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MATH3075 Financial Derivatives (Mainstream)
创建日期：2024-10-09 11:46:19
所属分类：金融Finance
标签：
Financial Derivatives (Mainstream)
MATH3075 Financial Derivatives (Mainstream)
Due by 11:59 p.m. on Sunday, 20 October 2024
1. [10 marks] CRR model: American call option. Assume the CRR model with
T = 2, the stock price S0 = 45, Su1 = 49.5, Sd1 = 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 ∈ {ω1, ω2}, K1(ω) = 38.5 for ω ∈ {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 P˜.
(b) Compute the price process Ca for this option using the recursive relationship
Cat = max
{
(St −Kt)+, (1 + r)−1 EP˜
(
Cat+1 | Ft
)}
with the terminal condition Ca2 = (S2 −K2)+.
(c) Find the rational exercise time τ ∗0 for the holder of this option.
(d) Find the issuer’s replicating strategy ϕ for the option up to the rational exer-
cise time τ ∗0 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. [10marks] Black-Scholesmodel: European claim. We place ourselves within
the setup of the Black-Scholes market modelM = (B, S) with a unique martingale
measure P˜. Consider a European contingent claim X with maturity T and the
following payoff
X = max (K,ST )− LST
where K = erTS0 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 pit(X) at time 0 ≤ t < T in
terms of Ft := ertS0, St and S0. Then compute the price pi0(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→0 pi0(X).
(d) Find the limit limσ→∞ pi0(X).
(e) Explain why the price of pi0(X) is positive when L = 1 by analysing the payoff
X when L = 1.
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