Constructing Arbitrage-free Binomial Models

In the past decades several versions of the binomial model for option pricing, originallyintroduced by COX, ROSS, AND RUBINSTEIN, have been discussed in the financeliterature. Some of these approaches model an arbitrage-free market in the discrete setupwhereas others attain this property only in the limit. We analyze the interrelation betweenthe drift coefficient of price processes on arbitrage-free financial markets and the correspondingtransition probabilities induced by a martingale measure. As a result, we obtain aflexible setting that encompasses most arbitrage-free binomial models and provides modificationsfor those that offer arbitrage opportunities. It is argued that the knowledge of thelink between drift and transition probabilities may be useful for pricing derivatives such asbarrier options. A simple example is presented to illustrate this idea.