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

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 Lévy 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 pathdependence) under any Lévy process with a known characteristic function.

Spitzer identity, Wiener-Hopf factorization and pricing of discretely monitored exotic options

“…The option price is obtained discounting the expectation value of the undamped payoff with respect to the appropriate distribution; this expectation can conveniently be computed through the Parseval/Plancherel relation [45] by a product in Fourier space and an inverse Fourier transform, (34) where p(x) = p M (x, N ) or p m (x, N ) for lookback options (to be synthetic, in the following we will consider only fixed-strike lookback options written on the minimum), p(x) = p X,M (x, N ) or p X,m (x, N ) for single-barrier options, and p = p X,m,M (x, N ) for double-barrier options. The introduction of a damping factor in the payoff is compensated by a shift of the Fourier transform of the probability density function.…”

Option pricing with Lévy-Stable processes generated by Lévy-Stable integrated variance

“…Another route to price vanilla options for stock prices that follow a geometric Lévy-Stable processes is to compute the option value as an integral in Fourier space, using Complex Fourier Transform techniques [Lew01], [CM99].…”

The COS method for option valuation under the SABR dynamics

“…In the last few decades, many pricing methods have been introduced, for example, convenient formulas [3,17,19], Monte Carlo methods [5,15,20], finite difference methods [10,18,22], quadrature methods [7] and also Fourier methods [6,12,25,26,28,33]. A variety of option pricing methods is compared in [37], including several Fourier methods.…”