ABSTRACT
The binomial model of a (B,S)-market was introduced in the previous chapter. Sometimes this model is also referred to as the Cox-Ross-Rubinstein model. Recall that the dynamics of the market are represented by equations
ΔBn = rBn−1 , B0 = 1 , (2.1) ΔSn = ρnSn−1 , S0 > 0 ,
where r ≥ 0 is a constant rate of interest with −1 < a < r < b, and profitabilities or risky asset returns
ρn = {
b with probability p ∈ [0, 1] a with probability q = 1− p , n = 1, . . . , N ,
form a sequence of independent identically distributed random variables. The stochastic basis in this model consists of Ω = {a, b}N , the space of sequences ω = (ω1, . . . , ωN ) of length N whose elements are equal to either a or b; F = 2Ω, the set of all subsets of Ω. The probability P has Bernoulli probability distribution with p ∈ [0, 1], so that
P ({ω}) = pPNi=1 I{b}(ωi) (1− p)PNi=1 I{a}(ωi) .
