Module Code
MAT00015H
BA, BSc and MMath Examinations 2019/20
Department:
Mathematics
Title of Exam:
Mathematical Finance I
Time Allowed:
2 hours
Allocation of Marks:
Question 1 carries 30 marks.
Question 2 carries 35 marks.
The marking scheme shown on each question is indicative only.
Instructions for Candidates:
Answer all questions.
Please write your answers in ink; pencil is acceptable for graphs and diagrams.
Do not use red ink.
Materials Supplied:
Green booklet
Calculator
Do not write on this booklet before the exam begins.
Do not turn over this page until instructed to do so by an invigilator.
Page 1 (of 4)
MAT00015H
1 (of 3).
(a) Consider a market consisting of two risky securities with expected returns and
covariances between the returns given respectively by the matrices
m =
(
µ1 µ2
)
=
(
2 3
)
and C =
(
2 1
1 4
)
.
(i) Consider a portfolio V with weights w = (w1, w2) = (0.4, 0.6). Compute
the expected return of V and the standard deviation of return of V . [6]
(ii) Find the minimum variance portfolio. [6]
(iii) Let V¯ be a portfolio on the efficient frontier. If w¯ denotes the weight vector
of V¯ , and C and m are as above, find γ such that
γ w¯C = m− µu
with µ = 0.5 and u = (1, 1). [4]
(b) Suppose a market consists of n risky securities S1, . . . , Sn with expected returns
µ1, . . . , µn and standard deviations of returns σ1, . . . , σn, respectively. Let C be
the covariance matrix of returns. Assume also that the risk-free return is R > 0.
(i) Explain what is meant by the market portfolio in the Capital Asset Pricing
Model. [3]
(ii) Write down the equation for the Capital Market Line (CML) under the as-
sumptions of the Capital Asset Pricing Model. [3]
(iii) Suppose that portfolios V1 and V2 have expected returns µV1 = 8 and µV2 = 5,
and standard deviations of returns σV1 = 2 and σV2 = 1, respectively. If V1
and V2 are both on the CML, find the risk-free return R. [8]
Page 2 (of 4)
MAT00015H
2 (of 3). (a) Let the market consist of three securities S1, S2, S3. Suppose the correspond-
ing vector of expected returns is m = (2, 1,−1). Let C be the covariance
matrix of returns. Suppose C and C−1 are respectively given by
C =
1
20
15 −5 5−5 7 −3
5 −3 7
 and C−1 =
 2 1 −11 4 1
−1 1 4
 .
Assume also that the risk-free return is R = 0.4.
(i) Compute the weights of the market portfolio. [10]
(ii) Compute the expected return and the standard deviation of the return on
the market portfolio. [8]
(iii) Compute the beta factor and the risk premium for a portfolio with weights
w1 = 0.7, w2 = −0.1, w3 = 0.4. [5]
(b) Consider a market which consists of a stock with price S(t), for t = 0, 1, 2,
. . . , T , and a risk-free asset A(t) with constant one-step return R > 0.
(i) State the no-arbitrage principle. [4]
(ii) Suppose we have a forward contract written on a stock S with initial
value S(0) and terminal value S(T ). Suppose also that we have a risk-
free asset with initial value A(0) and terminal value A(T ). Let F (0, T )
be the forward price. Show that if
