Reference no: EM131306861
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(Sr) = 1/ST1[k,∞)(ST).
Show that the arbitrage price Π0(X) at time t = 0 equals
Π0(X) = 1/S0 .T∑k=k^ (Tk)αkβT-k
for some constants a and 0. Give explicit formulae for α, β and k^ in terms of So, 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^ (Tk)(0.2)k(0.8βT-k
(c) We maintain the assumptions of part (b). Consider the European claim Y where
Y = h(ST) =1/ST1(-∞, k)(ST).
Find the arbitrage price Π0(Y) and show that Π0(Y) + Π0(Y) = 1/So
(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 Πt(Z1) = Πt(Z2) necessarily holds for every 0 ≤ t ≤ T?
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