Random Cox-Ross-Rubinstein Model and Plain Vanilla Options

In this paper we introduce and study random Cox-Ross-Rubinstein (CRR) model. The CRR model is a natural bridge, overture to continuous models for which it is possible to derive the Black Scholes option pricing formula. An attractive property of CRR model is that the binomial tree for geometric Brownian motion is consistent with the standard Black-Scholes formula for European options in that no mismatch occurs if a tree is used to price an American option where no closed form solutions exist, as well as several exotic options. The usual CRR model is prescribed by pair of numbers (u, d), where u denotes the increase rate of the stock over the fixed period of time and d denotes the decrease rate, with 0< d < 1< u. We call the pair (u, d) an environment of the CRR model. A pair (U , D ), where {U } and {D } are the sequences of independent, n n n n identically distributed random variables with 0< D < 1< U for all n, is called a random environment and binomial n n tree model with random environment is called random CRR model. In this paper we define and study a random CRR model for pricing of plain vanilla options.