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_r) = 1/S_T1[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 a and 0. 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(0.8β^T-k

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

Y = h(S_T) =1/S_T1(-∞, k)(S_T).

Find the arbitrage price Π₀(Y) and show that Π₀(Y) + Π₀(Y) = 1/S_o

(d) Let Z₁ and Z₂ be two European claims with maturity T in the CRR model. Assume that the equality Π₁(Z₁) = Π₂(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?