Implied Binomial Trees

Abstract

Despite its success, the Black-Scholes formula has become increasingly unreliable over time in the very markets where one would expect it to be most accurate. In addition, attempts by financial economists to extract probabilistic information from option prices have been puny in comparison to what is clearly possible. This paper 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). If specified exogenously, the model can also accommodate local interest rates and underlying asset payout rates which are general functions of the concurrent underlying asset price and time. In a 200 step lattice, for example, there are a total of 60,301 unknowns: 40,200 potentially different move sizes, 20,100 potentially different move probabilities, and 1 interest rate to be determined from 60,301 independent equations, many of which are non-linear in the unknowns. Despite this, a backwards recursive solution procedure exists which is only slightly more time-consuming than for a standard binomial tree with given constant move sizes and move probabilities. Moreover, closed-form expressions exist for the values and hedging parameters of European options maturing with or before the end of the tree. The tree can also be used to value and hedge American and several types of exotic options. Interpreted in terms of continuous-time diffusion processes, the model here assumes that the drift and local volatility are at most functions of the underlying asset price and time. But instead of beginning with a parameterization of these functions (as in previous research), the model derives these functions endogenously to fit current option prices. As a result, it can be thought of as an attempt to exhaust the potential for single state-variable path-independent diffusion processes to rectify problems with the Black- Scholes formula that arise in practice.