Look-Back Option Pricing Using the Fourier Transform B-Spline Method

Abstract

We derive a new, efficient closed-form formula approximating the price of discrete look-back options, whose underlying asset price is driven by an exponential semi-martingale process including (jump) diffusions, Levy 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 look-back 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 where the Spitzer formula is utilized. We illustrate the numerical efficiency of our algorithm by applying it in pricing fixed and floating discrete look-back options under Brownian motion, jump diffusion models, and the variance gamma process.A new efficient and robust methodology is presented for pricing discrete look-back options whose underlying asset price is driven by an exponential semi-martingale process and no analytical pricing formula exists. Using B-spline interpolation we obtain an accurate closed-form representation of the look-back option price under an inverse generalized Fourier transform. This provides look-back option prices across the quantum of strike prices with greater efficiency than for a single strike price under existing methods.We derive an explicit representation for the characteristic function of the maximum of a discretely observed stochastic process, which provides a significant improvement in terms of numerical efficiency over the Spitzer recurrence formula. This is of fundamental importance and could have a wide range of applications where the Spitzer formula is utilized. Examples considered include pricing fixed and floating discrete look-back options under Brownian motion, jump diffusion models, and the Variance Gamma process.