Advanced Modelling in Finance Using Excel and VBA

Chapter 10. Binomial Trees

Binomial trees are introduced first because they provide an easy way to understand the Black–Scholes analysis. The binomial tree provides a convenient way to model the share price process using a discrete binomial distribution to approximate the normal distribution of log returns assumed in the Black–Scholes analysis.

Since we can create hedge portfolios, we value options as if we are in a risk-neutral world. There remains the calculation of the expected value of the option payoff. We compare and contrast three different ways of configuring the binomial tree using different parameter choices for the tree (referred to as JR, CRR and LR trees according to their inventors: Jarrow and Rudd; Cox, Ross and Rubinstein; Leisen and Reimer). Each involves different combinations of incremental price changes and probabilities for the tree. Starting with European options, valuation via the JR tree is outlined, and then the better-known CRR tree is described. After discussing the convergence of the valuation approximations derived from trees to the underlying Black–Scholes value, the lesser known LR tree is outlined. Choosing between the different combinations of parameters may seem rather esoteric, but the best choice of parameters (LR) turns out to be much more efficient than the other choices. Experimentation with the different models for European options suggests that the LR tree provides values that match the Black–Scholes values very closely, using only a small ...