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

25 Pages Posted: 6 Sep 2001

Date Written: September 2001

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 expression 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 Levy process with a known characteristic function.

JEL Classification: G13

Suggested Citation

Lewis, Alan L., A Simple Option Formula for General Jump-Diffusion and Other Exponential Levy Processes (September 2001). Available at SSRN: https://ssrn.com/abstract=282110 or http://dx.doi.org/10.2139/ssrn.282110

Alan L. Lewis (Contact Author)

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