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

Alan Lewis

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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

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 (search for similar items in EconPapers)
JEL-codes: G13 (search for similar items in EconPapers)
Pages: 25 pages
Date: 2001-08-06
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (82)

Published in Option Valuation under Stochastic Volatility II, 2016, Ch 14

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Persistent link: https://EconPapers.repec.org/RePEc:vsv:svpubs:explevy

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