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MATH3075 Financial Derivatives

创建日期：2024-06-05 17:53:40

所属分类：数学mathematics

标签： MATH3075

Financial Derivatives

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MATH3075 Financial Derivatives (Mainstream)
1. [10 marks] CRR model: American put option. Consider the CRR model with
T = 2 and S0 = 100, Su1 = 120, Sd1 = 90. Assume that the interest rate r = 0.
Consider an American put option with reward process g(St, t) = (Lt − St)+ and
variable strike price L0 = 105, L1 = 116, L2 = 111.
(a) Find parameters u, d, the stock price at time T = 2, and a martingale measure
P˜ on (Ω,F2).
(b) Compute the price process P a of this option using the recursive relationship
P at = max
{
(Lt − St)+, (1 + r)−1
(
p˜P aut+1 + (1− p˜)P adt+1
)}
with the terminal condition P a2 = (L2 − S2)+.
(c) Find the holder’s rational exercise time τ ∗0 .
(d) Find the replicating strategy for the option up to the exercise time τ ∗0 .
(e) Check whether the arbitrage price process (P at ; t = 0, 1, 2) is a martingale or a
supermartingale under P˜ with respect to the filtration F.
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 Y with maturity T and the
following payoff
Y = γST −max (ST , L)
where L = erTS0 and γ < 0 is a real number. We take for granted the Black-Scholes
pricing formulae for the call and put options.
(a) Sketch the profile of the payoff Y as a function of the stock price ST at time T
and show that Y admits the representation Y = γST −CT (L)−L where CT (L)
denotes the payoff at time T of the European call option with strike L.
(b) Find an explicit expression for the arbitrage price pit(Y ) at time 0 ≤ t < T in
terms of Ft := ertS0, St and S0. Then compute the price pi0(Y ) in terms of S0
and use the equality N(x)−N(−x) = 2N(x)− 1 to simplify your result.
(c) Compute and describe the hedging strategy at time 0 for the claim Y .
(d) Find the limits limσ→0 pi0(Y ) and limσ→∞ pi0(Y ).
(e) Explain why the price pi0(Y ) is negative when γ = 1 by analysing the payoff Y .

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