How Precise Are Price Distributions Predicted by Implied Binomial Trees?

Abstract

In recent years, especially after the 1987 market crash, it became clear that the prices of the underlying asset do not exactly follow the Geometric Brow-nian Motion (GBM) model of Black and Scholes. The GBM model with constant volatility leads to a log-normal price distribution at any expiration date: All options on the underlying must have the same Black-Scholes (BS) implied volatility, and the Cox-Ross-Rubinstein (CRR) binomial tree makes use of this fact via the construction of constant transition probability from one node to the corresponding node at the next level in the tree. In contrast, the implied binomial tree (IBT) method simply constructs a numerical procedure consistent with the volatility smile. The empirical fact that the market implied volatilities decrease with the strike level, and increase with the time to maturity of options is better reflected by this construction. The algorithm of the IBT is a data adaptive modification of the CRR method.