Lattice-based model for pricing contingent claims under mixed fractional Brownian motion

Abstract

We propose a lattice-based model to approximate the dynamics of an asset with diffusion driven by a mixed fractional Brownian motion. Being it defined as the sum of a fractional Brownian motion and an independent Brownian motion, the method starts by discretizing separately the two processes. For the first process, we develop a binomial approach that is able to replicate the process variance at each discrete epoch, while a recombining binomial lattice is used to discretize the Brownian motion. Given the fractional Brownian motion path-dependency, the binomial approach is not recombining. Hence, to reduce the computational complexity, we establish a grid of representative values whose elements cover the range of possible asset values at each time slice. For each value of the grid, the algorithm identifies four successors in a bivariate environment. The successors may not appear among the generated asset values at the next epoch. In such cases, interpolation techniques are needed when solving backward to compute the initial value of the contingent claim. The proposed discretization allows to evaluate both European and American derivatives and numerical experiments confirm its accuracy and efficiency.