Fractional Binomial Trees

Binomial trees are discrete approximations of stochastic processes where at every discrete point in time the process has two possibilities: it either moves upwards or descends to a certain extent. Each alternative occurs with a certain probability adding up to 1. Consequently, two factors determine the characteristics of the resulting discrete process: The probability distributions of the single steps as well as the extent of the two possible shifts at each step. The binomial tree approach for classical Brownian motion is well-developed and leads to intuitive insights concerning the understanding of Brownian motion as the limit of an uncorrelated random walk. Cox et al. (1979) extended the very setting and defined a binomial stock price model converging weakly to the lognormal diffusion of geometric Brownian motion. Other processes of several important continuous time models in finance have been modeled successfully in a similar fashion (see e.g. Nelson and Ramaswamy (1990)). Hence, one might expect that a comparable approach for fractional Brownian motion should also be possible. However, as we will see in this chapter, things are a little bit harder to work out. This is mainly due to the property of serial correlation