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A Simple Option Formula for General Jump-Diffusion and other Exponential Levy Processes



Option values are well-known to be the integral of a discounted transition density times a payoff function; this is just martingale pricing. It's usually done in 'S-space', where S is the terminal security price. But, for Levy processes the S-space transition densities are often very complicated, involving many special functions and infinite summations. Instead, we show that it's much easier to compute the option value as an integral in Fourier space - and interpret this as a Parseval identity. The formula is especially simple because (i) it's a single integration for any payoff and (ii) the integrand is typically a compact expressions with just elementary functions. Our approach clarifies and generalizes previous work using characteristic functions and Fourier inversions. For example, we show how the residue calculus leads to several variation formulas, such as a well-known, but less numerically efficient, 'Black-Scholes style' formula for call options. The result applies to any European-style, simple or exotic option (without path-dependence) under any Lévy process with a known characteristic function

Suggested Citation

  • Alan L. Lewis, 2001. "A Simple Option Formula for General Jump-Diffusion and other Exponential Levy Processes," Related articles explevy, Finance Press.
  • Handle: RePEc:vsv:svpubs:explevy

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    References listed on IDEAS

    1. Dilip B. Madan & Peter P. Carr & Eric C. Chang, 1998. "The Variance Gamma Process and Option Pricing," Review of Finance, European Finance Association, vol. 2(1), pages 79-105.
    2. Bates, David S, 1991. " The Crash of '87: Was It Expected? The Evidence from Options Markets," Journal of Finance, American Finance Association, vol. 46(3), pages 1009-1044, July.
    3. David B. Colwell & Robert J. Elliott, 1993. "Discontinuous Asset Prices And Non-Attainable Contingent Claims," Mathematical Finance, Wiley Blackwell, vol. 3(3), pages 295-308.
    4. Eberlein, Ernst & Keller, Ulrich & Prause, Karsten, 1998. "New Insights into Smile, Mispricing, and Value at Risk: The Hyperbolic Model," The Journal of Business, University of Chicago Press, vol. 71(3), pages 371-405, July.
    5. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, June.
    6. Alan L. Lewis, 1998. "Applications of Eigenfunction Expansions in Continuous-Time Finance," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 349-383.
    7. Bakshi, Gurdip & Madan, Dilip, 2000. "Spanning and derivative-security valuation," Journal of Financial Economics, Elsevier, vol. 55(2), pages 205-238, February.
    8. Naik, Vasanttilak & Lee, Moon, 1990. "General Equilibrium Pricing of Options on the Market Portfolio with Discontinuous Returns," Review of Financial Studies, Society for Financial Studies, vol. 3(4), pages 493-521.
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    Cited by:

    1. Roger Lord & Christian Kahl, 2006. "Optimal Fourier Inversion in Semi-analytical Option Pricing," Tinbergen Institute Discussion Papers 06-066/2, Tinbergen Institute, revised 05 Jun 2007.
    2. Peng Cheng & Olivier Scaillet, 2002. "Linear-Quadratic Jump-Diffusion Modeling with Application to Stochastic Volatility," FAME Research Paper Series rp67, International Center for Financial Asset Management and Engineering.
    3. Tobias Lipp & Grégoire Loeper & Olivier Pironneau, 2013. "Mixing Monte-Carlo and Partial Differential Equations for Pricing Options," Post-Print hal-01558826, HAL.
    4. repec:eee:apmaco:v:317:y:2018:i:c:p:68-84 is not listed on IDEAS
    5. Marcelo G. Figueroa, 2006. "Pricing Multiple Interruptible-Swing Contracts," Birkbeck Working Papers in Economics and Finance 0606, Birkbeck, Department of Economics, Mathematics & Statistics.
    6. Philipp Mayer & Natalie Packham & Wolfgang Schmidt, 2015. "Static hedging under maturity mismatch," Finance and Stochastics, Springer, vol. 19(3), pages 509-539, July.
    7. repec:wsi:ijtafx:v:16:y:2013:i:06:n:s0219024913500349 is not listed on IDEAS
    8. Carolyn E. Phelan & Daniele Marazzina & Gianluca Fusai & Guido Germano, 2017. "Hilbert transform, spectral filtering and option pricing," Papers 1706.09755,
    9. Cartea, Álvaro & del-Castillo-Negrete, Diego, 2007. "Fractional diffusion models of option prices in markets with jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 374(2), pages 749-763.
    10. Kim, Young Shin & Lee, Jaesung & Mittnik, Stefan & Park, Jiho, 2015. "Quanto option pricing in the presence of fat tails and asymmetric dependence," Journal of Econometrics, Elsevier, vol. 187(2), pages 512-520.
    11. Carolyn E. Phelan & Daniele Marazzina & Gianluca Fusai & Guido Germano, 2017. "Fluctuation identities with continuous monitoring and their application to price barrier options," Papers 1712.00077,
    12. Küchler, Uwe & Tappe, Stefan, 2014. "Exponential stock models driven by tempered stable processes," Journal of Econometrics, Elsevier, vol. 181(1), pages 53-63.
    13. Fusai, Gianluca & Germano, Guido & Marazzina, Daniele, 2016. "Spitzer identity, Wiener-Hopf factorization and pricing of discretely monitored exotic options," European Journal of Operational Research, Elsevier, vol. 251(1), pages 124-134.
    14. repec:wsi:ijtafx:v:16:y:2013:i:01:n:s0219024913500015 is not listed on IDEAS
    15. Omar El Euch & Mathieu Rosenbaum, 2016. "The characteristic function of rough Heston models," Papers 1609.02108,
    16. repec:wsi:ijtafx:v:16:y:2013:i:08:n:s0219024913500507 is not listed on IDEAS
    17. Noureddine Krichene, 2005. "Subordinated Levy Processes and Applications to Crude Oil Options," IMF Working Papers 05/174, International Monetary Fund.
    18. Wong, Hoi Ying & Lo, Yu Wai, 2009. "Option pricing with mean reversion and stochastic volatility," European Journal of Operational Research, Elsevier, vol. 197(1), pages 179-187, August.
    19. Eduardo Abi Jaber & Omar El Euch, 2018. "Multi-factor approximation of rough volatility models," Working Papers hal-01697117, HAL.
    20. 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.
    21. Fajardo, José, 2015. "Barrier style contracts under Lévy processes: An alternative approach," Journal of Banking & Finance, Elsevier, vol. 53(C), pages 179-187.
    22. Lorenzo Torricelli, 2016. "Valuation of asset and volatility derivatives using decoupled time-changed Lévy processes," Review of Derivatives Research, Springer, vol. 19(1), pages 1-39, April.

    More about this item


    option pricing; jump-diffusion; Levy processes; Fourier; characteristic function; transforms; residue; call options; discontinuous; jump processes; analytic characteristic; Levy-Khintchine; infinitely divisible; independent increments;

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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