Suppose that a binomial tree has n steps, and the stock has initial price S0 and then at each step, its price can only move up by a factor u or down by a factor d. Let Sk, k = 0, 1, , n, be the price of the stock at the end of the k-th step. Denote by time length between consecutive steps, and r the risk-free interest rate. Consider a call option with strike price K with maturity n .

(a) In the risk-neutral world, what is the probability that the stock moves down at each step?

(b) For n = 3, calculate the fair price for the option at current time corresponding to the initial node of the tree (please write out explicit formula).

Assume n = 10, = 1, r = 6%, S0 = 100, u = 1.1, d = 0.9, K = 110.

(c) In the risk-neutral world, find the probability that the stock price moves up five times and down four times in the first nine steps. What is the corresponding price value?