Lookback option pricing using the Fourier transform B-spline method

Abstract

We derive a new, efficient closed-form formula approximating the price of discrete lookback options, whose underlying asset price is driven by an exponential semimartingale process, which includes ( jump) diffusions, Lévy models, affine processes and other models. The derivation of our pricing formula is based on inverting the Fourier transform using B-spline approximation theory. We give an error bound for our formula and establish its fast rate of convergence to the true price. Our method provides lookback option prices across the quantum of strike prices with greater efficiency than for a single strike price under existing methods. We provide an alternative proof to the Spitzer formula for the characteristic function of the maximum of a discretely observed stochastic process, which yields a numerically efficient algorithm based on convolutions. This is an important result which could have a wide range of applications in which the Spitzer formula is utilized. We illustrate the numerical efficiency of our algorithm by applying it in pricing fixed and floating discrete lookback options under Brownian motion, jump diffusion models, and the variance gamma process.

Keywords:

• Lookback option pricing
• Fourier transform
• B-spline interpolation
• Spitzer formula
• Jump diffusion
• Variance gamma

JEL Classification:

• G10
• G13

Acknowledgments

We would like to thank the two anonymous referees for their valuable comments and suggestions and in particular for pointing to us the papers by Lipton (2002a, b) and Wendel (1958) which helped to significantly improve the revised version of our paper. We are also grateful to Dimitrina Dimitrova and Zvetan Ignatov for valuable comments and suggestions on earlier versions of the paper.