ASSIGNMENT 2 MATH3075 Financial Derivatives (Mainstream) Due by 10 p.m. on Friday, 29 October 2021 1. [5 marks] CRR model: American put option. Consider the CRR model with T = 2 and S0 = 100, Su1 = 120, Sd1 = 90. Assume that the interest rate r = 0. Consider an American put option with reward process g(St, t) = (Kt − St)+ and variable strike price K0 = 105, K1 = 116, K2 = 111. (a) Find parameters u, d, the stock price at time T = 2, and a martingale measure P˜ on (Ω,F2). (b) Compute the price process P a of this option using the recursive relationship P at = max { (Kt − St)+, (1 + r)−1 ( p˜P aut+1 + (1− p˜)P adt+1 )} with the terminal condition P a2 = (K2 − S2)+. (c) Find the holder’s rational exercise time τ ∗0 . (d) Find the replicating strategy for the option up to the exercise time τ ∗0 . (e) Check whether the arbitrage price process (P at ; t = 0, 1, 2) is a martingale or a supermartingale under P˜ with respect to the filtration F. 2. [5 marks] Black-Scholes model: 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 a real number. 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 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) Compute and describe the hedging strategy at time 0 for the claim X. (d) Find the limits limσ→∞ pi0(X) and limT→0 pi0(X). (e) Explain why the price pi0(X) is positive when L = 1 by analysing the payoff X.