Skip to main content
Log in

Cite this article


Several authors have used Fourier inversion to compute prices of puts and calls, some using Parseval’s theorem. The expected value of max (SK, 0) also arises in excess-of-loss or stop-loss insurance, and we show that Fourier methods may be used to compute them. In this paper, we take the idea of using Parseval’s theorem further: (1) formulas requiring weaker assumptions; (2) relationship with classical inversion theorems for probability distributions; (3) formulas for payoffs which occur in insurance. Numerical examples are provided.

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

Access this article

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


  • M. Abramowitz, and I. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs and Mathematical Tables, Dover: New York, 1970.

    Google Scholar 

  • T. M. Apostol, Mathematical Analysis, Second Edition. Addison-Wesley: Reading, 1974.

    MATH  Google Scholar 

  • G. Bakshi, and D. B. Madan, “Spanning and derivative-security valuation,” Journal of Financial Economics vol. 55 pp. 205–238, 2000.

    Article  Google Scholar 

  • K. Borovkov, and A. Novikov, “A new approach to calculating expectations for option pricing,” Journal of Applied Probability vol. 39 pp. 889–895, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  • P. Carr, and D. B. Madan, “Option valuation using the fast Fourier transform,” Journal of Computational Finance vol. 2 pp. 61–73, 1999.

    Google Scholar 

  • F. Dufresne, and H. U. Gerber, “Risk theory for the compound Poisson process that is perturbed by diffusion,” Insurance: Mathematics and Economics vol. 10 pp. 51–59, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  • H. J. Furrer, “Risk processes perturbed by a α–stable Lévy motion,” Scandinavian Actuarial Journal vol. 1998 pp. 59–74, 1998.

    MATH  MathSciNet  Google Scholar 

  • S. L. Heston, “A closed-form solution for options with stochastic volatility with application to bond and currency options,” Review of Financial Studies vol. 6 pp. 327–343, 1993.

    Article  Google Scholar 

  • M. Kendall, and A. Stuart, The Advanced Theory of Statistics, Fourth Edition. Griffin: London, 1977.

    MATH  Google Scholar 

  • N. N. Lebedev, Special Functions and Their Applications, Dover: New York, 1972.

    MATH  Google Scholar 

  • R. W. Lee, “Option pricing by transform methods: extensions, unification, and error control,” Journal of Computational Finance vol. 7 pp. 51–86, 2004.

    Google Scholar 

  • A. L. Lewis, “A simple option formula for general jump–diffusion and other exponential Lévy processes,” publications: , 2001.

  • E. Lukacs, Characteristic Functions, Fourth Edition. Griffin: London, 1970.

    MATH  Google Scholar 

  • P. Malliavin, Integration and Probability, Springer Verlag: New York, 1995.

    MATH  Google Scholar 

  • S. Raible, Lévy Processes in Finance: Theory, Numerics, and Empirical Facts. Ph.D. Dissertation, Faculty of Mathematics, University of Freiburg, Germany, 2000.

  • G. Samorodnitsky, and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman & Hall: New York, 1994.

    MATH  Google Scholar 

  • K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press: Cambridge, 1999.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Daniel Dufresne.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Dufresne, D., Garrido, J. & Morales, M. Fourier Inversion Formulas in Option Pricing and Insurance. Methodol Comput Appl Probab 11, 359–383 (2009).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


AMS 2000 Subject Classification