9.1. Consider our N period binomial model with usual parameters u, d, r such that 0 < d <1+r < u. Further, assume r

9.1. Consider our N period binomial model with usual parameters u, d, r such that 0 < d 0. For each n, let G, be a random variable that depends on the first n coin tosses. Consider the following American-type security that can be exercised at any time n < N for payoff Gn, but must be exercised at time N if it has not been exercised before that time. In particular, this security must be exercised at some point. Let V be the price of this security at time n. Note that VN = GN because at time N, if the security had not been exercised yet, then it must be exercised. Further, the price process of this security satisfies the following recursion: Vn (wi... wn) ) = max (G(w ... wn), 1 + 7. (PVn+1(W ... WnII) + GVn+1 (W ...wT) wI))) (1) In particular, the price process Vo, V, ..., Vy satisfies the same recursion (1) as an usual American option, but not the terminal condition at time N. Finally, this price process can also be described in terms of stopping times. In particular, V = max TCSn,TN E GT (1+r) A. Let's consider a put version of this security with G = K - Sn. (a) Prove by induction that for all n, V = K - Sn. (b) Using part (a), show that from time 0, the optimal time to exercise the option is immediately (that is, time 0). B. Let's consider a call version of this security with G = Sn - K. N-n K (1+r (a) Prove by induction that for all n, V = Sn (b) Using part (a), show that from time 0, the optimal time to exercise the option is at expiration (that is, time N).