Binomial Tree Methods

We introduce the binomial algorithm for pricing based upon the martingale characterisation of the value of an option. We show how the underlying asset may be modelled in discrete time as a random walk on a recombining binomial tree, and investigate how to calibrate the parameters of this discrete model to market data. We look at how to modify the approach for dividend-paying assets, and conduct an error analysis in the case of European options by comparing valuations on binomial trees with an increasing number of steps against benchmark Black-Scholes valuations. We show how to implement a modified framework for the treatment of options with an early exercise facility, such as American or Bermudan options via the Bellman equations associated with the pricing problem. Finally we use conditional indexing in Python to efficiently value both knock-out- and knock-in- style barrier options.