Implied Binomial Trees



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    • University of California at Berkeley. Presidential address to the American Finance Association, January 1994, Boston, Massachusetts. I would like to give special thanks to Jack Hirshleifer, who while he has not commented specifically on this article, nonetheless as my mentor in my formative years, propelled me in its direction. William Keirstead, while he has been a Ph.D. student at Berkeley, has helped implement the nonlinear programming algorithms described in this article. I am also grateful for recent conversations with Hua He, John Hull, Hayne Leland, and Alan White, and earlier conversations with Ray Hawkins and David Shimko.


This article develops a new method for inferring risk-neutral probabilities (or state-contingent prices) from the simultaneously observed prices of European options. These probabilities are then used to infer a unique fully specified recombining binomial tree that is consistent with these probabilities (and, hence, consistent with all the observed option prices). A simple backwards recursive procedure solves for the entire tree. From the standpoint of the standard binomial option pricing model, which implies a limiting risk-neutral lognormal distribution for the underlying asset, the approach here provides the natural (and probably the simplest) way to generalize to arbitrary ending risk-neutral probability distributions.