F (0, T ) <
A(T )
A(0)
S(0)
then there is an arbitrage opportunity. [8]
Page 3 (of 4) Turn over
MAT00015H
3 (of 3). (a) Explain what is meant by a European call option with the strike price X and
expiry time T written on a stock S. What is the payoff of a European call
option? [5]
(b) If CE(0) denote the initial price of a call option written on stock with strike
price X and expiry time T . Assume also that the risk-free rate is R > 0.
Show that
max
{
0, S(0)− X
(1 +R)T
} ≤ CE(0) .
[10]
(c) Let CE(0) and PE(0) be the initial prices of a European call and put option,
respectively, written on a stock S with strike price X and expiry time T = 1.
Assume also that the risk-free rate is R > 0. Show that there is an arbitrage
opportunity if
CE(0)− PE(0) > S(0)− X
(1 +R)
.
[20]
Page 4 (of 4) End of examination.
SOLUTIONS: MAT00015H
1. (a) (i) The expected return of V is given by
µV = wm
T =
(
0.4 0.6
)(2
3
)
= 0.4× 2 + 0.6× 3 = 2.6 .
The variance of return of V is given by
σ2V = wCw
T =
(
0.4 0.6
)(2 1
1 4
)(
0.4
0.6
)
=
(
0.4 0.6
)(1.4
2.8
)
= 2.24 .
So the standard deviation is σV =
√
2.24 ≈ 1.497.
6 Marks
(ii) The weights of the minimum variance portfolio is given by
w =
uC−1
uC−1uT
,
where u = (1, 1). Note that if
C =
(
2 1
1 4
)
then
C−1 =
1
det(C)
(
4 −1
−1 2
)
=
1
7
(
4 −1
−1 2
)
.
Therefore the minimum variance portfolio has weights
w =
1
7
(
1 1
)( 4 −1
−1 2
)
1
7
(
1 1
)( 4 −1
−1 2
)(
1
1
)
=
(
3 1
)
(
3 1
)(1
1
) = (3/4 1/4) = (0.75 0.25) .
6 Marks
(iii) Multiplying the equation
γ w¯C = m− µu
on the right by C−1 gives
γ w¯ = (m− µu)C−1 .
Now multiplying on the right by uT and noting that w¯uT = 1 we obtain
γ = (m− µu)C−1uT .
5
SOLUTIONS: MAT00015H
Hence
γ =
1
7
((
2 3
)− 0.5 (1 1))( 4 −1−1 2
)(
1
1
)
=
1
7
(
1.5 2.5
)( 4 −1
−1 2
)(
1
1
)
=
1
7
(
1.5 2.5
)(3
1
)
=
7
7
= 1 .
4 Marks
(b) (i) The Market Portfolio is a portfolio which consists of all risky securities with
weights equal to their relative share in the whole market.
Alternative definition: The Market Portfolio is the tangency point of the efficient
frontier and the capital market line. (A diagram showing the tangency point in the
(σ, µ)-plane is also an acceptable answer.)
3 Marks
(ii) The equation for the Capital Market Line is
µV = R +
µM −R
σM
σV ,
where µV is the expected return of the portfolio V , σV is the standard deviation of
return on the portfolio V , µM is the expected return of the market portfolio, and
σM is the standard deviation of return on the market portfolio.
3 Marks
(iii) We have
µV1 = R +
µM −R
σM
σV1 ,
and
µV2 = R +
µM −R
σM
σV2 .
From the above we obtain
µV1 −R
σV1
=
µM −R
σM
,
and
µV2 −R
σV2
=
µM −R
σM
.
Therefore
µV1 −R
σV1
=
µV2 −R
σV2
,
which gives
R =
µV2σV1 − µV1σV2
σV1 − σV2
=
5× 2− 8× 1
2− 1 = 2 .
6
SOLUTIONS: MAT00015H
8 Marks Total: 30 Marks
2. (a) (i) We have m = (2, 1,−1), R = 0.4 and let u = (1, 1, 1). The weights of the
market portfolio are given by
wM =
(m−Ru)C−1
(m−Ru)C−1uT
=
((
2 1 −1)− (0.4 0.4 0.4))
 2 1 −11 4 1
−1 1 4

