A Simple Option Formula for General Jump-Diffusion and other Exponential Levy Processes
AbstractOption 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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Bibliographic InfoPaper provided by Finance Press in its series Related articles with number explevy.
Length: 25 pages
Date of creation: 06 Aug 2001
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option pricing; jump-diffusion; Levy processes; Fourier; characteristic function; transforms; residue; call options; discontinuous; jump processes; analytic characteristic; Levy-Khintchine; infinitely divisible; independent increments;
Find related papers by JEL classification:
- G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
This paper has been announced in the following NEP Reports:
- NEP-ALL-2001-09-10 (All new papers)
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- Peng Cheng & Olivier Scaillet, 2002.
"Linear-Quadratic Jump-Diffusion Modeling with Application to Stochastic Volatility,"
FAME Research Paper Series
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- Noureddine Krichene, 2005. "Subordinated Levy Processes and Applications to Crude Oil Options," IMF Working Papers 05/174, International Monetary Fund.
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