Binomial lattice model for stock prices

Here we model the price of a stock in discrete time by a Markov chain of the recursive form Sn+1 = SnYn+1, n ≥ 0, where the {Yi} are iid with distribution P (Y = u) = p, P (Y = d) = 1 − p. Here 0 < d < 1 + r < u are constants with r the risk-free interest rate ((1 + r)x is the payoff you would receive one unit of time later if you bought $x worth of the risk-free asset (a bond for example, or placed money in a savings account at that fixed rate) at time n = 0). Given the value of Sn, Sn+1 = { uSn, w.p. p; dSn, w.p. 1− p, n ≥ 0,