Question

1. [10 marks] CRR model: European contingent claim. Consider the CRR model M = (B, S) with the horizon date T = 2, the interest rate r = 0.2, and the stock price S0 = 80, S_1^u = 104, S_1^d = 88. Let X be the European contingent claim maturing at T = 2 with the payoff given by the following expression with K = 116 X = |S2 ? K|?_{|S2 ? K|>10} = { |S2 ? K|, on the event {|S2 ? K| > 10}, 0, on the event {|S2 ? K| ? 10}. (a) Find the parameters u and d, compute the stock price at time t = 2, and find the risk-neutral probability ?. (b) Compute the arbitrage price of X using the risk-neutral valuation formula ?t(X) = Bt E?(X B_T^-1 | Ft), t = 0, 1, 2. (c) Find the replicating strategy ?1 = (?_1^0, ?_1^1) at time t = 1 for the claim X and check that V1(?) = ?1(X). (d) Find the replicating strategy ?0 = (?_0^0, ?_0^1) at time t = 0 for the claim X and check that V0(?) = ?0(X). (e) Show that the following equality is always satisfied in the CRR model (do not use numbers to derive this equality) E?(S2 ? S1) = r(1 + r)S0. (1) Compute the price of Y = |S2 ? S1| at time 0 using equation (1).

          1. [10 marks] CRR model: European contingent claim. Consider the CRR model M = (B, S) with the horizon date T = 2, the interest rate r = 0.2, and the stock price S0 = 80, S_1^u = 104, S_1^d = 88. Let X be the European contingent claim maturing at T = 2 with the payoff given by the following expression with K = 116
X = |S2 ? K|?_{|S2 ? K|>10} = { |S2 ? K|, on the event {|S2 ? K| > 10}, 0, on the event {|S2 ? K| ? 10}.
(a) Find the parameters u and d, compute the stock price at time t = 2, and find the risk-neutral probability ?.
(b) Compute the arbitrage price of X using the risk-neutral valuation formula
?t(X) = Bt E?(X B_T^-1 | Ft), t = 0, 1, 2.
(c) Find the replicating strategy ?1 = (?_1^0, ?_1^1) at time t = 1 for the claim X and check that V1(?) = ?1(X).
(d) Find the replicating strategy ?0 = (?_0^0, ?_0^1) at time t = 0 for the claim X and check that V0(?) = ?0(X).
(e) Show that the following equality is always satisfied in the CRR model (do not use numbers to derive this equality)
E?(S2 ? S1) = r(1 + r)S0. (1)
Compute the price of Y = |S2 ? S1| at time 0 using equation (1).

1. [10 marks] CRR model: European contingent claim. Consider the CRR model M = (B, S) with the horizon date T = 2, the interest rate r = 0.2, and the stock price S0 = 80, S1^u = 104, S1^d = 88. Let X be the European contingent claim maturing at T = 2 with the payoff given by the following expression with K = 116
X = |S2 ? K|?|S2 ? K|>10 = |S2 ? K|, on the event |S2 ? K| > 10, 0, on the event |S2 ? K| ? 10.
(a) Find the parameters u and d, compute the stock price at time t = 2, and find the risk-neutral probability ?.
(b) Compute the arbitrage price of X using the risk-neutral valuation formula
?t(X) = Bt E?(X BT^-1 | Ft), t = 0, 1, 2.
(c) Find the replicating strategy ?1 = (?1^0, ?1^1) at time t = 1 for the claim X and check that V1(?) = ?1(X).
(d) Find the replicating strategy ?0 = (?0^0, ?0^1) at time t = 0 for the claim X and check that V0(?) = ?0(X).
(e) Show that the following equality is always satisfied in the CRR model (do not use numbers to derive this equality)
E?(S2 ? S1) = r(1 + r)S0. (1)
Compute the price of Y = |S2 ? S1| at time 0 using equation (1).

Added by Deborah G.

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