A Note on Constructing 50-50 Step Probability Binomial Lattices to Replicate Wiener Diffusion

Binomial lattices are sequences of discrete distributions commonly used to approximate the future value states of a financial claim, such as a stock price, when the instantaneous rate of return is assumed to be governed by a Wiener diffusion process. In that case, both pedagogical and professional conventions generally follow the lattice construction methodology used by Cox, Ross, and Rubinstein ("CRR") in their classical article. In some applications, it is more convenient to replace the "implied" branching probabilities of that construction with a more natural and tractable alternative: that is, with the probability of "up" and "down" branching being exactly one-half, or, vernacularly, with a "50-50 step" probability. This elementary note reviews such an alternative formulation for constructing a binomial lattice, which can be viewed as simply entailing multiplicative shifts of every state value on a CRR-constructed binomial lattice. This transformation maintains (in fact, improves) the equivalence of the lattice values' moments to those arising from the replicated diffusion. The expression of that transform is derived, and the effect on the lattice values' moments and orders of convergence to the limit imposed by the continuous process are given. To show the absence of numerical effect, the values of some simple European options obtained from the two alternative binomial lattice constructions are compared against the limiting Black Scholes values.