Binomial Lattice

The Binomial Model framework

Don Chance (2007) and (1999) observed that option pricing theory had become one of the most powerful tools in economics and finance. The celebrated Black-Scholes-Merton model not only netted a Nobel Prize for Scholes and Merton but also completely re-invented the financial industry, Chicago Markets and even banking. Its discrete time analogue, the binomial model, has also attracted similar levels of attention in part because it furnishes a clear illustration of the essential concepts behind option pricing theory with a minimum of mathematical fuss. The binomial framework proves to offer greater flexibility in accommodating many path-dependent options. In particular, the binomial framework can be used to price American options. To date the downside to this approach has been the sheer computational intensity embedded in the lattice approach. Analysts have tended to abandon the Cox, Ross and Rubinstein model and use closed form approximations for valuing American Options especially when a book of contracts have to be priced or estimating implied volatility on platforms that have to be refreshed in real time.

We will show in the following pages that lattices models can be optimized so as to become faster and more accurate relative to analytic analogues. We achieve this by applying Operations Research techniques which leads to a useful intersection of mathematics and computer science with finance. This intersection has led to an approach that we refer to as Intelligent Lattice Search. We initially take the Cox, Ross and Rubinstein (1979) model and introduce a relatively simple AI algorithm that pares back unnecessary blanket calculations embedded in standard lattices. We reduce computational runtime from over 18 minutes down to less than 3 seconds to estimate a 15,000-step binomial tree. Delta and Implied Volatility can also be accelerated relative to standard models. Lattice estimation, in general, is considered to be slow and not practical for valuing large books of options. A major plus here is that Intelligent lattice Search can be applied to all the major binomial frameworks. In each instance, we provide dramatic improvements in the estimation speed and accuracy and this can be benchmarked against the leading closed form solutions for American Options.

The origins of the binomial model are slightly convoluted. Around 1975 William Sharpe, (also a Nobel Laureate), intimated to Mark Rubinstein that a binomial framework would be feasible for valuation. Sharpe subsequently formalized the idea in the first edition of his textbook. The better-known original paper on the model is Cox, Ross, and Rubinstein (1979), but contemporaneously, Rendleman and Bartter (1979) presented the same model in a slightly different manner.