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Spitzer identity, Wiener-Hopf factorization and pricing of discretely monitored exotic options

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  • Fusai, Gianluca
  • Germano, Guido
  • Marazzina, Daniele

Abstract

The Wiener-Hopf factorization of a complex function arises in a variety of fields in applied mathematics such as probability, finance, insurance, queuing theory, radio engineering and fluid mechanics. The factorization fully characterizes the distribution of functionals of a random walk or a Lévy process, such as the maximum, the minimum and hitting times. Here we propose a constructive procedure for the computation of the Wiener-Hopf factors, valid for both single and double barriers, based on the combined use of the Hilbert and the z-transform. The numerical implementation can be simply performed via the fast Fourier transform and the Euler summation. Given that the information in the Wiener-Hopf factors is strictly related to the distributions of the first passage times, as a concrete application in mathematical finance we consider the pricing of discretely monitored exotic options, such as lookback and barrier options, when the underlying price evolves according to an exponential Lévy process. We show that the computational cost of our procedure is independent of the number of monitoring dates and the error decays exponentially with the number of grid points.

Suggested Citation

  • Fusai, Gianluca & Germano, Guido & Marazzina, Daniele, 2016. "Spitzer identity, Wiener-Hopf factorization and pricing of discretely monitored exotic options," LSE Research Online Documents on Economics 67564, London School of Economics and Political Science, LSE Library.
  • Handle: RePEc:ehl:lserod:67564
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    1. Jin, Xing & Li, Xun & Tan, Hwee Huat & Wu, Zhenyu, 2013. "A computationally efficient state-space partitioning approach to pricing high-dimensional American options via dimension reduction," European Journal of Operational Research, Elsevier, vol. 231(2), pages 362-370.
    2. Lord, Roger & Fang, Fang & Bervoets, Frank & Oosterlee, Kees, 2007. "A fast and accurate FFT-based method for pricing early-exercise options under Lévy processes," MPRA Paper 1952, University Library of Munich, Germany.
    3. Peter Carr & John Crosby, 2010. "A class of Levy process models with almost exact calibration to both barrier and vanilla FX options," Quantitative Finance, Taylor & Francis Journals, vol. 10(10), pages 1115-1136.
    4. Kay Giesecke & Dmitry Smelov, 2013. "Exact Sampling of Jump Diffusions," Operations Research, INFORMS, vol. 61(4), pages 894-907, August.
    5. Xiaoqun Wang & Ken Seng Tan, 2013. "Pricing and Hedging with Discontinuous Functions: Quasi-Monte Carlo Methods and Dimension Reduction," Management Science, INFORMS, vol. 59(2), pages 376-389, July.
    6. Olivier Le Courtois & François Quittard-Pinon, 2008. "The optimal capital structure of the firm with stable Lévy assets returns," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 31(1), pages 51-72, May.
    7. Alan L. Lewis, 2001. "A Simple Option Formula for General Jump-Diffusion and other Exponential Levy Processes," Related articles explevy, Finance Press.
    8. Date, Paresh & Islyaev, Suren, 2015. "A fast calibrating volatility model for option pricing," European Journal of Operational Research, Elsevier, vol. 243(2), pages 599-606.
    9. Dingeç, Kemal Dinçer & Hörmann, Wolfgang, 2012. "A general control variate method for option pricing under Lévy processes," European Journal of Operational Research, Elsevier, vol. 221(2), pages 368-377.
    10. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    11. Gianluca Fusai & I. Abrahams & Carlo Sgarra, 2006. "An exact analytical solution for discrete barrier options," Finance and Stochastics, Springer, vol. 10(1), pages 1-26, January.
    12. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    13. Soeren Asmussen & Dilip Madan & Martijn Pistorius, 2007. "Pricing Equity Default Swaps under an approximation to the CGMY L\'{e}% vy Model," Papers 0711.2807, arXiv.org.
    14. Gianluca Fusai & Daniele Marazzina & Marina Marena & Michael Ng, 2012. "Z -Transform and preconditioning techniques for option pricing," Quantitative Finance, Taylor & Francis Journals, vol. 12(9), pages 1381-1394, November.
    15. Liming Feng & Vadim Linetsky, 2008. "Pricing Discretely Monitored Barrier Options And Defaultable Bonds In Lévy Process Models: A Fast Hilbert Transform Approach," Mathematical Finance, Wiley Blackwell, vol. 18(3), pages 337-384, July.
    16. Gerber, Hans U. & Shiu, Elias S.W. & Yang, Hailiang, 2013. "Valuing equity-linked death benefits in jump diffusion models," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 615-623.
    17. Asmussen, Søren & Avram, Florin & Pistorius, Martijn R., 2004. "Russian and American put options under exponential phase-type Lévy models," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 79-111, January.
    18. Fusai, Gianluca & Recchioni, Maria Cristina, 2007. "Analysis of quadrature methods for pricing discrete barrier options," Journal of Economic Dynamics and Control, Elsevier, vol. 31(3), pages 826-860, March.
    19. Sesana, Debora & Marazzina, Daniele & Fusai, Gianluca, 2014. "Pricing exotic derivatives exploiting structure," European Journal of Operational Research, Elsevier, vol. 236(1), pages 369-381.
    20. S. I. Boyarchenko & S. Z. Levendorskii, 2002. "Pricing of perpetual Bermudan options," Quantitative Finance, Taylor & Francis Journals, vol. 2(6), pages 432-442.
    21. Liming Feng & Vadim Linetsky, 2008. "Pricing Options in Jump-Diffusion Models: An Extrapolation Approach," Operations Research, INFORMS, vol. 56(2), pages 304-325, April.
    22. Liming Feng & Vadim Linetsky, 2009. "Computing exponential moments of the discrete maximum of a Lévy process and lookback options," Finance and Stochastics, Springer, vol. 13(4), pages 501-529, September.
    23. Marc Jeannin & Martijn Pistorius, 2008. "A transform approach to compute prices and greeks of barrier options driven by a class of Levy processes," Papers 0812.3128, arXiv.org, revised Mar 2009.
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    More about this item

    Keywords

    Path-dependent options; Hilbert transform; Lévy process; Spitzer identity; Wiener-Hopf factorization;
    All these keywords.

    JEL classification:

    • J1 - Labor and Demographic Economics - - Demographic Economics

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