MATH3075 Financial Derivatives (Mainstream) ASSIGNMENT 2
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ASSIGNMENT 2
MATH3075 Financial Derivatives (Mainstream)
2022
1. [10 marks] CRR model: American call option. Assume the CRR model with T = 2, the stock price S0 = 45, S1(u) = 49.5, S1(d) = 40.5 and the interest rate r = -0.05. Consider the American call option with the reward process g(St , t) = (St - Kt )+ for t = 0, 1, 2 where the random strike price satisfies K0 = 40, K1 (ω) = 35.5 for ω e {ω1 , ω2 }, K1 (ω) = 38.5 for ω e {ω3 , ω4 } and K2 = 36.45.
(a) Find the parameters u and d, compute the stock price at time t = 2 and find the unique martingale measure .
(b) Compute the price process Ca for this option using the recursive relationship
Ct(a) = max {(St - Kt )+ , (1 + r)_1 E ╱ C | 于t、}
with the terminal condition C2(a) = (S2 - K2 )+ .
(c) Find the rational exercise time τ0(*) for the holder of this option.
(d) Find the issuer’s replicating strategy ϕ for the option up to the rational exer- cise time τ0(*) and show that the wealth of the replicating strategy matches the price computed in part (b).
(e) Compute the profit of the issuer at time T if the holder decides to exercise the option at time T.
2. [10 marks] Black-Scholes model: European claim. We place ourselves within the setup of the Black-Scholes market model M = (B, S) with a unique martingale measure . Consider a European contingent claim X with maturity T and the following payoff
X = max (K, ST ) - LST
where K = erT S0 and L > 0 is an arbitrary constant. 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 following 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 πt (X) at time 0 < t < T in terms of Ft := ert S0 , St and S0 . Then compute the price π0 (X) in terms of S0 and use the equality N(x) - N(-x) = 2N(x) - 1 to simplify your result.
(c) Find the limit limT二0 π0 (X).
(d) Find the limit limσ 二+ π0 (X).
(e) Explain why the price of π0 (X) is positive when L = 1 by analysing the payoff X when L = 1.
2022-10-12