Random binomial tree models and pricing options

Abstract

The binomial tree model is a natural bridge, overture to continuous models for which it is possible to derive the Black-Scholes option pricing formula. In turn a binomial branch model is the simplest possible non–trivial model which theory is based on the principle of no arbitrage works. The binomial tree model is defined by a pair of real numbers (u,d) such that the stock can move up from S0 to a new level, uS0 or down from S0 to a new level, dS0, where u > 1; 0 < d < 1. We shall call pair (u,d) the environment of the binomial tree model. The binomial tree model is called a random binomial tree model, if the corresponding environment is random. We introduce a simplest random binomial tree model, illustrating that risk – neutral valuation gives the same results as no-arbitrage arguments and describe some properties of the random binomial tree models. The random binomial tree model produces results which are a reflect of the real market better than the binomial tree model when fewer time steps are modelled. The model is solvable and there exist analytic pricing formulae for various options. In this thesis we produce these formulas for a European call and put options and also an American call and put options for a single period, a two periods and an arbitrary N-period time steps.