Randomized quasi Monte Carlo methods for pricing of barrier options under fractional Brownian motion

Abstract

Randomized quasi-Monte Carlo (RQMC) method is presented to compute the problem of a barrier option pricing. It is assumed that stock prices are modeled with a fractional Brownian motion (FBM). The FBM is a Gaussian process with dependent and stationary increments except H = ½. The FBM can model stock prices with short or long memory. We propose a trajectory generation technique based on fast Fourier transforms to simulate stock prices modeled by FBM. A stock price trajectory is utilized to predict pricing of barrier options. Barrier options are options whose payoff function depend on the stock prices during the option’s lifetime. Using the results of the stock price trajectory and RQMC method can be determined the price of a barrier option under FBM. We conclude that RQMC is an efficient technique for calculating the price of barrier options rather than a standard Monte Carlo (MC).