Skip to main content
SpringerLink
Log in
Book cover

Hidden Markov Models in Finance pp 211–226Cite as

An Exact Formula for Pricing American Exchange Options with Regime Switching

  • Chapter
  • First Online:

  • 3117 Accesses

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 209))

Abstract

This paper investigates the pricing of American exchange options when the price dynamics of each underlying risky asset are assumed to follow a Markov-modulated Geometric Brownian motion; that is, the appreciation rate and the volatility of each underlying risky asset depend on unobservable states of the economy described by a continuous-time hidden Markov process. We show that the price of an American exchange option can be reduced to the price of an American option. Then, we modify the result of Zhu and Chan (An analytic formula for pricing American options with regime switching. Submitted for publication, 2012), a closed-form analytical pricing formula for the American exchange option is given.

This is a preview of subscription content, 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 EPUB and 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

Learn about institutional subscriptions

References

  1. Benth, F.E., Di Nunno, G., Khedher, A., Schmeck, M.D.: Pricing of spread options on a bivariate jump market and stability to model risk. Working paper, University of Oslo (2012)

    Google Scholar 

  2. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Politi. Econ. 81, 637–659 (1973)

    Article  Google Scholar 

  3. Buffington, J., Elliott, R.J.: Regime switching and European options. In: Pasik-Duncan B. (eds.) Stochastic Theory and Control, Proceedings of a Workshop, Lawrence, pp. 73–81. Springer, Berlin (2002)

    Google Scholar 

  4. Buffington, J., Elliott R.J.: American options with regime switching. Int. J. Theor. Appl. Financ. 5, 497–514 (2002)

    Article  Google Scholar 

  5. Carmona, R., Durrleman, V.: Pricing and hedging spread options. SIAM Rev. 45(4), 627–685 (2003)

    Article  Google Scholar 

  6. Cheang, G.H.L., Chiarella, C.: Exchange options under jump-diffusion dynamics. Appl. Math. Financ. 18(3), 245–276 (2011)

    Article  Google Scholar 

  7. Dempster, M., Hong, S.: Spread option valuation and the fast Fourier transform. In: Geman, H., Madan, D., Pliska, S.R., Vorst, T. (eds.) Mathematical Finance, Bachelier Congress, vol. 1, pp. 203–220. Springer, Berlin (2000)

    Google Scholar 

  8. Deng, S., Li, M., Zhou, J.: Closed-form approximations for spread option prices and greeks. J. Deriv. 16(4), 58–80 (2008)

    Google Scholar 

  9. Elliott, R.J., Kopp, P.E.: Mathematics of Financial Markets. Springer, Berlin/Heidelberg/New York (1999)

    Book  Google Scholar 

  10. Elliott, R.J., van der Hoek, J.: An application of hidden Markov models to asset allocation problems. Financ. Stoch. 3, 229–238 (1997)

    Article  Google Scholar 

  11. Elliott, R.J., Aggoun, L., Moore J.B.: Hidden Markov Models: Estimation and Control. Springer, Berlin/Heidelberg/New York (1994)

    Google Scholar 

  12. Elliott, R.J., Hunter, W.C., Jamieson, B.M.: Financial signal processing. Int. J. Theor. Appl. Financ. 4, 567–584 (2001)

    Article  Google Scholar 

  13. Elliott, R.J., Malcolm, W.P., Tsoi, A.H.: Robust parameter estimation for asset price models with Markov modulated volatilities. J. Econ. Dyn. Control 27(8), 1391–1409 (2003)

    Article  Google Scholar 

  14. Elliott, R.J., Chan, L.L., Siu, T.K.: Option pricing and Esscher transform under regime switching. Ann. Financ. 1(4), 423–432 (2005)

    Article  Google Scholar 

  15. Elliott, R.J., Chan, L., Siu, T.K.: Option valuation under a regime-switching constant elasticity of variance process. Appl. Math. Comput. 219(9), 4434–4443 (2013)

    Article  Google Scholar 

  16. Eydeland, A., Wolyniec, K.: Energy and Power Risk Management: New Developments in Modeling, Pricing and Hedging. Wiley, Hoboken (2003)

