. Connections between entropic and linear projections in asset pricing estimation. Journal of Econometrics 107, 159–174 (2002)

    Google Scholar 

  31. McLeish, D. L., Reesor, R. M.: Risk, entropy, and the transformation of distributions. North American Actuarial Journal 7(2), 128–144 (2003)

    Google Scholar 

  32. Merton, R. C.: The theory of rational option pricing. Bell Journal of Economics and Management Science 4, 141–183 (1973)

    Google Scholar 

  33. Miyahara, Y.: Geometric Lévy process and MEMM: pricing model and related estimation problems. Asia-Pacific Financial Markets 8, 45–60 (2001)

  34. Pafumi, G.: A study of a family of equivalent martingale measures to price an option with an application to the Swiss market. Bulletin of the Swiss Association of Actuaries 159–194 (1997)

  35. Pliska, S.: Introduction to mathematical finance: discrete time models. Oxford: Blackwell 1997

  36. Schweizer, M.: Approximation pricing and the variance-optimal martingale measure. Annals of Probability 24, 206–236 (1996)

    Google Scholar 

  37. Shiryaev, A. N.: Essentials of stochastic finance: facts, models, theory. Singapore: World Scientific 1999

  38. Siu, T. K., Tong, H., Yang, H.: Bayesian risk measures for derivatives via random Esscher transform. North American Actuarial Journal 5(3), 78–91 (2001)

    Google Scholar 

  39. Siu, T. K., Tong, H., Yang, H.: On pricing derivatives under GARCH models: a dynamic Gerber-Shiu approach. North American Actuarial Journal 8(3), 17–31 (2004)

    Google Scholar 

  40. Stutzer, M.: A simple non-parametric approach to derivative security valuation. Journal of Finance 51(5), 1633–1652 (1996)

    Google Scholar 

  41. Yang, H.: The Esscher transform. In: Teugels, J., Sundt, B. (eds.) Encyclopedia of actuarial science vol. 2, pp. 617–621. New York: Wiley 2004

  42. Yao, Y.: State price density, Esscher transforms, and pricing options on stocks, bonds, and foreign exchange rates. North American Actuarial Journal 5(3), 104–117 (2001)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Haskayne School of Business, University of Calgary, Calgary, Alberta, CANADA

    Robert J. Elliott

  2. Department of Mathematics and Statistics, University of Calgary, CANADA

    Leunglung Chan

  3. Department of Actuarial Mathematics and Statistics, School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, UK

    Tak Kuen Siu

Authors
  1. Robert J. Elliott
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leunglung Chan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tak Kuen Siu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert J. Elliott.

Additional information

We would like to thank the referees for many helpful and insightful comments and suggestions.

Correspondence to: R. J. Elliott

Rights and permissions

Reprints and permissions

About this article

Cite this article

Elliott, R., Chan, L. & Siu, T. Option pricing and Esscher transform under regime switching. Annals of Finance 1, 423–432 (2005). https://doi.org/10.1007/s10436-005-0013-z

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10436-005-0013-z

Keywords and Phrases

JEL Classification Numbers

Search

Navigation