Options and mathematics:

Do not skip calculations in the exercises and write as clear as possible. If some portion of the solution is not clearly readable, it will be assumed to be wrong.

Consider a 3 period binomial model with the following parameters:

S(0) = 8, u = log 2, d = log 2, r = 0, p (0, 1).

Consider also a European style derivative U in this market such that U expires worthless if the stock price exceeds Q = 30 or falls below L = 3/2 at some time t {1, 2, 3} and which otherwise pays the amount

Y = max{S(t), t = 0, 1, 2, 3} min{S(t), t = 0, 1, 2, 3}.

Compute the binomial price of the derivative at time t = 0. Find the value of p which maximizes the probability that U expires in the money and the value of p which maximizes the probability that the return on 1 share of U be positive.