An Efficient Implementation of Implied Binomial Trees

Implied binomial trees are constructed by fitting a risk-neutral density (in the form of ending nodal probabilities) to observed option prices. Since there are usually not enough options traded in the marketplace, a quadratic program is often used to extract the ending nodal probabilities from observed option prices. A problem with this commonly used approach is that the quadratic program is usually a high dimensional one and can be difficult or inefficient to solve. This is because numerical accuracy may require the construction of a large binomial tree, resulting in a high dimensional quadratic program as the number of unknowns (i.e., the number of ending nodal probabilities) is proportional to the number of binomial periods. In this paper, we propose a more efficient implementation of implied binomial trees by incorporating cubic spline smoothing in the quadratic program. Only a selected subset of ending nodal probabilities is treated as unknowns while the remainder is interpolated using cubic splines. This allows us to substantially reduce the dimension of the quadratic program and thus improve the efficiency of its numerical implementation. We demonstrate the effectiveness of the smoothed implied binomial tree method by evaluating its numerical accuracy in estimating the risk-neutral density and moments such as volatility, skewness and kurtosis. JEL classifications: G13