Fast Fourier Transform Option Pricing: Efficient Approximation Methods Under Multi-Factor Stochastic Volatility and Jumps

  • J. P. F. CharpinEmail author
  • M. Cummins
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 19)


Fourier option-pricing methods are popular due to the dual benefits of wide applicability and computational efficiency. The literature tends to focus on a limited subset of models with analytic conditional characteristic functions (CCFs). Models that require numerical solutions of the CCF undermine the efficiency of Fourier methods. To tackle this problem, an ad hoc approximate numerical method was developed that provide CCF values accurately much faster than traditional methods. This approximation, based on averaging, Taylor expansions and asymptotic behaviour of the CCFs, is presented and tested for a range of affine models, with multi-factor stochastic volatility and jumps. The approximation leads to average run-time accelerations up to 50 times those of other numerical implementations, with very low absolute and relative errors reported.


Fast Fourier Transform Option Price Stochastic Volatility Strike Price Saddlepoint Approximation 
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.



Jean Charpin acknowledges the support of the Mathematics Applications Consortium for Science and Industry ( funded by the Science Foundation Ireland Mathematics Initiative grant 06/MI/005.


  1. 1.
    Bakshi, G., Cao, C., Chen, Z.: Empirical performance of alternative option pricing models. J. Financ. 52, 2003–2049 (1997)CrossRefGoogle Scholar
  2. 2.
    Bates, D.: Jumps and stochastic volatility: exchange rate processes implicit in deutsche mark options. Rev. Financ. Stud. 9, 69–107 (1996)CrossRefGoogle Scholar
  3. 3.
    Bates, D.: Post-’87 crash fears in the S&P 500 futures option market. J. Econometrics. 94, 181–238 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Benhamou, E.: Fast Fourier transform for discrete Asian options. J. Comput. Financ. 6, 49–61 (2002)Google Scholar
  5. 5.
    Carr, P., Madan, D.: Option valuation using the fast Fourier transform. J. Comput. Financ. 3, 463–520 (1999)Google Scholar
  6. 6.
    Carr, P., Madan, D.: Saddlepoint methods for option pricing. J. Comput. Financ. 13, 49–61 (2009)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Carr, P., Géman, H., Madan D., et al.: Option pricing using integral transforms. New York University. Available online. (2011). Last time accessed May 2012
  8. 8.
    Chacko, G., Das, S.R.: Pricing interest rate derivatives: a general approach. Rev. Financ. Stud. 15, 195–241 (2002)CrossRefGoogle Scholar
  9. 9.
    Das, S.R., Foresi, S.: Exact solutions for bond and option prices with systematic jump risk. Rev. Derivatives Res. 1, 7–24 (1996)CrossRefGoogle Scholar
  10. 10.
    Dempster, M.A.H., Hong, S.S.G.: Spread option valuation and the fast Fourier transform. In: German, H., Madan, D., Pliska, S.R., Vorst, T. (ed.) Mathematical Finance – Bachelier Congress 2000: Selected Papers form the World Congress of the Bachelier Finance Society. Springer-Verlag, Berlin (2001)Google Scholar
  11. 11.
    Duffie, D., Pan, J., Singleton, K.: Transform analysis and option pricing for affine jump-diffusions. Econometrica 68, 1343–1376 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Eraker, B.: Do stock prices and volatility jump? Reconciling evidence from spot and option prices. J. Financ. 59, 1367–1404 (2004)Google Scholar
  13. 13.
    Filipovic, D., Mayerhoffer, E., Schneider, P.: Density approximations for multivariate affine jump-diffusion processes [Working Paper]. Available via arXiv. (2011). Last time accessed May 2012
  14. 14.
    Glasserman, P., Kim, K-K.: Saddlepoint approximations for affine jump-diffusion models. J. Econ. Dyn. Control. 33, 15–36 (2009)Google Scholar
  15. 15.
    Heston, S.L.: A closed-form solution for options with stochastic volatility and applications to bond and currency options. Rev. Financ. Stud. 6, 327–343 (1993)CrossRefGoogle Scholar
  16. 16.
    Hurd, T.R., Zhou, Z.: A Fourier transform method for spread option pricing [Working Paper]. McMaster University. Available via arXiv.\_cache/arxiv/pdf/0902/0902.3643v1.pdf (2009). Last time accessed May 2012
  17. 17.
    Jackson, K.R., Jaimungal, S., Surkov, V.: Fourier space time-stepping for option pricing with Levy models. J. Comput. Financ. 12, 1–29 (2008)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Jaimungal, S., Surkov, V.: Fourier space time-stepping for option pricing with Levy models [Working Paper]. University of Toronto. Available via Social Science Research Network.\_id=1302887 (2009). Last time accessed May 2012
  19. 19.
    Lee, R.W.: Option pricing by transform methods: Extensions, unification, and error control. J. Comput. Financ. 7, 51–86 (2004)Google Scholar
  20. 20.
    Lord, R., Fang, F., Bervoets, F., et al.: A fast and accurate FFT-based method for pricing early-exercise options under Levy processes. SIAM J. Sci. Comput. 30, 1678–1705 (2008)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lord, R., Kahl, C.: Optimal Fourier inversion in semi-analytical option pricing. J. Comput. Financ. 10, 1–30 (2007)Google Scholar
  22. 22.
    Minenna, M., Verzella, P.: A revisited and stable Fourier transform method for affine jump diffusion models. J. Bank. Financ. 32, 2064–2075 (2008)CrossRefGoogle Scholar
  23. 23.
    Scott, L.O.: Pricing stock options in a jump-diffusion model with stochastic volatility and interest rates: application of Fourier inversion methods. Math. Financ. 7, 345–358 (1997)CrossRefGoogle Scholar
  24. 24.
    Takahashi, A., Takehara, K.: Fourier transform method with an asymptotic expansion approach: An application to currency options [Working Paper]. University of Tokyo. Available online. (2009). Last time accessed May 2012
  25. 25.
    Zhylyevskyy, O.: A fast Fourier transform technique for pricing American options under stochastic volatility. Rev. Derivatives Res. 13, 1–24 (2010)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, Mathematics Application Consortium for Science and Industry (MACSI), College of Science and EngineeringUniversity of LimerickLimerickIreland
  2. 2.DCU Business SchoolDublin City UniversityDublin 9Ireland

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