1. [10 marks] CRR model: American call option. Assume the CRR model with T = 2, the stock price S0 = 45, Su1 = 49.5, Sd1 = 40.5 and the interest rate r = −0.05. Consider the American call option with the reward process g(St, t) = (St − Kt)+ for t = 0, 1, 2 where the random strike price satisfies K0 = 40, K1(ω) = 35.5 for ω ∈ {ω1, ω2}, K1(ω) = 38.5 for ω ∈ {ω3, ω4} and K2 = 36.45.

 (a) Find the parameters u and d, compute the stock price at time t = 2 and find the unique martingale measure P˜. (b) Compute the price process Ca for this option using the recursive relationship Cat = max { (St −Kt)+, (1 + r)−1 EP˜ ( Cat+1 | Ft )} with the terminal condition Ca2 = (S2 −K2)+. (c) Find the rational exercise time τ ∗0 for the holder of this option. (d) Find the issuer’s replicating strategy ϕ for the option up to the rational exer- cise time τ ∗0 and show that the wealth of the replicating strategy matches the price computed in part 

(b). (e) Compute the profit of the issuer at time T if the holder decides to exercise the option at time T . 

2. [10marks] Black-Scholesmodel: European claim. We place ourselves within the setup of the Black-Scholes market modelM = (B, S) with a unique martingale measure P˜. Consider a European contingent claim X with maturity T and the following payoff X = max (K,ST )− LST where K = erTS0 and L > 0 is an arbitrary constant. We take for granted the Black-Scholes pricing formulae for the call and put options. 

(a) Sketch the profile of the payoff X as a function of the stock price ST at time T and show that X admits the following representation X = K + CT (K)− LST where CT (K) denotes the payoff at time T of the European call option with strike K.

 (b) Find an explicit expression for the arbitrage price pit(X) at time 0 ≤ t < T in terms of Ft := ertS0, St and S0. Then compute the price pi0(X) in terms of S0 and use the equality N(x)−N(−x) = 2N(x)− 1 to simplify your result. 

(c) Find the limit limT→0 pi0(X). 

(d) Find the limit limσ→∞ pi0(X). (e) Explain why the price of pi0(X) is positive when L = 1 by analysing the payoff X when L =