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Department of Mathematics
University College London
MATH0094 Market Risk and Portfolio Theory
MSc Examination
2022-2023
TIME ALLOWED: 3 HOURS
The exam contains FOUR questions. All questions should be attempted. Each
question is worth 25 marks.
Calculators are permitted.
MATH0094 1 TURN OVER
Question 1. Consider a T-periods arbitrage-free market with n risky assets and
a risk-free asset S
0
. As usual, we assume that S0
0 = 1.
(a) Let θ and ξ be self-funding strategies such that S
θ
t = S
ξ
t
for some t ∈ {1, . . . , T}.
Show that S
θ
0 = S
ξ
0
, and explain the financial meaning of this property.
[7 marks]
Let K ∈ R+ and τ ∈ {1, . . . , T} be arbitrary but fixed values. A call option on the
asset i ∈ {1, . . . , n} with strike K and maturity τ is an asset with payoff (S
i
τ K)
+
where (.)
+ denotes positive part. Likewise, a put option on the asset i with strike
K and maturity τ is an asset with payoff (K S
i
τ
)
+.
In what follows, let p
i
c be an arbitrage-free price of a contingent call option on the
asset i with strike K and maturity τ (respectively let p
i
p be the arbitrage-free price
of the analogous put option). Further, assume that the risk-free asset is constant
and equal to 1, i.e. St
0 = 1, t = 0, . . . , T.
(b) Show that
p
i,K
c p
i,K
p = S
i
0 K.
Hint: Recall that a = (a)
+ ( a)
+.
[9 marks]
(c) Suppose that {θt}t=0,…,τ is a self-funding strategy (in terms of units) replicating
a call option on the asset i with strike K and maturity τ . Find a self-funding
strategy replicating the corresponding put option (i.e. same asset, strike, and
maturity).
[9 marks]
[Total: 25 marks]
MATH0094 2 CONTINUED
Question 2.
Consider a pure investor with CRRA utility function v with constant relative
risk aversion ρ = 1. The investor acts in a one-period market model with finite
probability, with n-risky assets, a risk-free asset, and m outcomes in the probability
space. The market is characterised by the random returns vector R.
(a) Let R ∈ R
(n+1)×m be the matrix of returns
R = {R1
i
(ωj )}i=0,…,n;j=1,…m.
Find conditions in terms of R guaranteeing existence and uniqueness of the
solution to the utility maximisation problem.
[8 marks]
Consider the investor-consumer problem
max
C0,π
E[v(C0) + δv(C1)]
s.t.
C1 = (w0 C0)(( R1 R
0
1
11) · π + R
0
1
);
for δ ∈ (0, 1).
(b) Suppose that the conditions in (a) hold. i) Find the optimal consumption
strategy; and ii) show that the optimal investment strategy satisfies
1
R0
1
E
”
Ri
1
(
R1 R0
1
11) · π
+ R0
1
#
= E
”
1
(
R1 R0
1
11) · π
+ R0
1
#
,
for any i = 0, . . . , n.
[11 marks]
(c) Find an SDF in this market in terms of the optimal investment strategy.
[6 marks]
[Total: 25 marks]
MATH0094 3 TURN OVER
Question 3. Let X be a continuous random variable with probability distribu_x005f tion function fx and cumulative density function FX. Assume further that FX is
invertible. Recall that expected shortfall is defined by
ESα
(X) = 1
1 α
Z α
1
V@Ru
(X)du.
(a) Show that
ESα
(X) = V@Rα
(X) + 1
1 α
E[( X V@Rα
(X))+]
where (a)
is the negative part operator, i.e. (a)
= ( a)
+.
[7 marks]
(b) Prove that
ESα
(X) = inf
z∈R
{z +
1
1 α
E[( X z)
+]}
Hint: Apply first order conditions to the optimization problem on the right hand side, and use the fact that -X has a density.
[9 marks]
(c) Use (b) to show that ESα
satisfies convexity, and give a financial interpretation
for this property.
[9 marks]
[Total: 25 marks]
MATH0094 4 CONTINUED
Question 4. Consider a market with three risky assets and a risk-free asset.
The risk-free asset has unit return, while the risky assets have mean return μ and
variance-covariance Σ given by
μ =

3
7
9

Σ =

1 1 1
1 5 5
1 5 13

. Note that Σ 1 =
1
8

10 2 0
2 3 1
0 1 1

.
(a) Find an expression that characterises all portfolios in the mean-variance fron tier excluding the risk-free asset in terms of the above data.
[ 8 marks]
(b) Find in this market:
i. the tangency portfolio; and
ii. the maximal Sharpe ratio (Smax) of the market including the risk-free
asset.
[ 9 marks]
(c) An investor evaluates risk using standard deviation. They would like to fix a
target average return of 3. Find their optimal portfolio in the market (including
risk-free asset).
[ 8 marks]
[Total: 25 marks]
MATH0094 5 TURN OVER