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

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    Abstract

    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

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    File URL: http://optioncity.net/pubs/ExpLevy.pdf
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    Bibliographic Info

    Paper provided by Finance Press in its series Related articles with number explevy.

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    Length: 25 pages
    Date of creation: 06 Aug 2001
    Date of revision:
    Handle: RePEc:vsv:svpubs:explevy

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    Related research

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

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    1. Bakshi, Gurdip & Madan, Dilip, 2000. "Spanning and derivative-security valuation," Journal of Financial Economics, Elsevier, vol. 55(2), pages 205-238, February.
    2. 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.
    3. 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.
    4. 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.
    5. Alan L. Lewis, 1998. "Applications of Eigenfunction Expansions in Continuous-Time Finance," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 349-383.
    6. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv.
    7. 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 1009-44, July.
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    Cited by:
    1. Roger Lord & Christian Kahl, 2006. "Why the Rotation Count Algorithm works," Tinbergen Institute Discussion Papers 06-065/2, Tinbergen Institute.
    2. Marcelo G. Figueroa, 2006. "Pricing Multiple Interruptible-Swing Contracts," Birkbeck Working Papers in Economics and Finance 0606, Birkbeck, Department of Economics, Mathematics & Statistics.
    3. 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.
    4. Küchler, Uwe & Tappe, Stefan, 2014. "Exponential stock models driven by tempered stable processes," Journal of Econometrics, Elsevier, vol. 181(1), pages 53-63.
    5. Noureddine Krichene, 2005. "Subordinated Levy Processes and Applications to Crude Oil Options," IMF Working Papers 05/174, International Monetary Fund.

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