Question 1. The CRR model: European claim. Consider the CRR model of stock price S with T periods and parameters d < 1 + r < n, where r is the one-period interest rate.

(a) Consider the European claim X with expiry date T and the payoff

X = g(S_T) = 1/S_T 1|K, ∞)(S_T)

Show that the arbitrage price Π₀(X) at time t = 0 equals

Π₀(X) = 1/S₀ ∑^T_{k =k^{^}} (^T _k) α^kβ^T-k

for some constants α and β. Give explicit formulae for α, β and k^{^}. in terms of S_o, u, d and r.

(b) Assume that p^~ = 0.5, r = 0.25 and d = u^-1. Compute u and d and deduce that

Π₀(X) = 1/S₀ ∑^T_{k =k^{^}} (^T _k)(0.2)^k(0.8)^T-k

(c) We maintain the assumptions of part (b). Consider the European claim Y where

Y = h(S_T) = 1/S_T 1(-∞, k)(S_T).

Find the arbitrage price Πo(Y) and show that Π_o(X) + Π_o(Y) = 1/S_o

(d) Let Z₁ and Z₂ be two European claims with maturity T in the CRR model. Assume that the equality Π_t(Z₁) = Π_t(Z₂) holds for every t ≤ U where U is a fixed date satisfying 0 < U < T. Does the equality Π_t(Z₁) = Π_t(Z₂) necessarily holds for every 0 ≤ t ≤ T?