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
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
- 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.
- 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.
- 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.
- 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.
- Alan L. Lewis, 1998. "Applications of Eigenfunction Expansions in Continuous-Time Finance," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 349-383.
- Bakshi, Gurdip & Madan, Dilip, 2000. "Spanning and derivative-security valuation," Journal of Financial Economics, Elsevier, vol. 55(2), pages 205-238, February.
When requesting a correction, please mention this item's handle: RePEc:vsv:svpubs:explevy. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Alan Lewis)
If references are entirely missing, you can add them using this form.