The Bino-Trinomial Tree: A Simple Model for Efficient and Accurate Option Pricing

Abstract

A model with a closed-form solution is the Holy Grail of derivatives valuation, because as computers have become increasingly powerful, exact answers to even very complicated formulas can typically be obtained almost instantaneously. Unfortunately, as derivative instruments have become increasingly complicated, closed-form valuation has become increasingly rare. Researchers are now trying to develop more efficient and accurate approximation techniques. Lattice models, such as the classic binomial model of Cox, Ross, and Rubinstein, are among the workhorses of this effort. Lattice models converge to accurate values as the number of node calculations increases, but convergence is often erratic and slow. The biggest problem is that the option payoff between two price nodes can be highly nonlinear, for example when the critical price barrier for a knock-out option falls between two layers of nodes, and a large jump in the computed option value occurs when a small change in the number of time steps causes the critical price to hop from one side of a node to the other. A variety of tricks have been proposed to deal with this problem by adapting the geometry of the lattice to make the nodes land directly on top of the critical prices. Generally this has required the additional flexibility afforded by a trinomial structure rather than the binomial, but that is costly in terms of efficiency. In this article, Dai and Lyuu introduce a new approach that achieves remarkable improvement in efficiency by combining binomial and trinomial structures. Here the trick is to construct a binomial lattice with nodes that land on top of the key prices, but to use a very small amount of trinomial lattice to connect the initial price—the model option value—to the binomial structure. This allows both the best placement of the tree relative to the critical areas and also great efficiency gains because binomial lattice probabilities for the terminal nodes can be computed directly using combinatorial results, skipping over calculations for all of the intermediate time steps.