Binomial Lattices

Abstract

In this chapter we explain binomial lattices beyond the standard case of a single source of uncertainty governed by a GBM process. We thus develop a binomial lattice which supports mean reversion in commodity prices. Then we show how to build a trinomial lattice, and also two-dimensional binomial lattices for a number of cases. We start from the basic binomial lattice for the spot price. If this discrete-time model is to approach the underlying continuous-time process, the size of the jumps at each time step along with their probabilities must fulfill certain properties. From the one-period lattice we then move to the n-period setting. We consider initially a non-dividend paying asset, since this is the simplest setting. Afterwards we extend the analysis to dividend paying assets. In addition to the usual lattice model, we develop this technique when the underlying variable is a futures price, and also the natural logarithm of a futures price; these lattices display particular features which can be useful in numerical implementations of the model. We fully develop a number of examples of finite-lived and perpetual options, both analytically and numerically. We also consider the case of two GBMs, two IGBMs, and one GBM alongside one IGBM.