    Google Scholar 

  17. Gounden, S., O’Hara, J.G.: An analytic formula for the price of an American-style Asian option of floating strike type. Appl. Math. Comput. 217, 2923–2936 (2010)

    Article  Google Scholar 

  18. Guo, X.: Information and option pricings. Quant. Financ. 1, 38–44 (2001)

    Article  Google Scholar 

  19. Hildebrand, F.B.: Advanced Calculus and Applications. Prentice Hall, Englewood Cliffs (1976)

    Google Scholar 

  20. Hurd, T., Zhou, Z.: A Fourier transform method for spread option pricing. SIAM J. Financ. Math. 1, 142–157 (2009)

    Article  Google Scholar 

  21. Kirk, E.: Correlation in energy markets. In: Kaminski, V. (ed.) Managing Energy Price Risk, pp. 71–78. Risk Publications, London (1996)

    Google Scholar 

  22. Liao, S.J.: Homotopy Analysis Method in Nonlinear Differential Equations. Springer, Heidelberg/New York/Higher Education Press, Beijing (2012)

    Book  Google Scholar 

  23. Liao, S.J., Zhu, S.P.: Solving the Liouville equation with the general boundary element method approach. Bound. Elem. Technol. XIII, 407–416 (1999)

    Google Scholar 

  24. Margrabe, W.: The value of an option to exchange one asset for another. J. Financ. 33(1), 177–186 (1978)

    Article  Google Scholar 

  25. Merton, R.: Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4, 141–183 (1973)

    Article  Google Scholar 

  26. Odibat, Z.M.: A study on the convergence of homotopy analysis method. Appl. Math. Comput. 217, 782–789 (2010)

    Article  Google Scholar 

  27. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic, New York (1970)

    Google Scholar 

  28. Ravindran, K.: Low-fat spreads. Risk 6(10), 56–57 (1993)

    Google Scholar 

  29. Shimko, D.: Options on futures spreads: hedging, speculation and valuation. J. Futures Mark. 14(2), 183–213 (1994)

    Article  Google Scholar 

  30. Siu, T.K., Erlwein, C., Mamon, R.: The pricing of credit default swaps under a Markov-modulated Mertons structural model. North Am. Actuar. J. 12(1), 19–46 (2008)

    Google Scholar 

  31. Venkatramanan, A., Alexander, C.: Closed form approximations for spread options. Appl. Math. Financ. 18(5), 447–472 (2011)

    Article  Google Scholar 

  32. Yuen, F.L., Yang, H.: Option pricing in a jump-diffusion model with regime-switching. ASTIN Bull. 39(2), 515–539 (2009)

    Article  Google Scholar 

  33. Yuen, F.L., Yang, H.: Pricing Asian options and equity-indexed annuities with regime switching by the trinomial tree method. North Am. Actuar. J. 14(2), 256–277 (2010)

    Article  Google Scholar 

  34. Zhu, S.P.: An exact and explicit solution for the valuation of American put options. Quant. Financ. 6(3), 229–242 (2006)

    Article  Google Scholar 

  35. Zhu, S.P., Chan, L.: An analytic formula for pricing American options with regime switching. Submitted for publication (2012)

    Google Scholar 

  36. Zhu, S.P., Badran, A., Lu, X.: A new exact solution for pricing European options in a two-state regime-switching economy. Comput. Math. Appl. 64, 2744–2755 (2012)

    Article  Google Scholar 

Download references

Acknowledgements

We wish to thank the referees for helpful comments.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

    Leunglung Chan

Authors
  1. Leunglung Chan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leunglung Chan .

Editor information

Editors and Affiliations

  1. Dept of Statistical & Actuarial Sciences, University of Western Ontario, London, Ontario, Canada

    Rogemar S. Mamon

  2. School of Mathematics, University of Adelaide, Adelaide, Australia

    Robert J. Elliott

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this chapter

Cite this chapter

Chan, L. (2014). An Exact Formula for Pricing American Exchange Options with Regime Switching. In: Mamon, R., Elliott, R. (eds) Hidden Markov Models in Finance. International Series in Operations Research & Management Science, vol 209. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-7442-6_9

Download citation

Publish with us

Policies and ethics

Search

Navigation