((
2 1 −1)− (0.4 0.4 0.4))
 2 1 −11 4 1
−1 1 4
11
1

=
(
1.6 0.6 −1.4)
 2 1 −11 4 1
−1 1 4

(
1.6 0.6 −1.4)
 2 1 −11 4 1
−1 1 4
11
1

=
(
5.2 2.6 −6.6)
(
5.2 2.6 −6.6)
11
1
 =
1
1.2
(
5.2 2.6 −6.6)
=
(
13/3 13/6 −11/2) ≈ (4.333 2.167 −5.5) . 10 Marks
(a) (ii) The expected return of the market portfolio which has weights wM =
(4.333, 2.167,−5.5) is given by
µM = wM m
T =
(
4.333 2.167 −5.5)
 21
−1
 = 49/3 ≈ 16.333 .
The variance of the return on the market portfolio is given by
σ2M = wMCw
T
M =
1
20
(
4.333 2.167 −5.5)
15 −5 5−5 7 −3
5 −3 7
4.3332.167
−5.5

=
1
20
(
26.66 10.004 −23.336)
4.3332.167
−5.5
 ≈ 265.5444
20
≈ 13.277 .
7
SOLUTIONS: MAT00015H
So the standard deviation of return on the market portfolio is σM =
√
13.277 ≈
3.6438.
8 Marks
(a) (iii) The expected return for the portfolio V with weightswV = (0.7,−0.1, 0.4)
is given by
µV = wV m
T =
(
0.7 −0.1 0.4)
 21
−1
 = 0.9 .
The Capital Asset Pricing Model states that
µV = R + β(µM −R)
and this gives the beta factor
β =
µV −R
µM −R =
0.9− 0.4
16.333− 0.4 =
0.5
15.933
≈ 0.0314 .
The risk premium is given by µV −R = 0.9− 0.4 = 0.5.
5 Marks
(b) (i) There is no admissible strategy such that the value of the portfolio at time
t = 0 is V (0) = 0 and the future value V (t) satisfies: V (t) ≥ 0 with probability 1
and V (t) > 0 with positive probability for some t > 0.
4 Marks
(b) (ii) We consider the following strategy
(−1, S(0)
A(0)
, 1)
i.e., borrow one risky security and with it to buy S(0)/A(0) risk free securities, and
take one long forward contract (to buy a risky security). Then the wealth at time
t = 0 is
−S(0) + S(0)
A(0)
A(0) = 0
and the wealth at time t = T is
W (T ) = −S(T ) + S(0)
A(0)
A(T ) + (S(T )− F (0, T )) = S(0)
A(0)
A(T )− F (0, T ) > 0
so we have an arbitrage opportunity.
8 Marks Total: 35 Marks
8
SOLUTIONS: MAT00015H
3. (a) A European call option is a contract which gives the holder the right to buy an
asset for a price X fixed in advance (the exercise or strike price) at a specified time
T (called the exercise or expiry time). The payoff for a European call option is
CE(T ) = (S(T )−X)+ ,
where S(T ) is the price of stock at time T and x+ = max{0, x}.
5 Marks
(b) Note that the contract is worth something so CE ≥ 0. Hence we need to show
that
CE ≥ S(0)− X
(1 +R)T
.
Recall from the put-call parity that
CE − PE = S(0)− X
(1 +R)T
,
which implies that
CE ≥ CE − PE = S(0)− X
(1 +R)T
and then (since PE(0) ≥ 0)
CE ≥ max{0, S(0)− X
(1 +R)T
} .
This proves the desired inequality.
10 Marks
(c) Consider the following strategy. At time t = 0, we:
• buy 1 share,
• sell a European call option,
• buy one European put option.
This costs us PE(0)−CE(0)+S(0) so we do this by borrowing PE(0)−CE(0)+
S(0) or equivalently investing CE(0)− S(0)− PE(0). At time T = 1 we have
(CE(0)− S(0)− PE(0))(1 +R) + S(1)
and our contracts.
(1) If S(1) < X we do not exercise the EC but the EP option will be exercised
giving X − S(1) and wealth
(CE(0)− S(0)− PE(0))(1 +R) + S(1) +X − S(1)
= (CE(0)− S(0)− PE(0))(1 +R) +X .
9
SOLUTIONS: MAT00015H
(2) If S(1) > X then the EC is exercised to give−(S(1)−X). The put option will
not be exercised. Again our wealth is
(CE(0)− S(0)− PE(0))(1 +R) + S(1)− (S(1)−X)
= (CE(0)− S(0)− PE(0))(1 +R) +X .
By the assumption we have
CE(0)− PE(0) > S(0)− X
(1 +R)
,
which is equivalent to
(CE(0)− S(0)− PE(0))(1 +R) +X > 0 .
So (1) and (2) above show that this strategy gives an arbitrage opportunity.
20 Marks Total: 35 Marks