Consider a binomial model. S0 = 4. In each period, the stock price doubles 1

with a probability of 1/2 and halves with a probability of 1/2. The interest rate is r = 1/ 2 . Suppose N = 2. An American contract yields: if exercised at time t = 0, a payment of $0; if exercised at time t = 1, a payment of $6 dollars if the stock price has decreased and of $0 otherwise; if exercised at time t = 2, a payment of $36 if the stock price has jumped twice, of $18 if the stock price is equal to $4 and of $0 otherwise.

What is the fair price of the contract at time t = 0?

Denote the price of the contract at time n, if it has not been exercised before, by Vn. You decide to exercise the contract if the stock decreases at time t = 1 or if the stock price has jumped twice; denote this trading strategy/stopping time by . Write down explicitly as a function and find E[V2 ].