Margrabe Revisited

Part of the Fields Institute Communications book series (FIC, volume 74)


We introduce a new representation of the bivariate normal distribution to first give a short derivation of the classic Margrabe exchange-option formula, using elementary integration methods. The second application is a new and simple technique to provide an accurate lower bound for the value of a spread option with a nonzero strike.


Canonical Formulation Case Type Exchange Option Strike Price Spot Price 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The simplified proof of the Margrabe formula, using elementary integration techniques, was developed in the context of a course on energy commodities that the author teaches in the Mathematical Finance Program at the University of Toronto. I thank the anonymous referee for comments and suggestions that helped to improve the presentation, and my wife Marguerite Martindale for a professional line edit.


  1. 1.
    Bjerksund, P., Stensland, G.: Closed form spread option valuation. Quant. Finance 14(10), 1785–1794 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Carmona, R., Durrleman, V.: Pricing and hedging spread options. SIAM Rev. 45(4), 627–685 (2003)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Carmona, R., Durrleman, V.: Pricing and hedging spread options in a log-normal model. Technical Report, Department of Operations Research and Financial Engineering. Princeton University, Princeton, NJ, 16 March (2003)Google Scholar
  4. 4.
    Clewlow, L., Strickland, C.: Energy Derivatives: Pricing and Risk Management. Lacima Publications, London (2000)Google Scholar
  5. 5.
    Geman, H.: Commodities and Commodity Derivatives: Modeling and Pricing for Agriculturals, Metals and Energy. Wiley, London (2005)Google Scholar
  6. 6.
    Li, M., Deng, S.-J., Zhou, J.: Closed-form approximations for spread option prices and Greeks. J. Deriv. 15(3), 58–80 (Spring 2008)CrossRefGoogle Scholar
  7. 7.
    Margrabe, W.: The value of an option to exchange one asset for another. J. Finance 33(1), 177–186 (1978)CrossRefGoogle Scholar
  8. 8.
    Pilipovic, D.: Energy Risk: Valuing and Managing Energy Derivatives, 2nd edn. McGraw-Hill, New York (2007)Google Scholar
  9. 9.
    Tong, Y.L.: The Multivariate Normal Distribution. Springer Series in Statistics. Springer, New York (1990)MATHCrossRefGoogle Scholar
  10. 10.
    van der Hoek, J., Korolkiewicz, M.W.: New analytic approximations for pricing spread options. In: Cohen, S.N., Madan, D., Siu, T.K.,Yang, H. (eds.) Stochastic Processes, Finance and Control: A Festschrift in Honor of Robert J. Elliott. Advances in Statistics, Probability and Actuarial Science, vol. 1, pp. 259–284. World Scientific, Singapore (2012)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Mathematical Finance ProgramUniversity of TorontoTorontoCanada

Personalised recommendations