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MATH3075 Financial Derivatives
创建日期：2024-04-20 17:41:23
所属分类：数学mathematics
标签： MATH3075
Mathematics and Statistics

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CONFIDENTIAL EXAM PAPER

School of Mathematics and Statistics
EXAMINATION
MATH3075 Financial Derivatives (Mainstream)
EXAM WRITING TIME: 2 hours
READING TIME: 10 minutes
EXAM CONDITIONS: This is an OPEN book examination
During reading time - writing is not permitted ONLY on the exam paper
Please tick the box to confirm that your examination paper is complete.
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ANONYMOUSLY MARKED
(Please do not write your name on this exam paper)
For Examiner Use Only
Q Mark
1
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MATH3075 Semester 2, 2020 Page 2 of 5
1. [20 marks] Single-period market model.
Consider a single-period security market model M = (B, S) on a finite
state space Ω = {ω1, ω2, ω3}. Assume that the savings account B equals
B0 = 1, B1 = 2 and the stock price S satisfies S0 = 15 and
S1 = (S1(ω1), S1(ω2), S1(ω3)) = (36, 27, 24).
The real-world probability P on Ω is such that P(ωi) = pi > 0 for i = 1, 2, 3.
(a) Check directly using only the definition of an arbitrage opportunity
that the modelM = (B, S) is arbitrage-free.
(b) Find the class M of all risk-neutral probability measures for the
modelM. Is the market modelM complete?
(c) Describe the class A of all attainable contingent claims.
(d) Show that the contingent claim X = (X(ω1), X(ω2), X(ω3)) = (10, 7, 6)
is attainable and compute its arbitrage price pi0(X) using the repli-
cating strategy for X.
(e) Consider the contingent claim X = (30, 21, 18). Show that the ex-
pected value
EQ
(
X
B1
)
does not depend on the choice of a risk-neutral probability measure
Q ∈M. Is this claim attainable?
2
MATH3075 Semester 2, 2020 Page 3 of 5
2. [20 marks] CRR model: European claim.
Consider the CRR model with T = 2 and S0 = 20, Su1 = 26, Sd1 = 22.
Assume that the interest rate r = 0.2. Let X be the European contingent
claim maturing at T = 2 with the payoff given by the formula
X =
(
S2 − S1
)
1{S2−S1>5} =
 S2 − S1, on the event {S2 − S1 > 5},0, on the event {S2 − S1 ≤ 5}.
(a) Show explicitly that the contingent claim X is path-dependent.
(b) Find the risk-neutral probability measure P˜ for the modelM = (B, S)
and compute the arbitrage price of X using the risk-neutral valua-
tion formula
pit(X) = Bt EP˜
(
XB−1T | Ft
)
, t = 0, 1, 2.
(c) Find the replicating portfolio (φ0, φ1) for the claim X and check that
the equality Vt(φ) = pit(X) is satisfied for t = 0, 1, 2.
(d) Show that in any CRR model
EP˜(S2 − S1) = r(1 + r)S0.
Let Y =
(
S2 − S1
)
1{S2−S1≤5}. Find the price of Y at time 0 using the
additivity of arbitrage prices and the fact that X+Y = S2−S1. Check
your result by computing
pi0(Y ) = EP˜
(
Y
B2
)
.
(e) Find the unique probability measure P̂ on (Ω,F2) such that the pro-
cess Bt/St, t = 0, 1, 2 is a martingale under P̂ with respect to the fil-
tration F = (Ft)t=0,1,2.
3
MATH3075 Semester 2, 2020 Page 4 of 5
3. [20 marks] CRR model: American option.
Consider the CRR model with T = 2 and S0 = 180, Su1 = 198, Sd1 = 162.
Assume that the interest rate r = 0. Consider the American call option
with the reward process (St−Kt)+ where the variable strike price satisfies
K0 = 148, K1 = 142, K2 = 145.8.
(a) Find the parameters u and d, compute the stock price at time t = 2,
and find the unique martingale measure P˜ for the model.
(b) Compute the arbitrage price process Ca for this option using the re-
cursive 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 the option.
(d) Find the replicating strategy ϕ for the option up to the rational exer-
cise time τ ∗0 and compute the initial wealth V0(φ).
(e) Find the arbitrage price C0 for the European call option with the
payoff C2 = (S2 −K2)+ at time T = 2 and compute the early exercise
premium Ca0 − C0.
4
MATH3075 Semester 2, 2020 Page 5 of 5
4. [20 marks] Black-Scholes model.
Assume that the stock price S is governed under the risk-neutral proba-
bility measure P˜ by the Black-Scholes stochastic differential equation
dSt = St
(
r dt+ σ dWt
)
where σ > 0 is the volatility and r is the short-term interest rate. Con-
sider the European contingent claim X with maturity T and the following
payoff
X = KST −min (ST , L)
where L = erTS0 and K > 0 is an arbitrary constant.
(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 − 1)ST + CT (L)
where CT (L) = (ST − L)+ is the payoff at time T of the call option
with the strike L.
(b) Using the Black-Scholes call option pricing formula, find an explicit
expression for the arbitrage price pi0(X) at time t = 0.
(c) Find the limit of the arbitrage price pi0(X) when T approaches 0.
(d) Find the limit of the arbitrage price pi0(X) when the volatility σ goes
to∞.
(e) Explain why the price of X is positive when K ≥ 1.
End of examination
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