Exam 3

QUANTITATIVE METHODS FOR FINANCE

Mock Exam 3 (Academic Year 2013-14)

[5 exercises; 31 points available; 90 minutes available]

1 Consider a stock that pays out the dividend 2 (X + 3)2 dt every �second� (with dX =

�k (X �m) dt+X�dz).

[8 points] Use the educated guess qX2+ uX + v (q, u, and v are constants to be determined) to

work out the equilibrium stock price S (X).

[7 points] Your initial capital is H = 2000 Euro. You retrieve an additional sum of 500 Euro by

selling short the stock and invest 2500 Euro in the riskless asset. Work out the total gain dH on your

portfolio.

2 [4 points] Consider a constrained log-utility investor whose problem is

maxw

Ehlog� fW � i

sub w � 250% , fW = 100 ( (1 + r) + w (er � r) ) ;

where

r = 0% ; er =

8><>:g > �3% with probability 1

2

�3% with probability 12

.

If the constrained optimal portfolio is w� = 2:5 and the shadow price is l� = 0, the up-state return on

the risky asset is:

a) g = 385;

b) g = 585;

c) g = 685;

d) g = 885.

Alessandro Sbuelz - SBFA, Catholic University of Milan 1

3 [4 points] A �rm produces two outputs x and y (they can be sold at the �xed prices 30

and 40, respectively). An embargo is imposed on the �rm�s average production:

1

2x+

1

2y � 30 .

Given that the production costs are C (x; y) = 12x2 + 1

2y2 � 1

3xy + 70, the constrained maximum

pro�t is:

a) 1048: 75;

b) 1248: 75;

c) 1848: 75;

d) 1448: 75.

4 [4 points] Consider the following one-period arbitrage-free market with a zero riskfree

rate (r = 0):

M =

26664�1:0 �3: 2 �4: 81 3 5

1 5 7

1 2 3

37775 .

The no-arbitrage price of a European call option written on the risky security 2 (the strike price is 3)

is:

a) 2: 70;

b) 1: 25;

c) 1:80;

d) 0:80.

5 [4 points] Consider the following one-period market with a zero riskfree rate (r = 0):

M =

26664�1:0 �1:45 �1:101 1 1

1 1 0

1 2 2

37775 .

The �nal proceed prevailing in the state !2 of the strategy #� that costs �10 cents (V#� (0) = � 10

100)

and pays o¤ nothing in the states !1 and !3 (V#� (1) (!1) = V#� (1) (!3) = 0) is:

a) V#� (1) (!2) = �27.

b) V#� (1) (!2) =27;

c) V#� (1) (!2) = �47;

d) V#� (1) (!2) =37.


SOLUTIONS

1 The equilibrium-valuation problem is

1

dtEt [dS] + 2X2 + 12X + 18 = Sr + SXX��

with1

dtEt [dS] = SX (�k (X �m)) +

1

2SXXX

2�2 .

The educated guess

S (X) = qX2 + uX + v

has derivatives

SX = 2qX + u and SXX = 2q

and must meet the equilibrium equation

(2qX + u) (�k (X �m)) + 122qX2�2 + 2X2 + 12X + 18�

�qX2 + uX + v

�r � (2qX + u)X�� = 0

m

X2��2qk + q�2 + 2� qr � 2q��

�| {z }= 0

+X(2qkm+ 12� uk � ur � u��)| {z }= 0

+ (ukm+ 18� vr)| {z }= 0

= 0 .

Hence

q =2

2k + r � �2 + 2�� ,

u =4km

(2k + r � �2 + 2��) (k + r + ��) +12

k + r + ��,

v =km

r

�4km

(2k + r � �2 + 2��) (k + r + ��) +12

k + r + ��

�+18

r.


The total gain on your portfolio is

dH = � 500S

�dS + 2 (X + 3)2 dt

�+ 2500rdt

= � 500 dS + 2 (X + 3)2 dt

S

!+ 2500rdt

= � 500��

r +SXX

S��

�dt +

SXX

S�dz

�+ 2500rdt

=

�Hr � 500

SXX

S��

�dt � 500

SXX

S�dz .

SXX

S(elasticity) =

2qX2 + uX

S

=

�4X2

2k+r��2+2�� +4kmX

(2k+r��2+2��)(k+r+��) +12X

k+r+��

�S

=S + 2X2

2k+r��2+2�� kmr

�4km

(2k+r��2+2��)(k+r+��) +12

k+r+��

�� 18

r

S

=S + qX2 � v

S.


SOLUTIONS

2 The correct answer is a).

The investor�s expected utility is

Ehlog� fW � i

= 0:5 ln (100 + w (100g � 0)) + 0:5 ln (100� 3w)

and the Lagrangian function is

L (w; l) = Ehlog� fW � i

� l ( w � 2:5 ) .

The Kuhn-Tucker First Order Conditions are:8>>>>>>>><>>>>>>>>:

L w = 0

l � 0

L l � 0

l � L l = 0 .

If l = 0 (we assume a painless constraint), the F.O.C.s become

Lw (w; l)jl=0 =1

2

100g

(100 + 100gw)+1

2

�3(100� 3w)

= � 12

�100g + 6gw + 3(100� 3w) (1 + gw)

= 0 ,

Ll (w; l)jl=0 = � (w � 2:5) � 0 ,

so that

Lw (w; l)j l=0; w=2:5 = 0 () � 100g + 15g + 3 = 0 = 0 () g =3

85= 3: 529 411 76% .


The graphical analysis (not required) follows, with8><>:100 + w

�30085� 0�> 0

100 + w (�3� 0) > 0() w 2

��853;100

3

�.

30 25 20 15 10 5 5 10 15 20 25 30 35

1

1

2

3

4

5

allocation w

expected utility


SOLUTIONS

3 The correct answer is d).

The problem is

maxx;yP (x; y) sub

1

2x+

1

2y � 30

with

P (x; y) = 30x+ 40y ��1

2x2 +

1

2y2 � 1

3xy + 70

�:

The First Order Conditions for constrained optimality will be su¢ cient because the constraint

function is linear (the feasible set�(x; y) 2 R2 : 1

2x+ 1

2y � 30

is convex) and the pro�t function P (x; y)

is strictly concave:

H =

264 Pxx Pxy

Pyx Pyy

375 =264 �1

13

13

�1

375 with Pxx = �1 < 0 and det (H) =8

9> 0 :

Given the Lagrangian function

L (x; y; l) = P (x; y)� l�1

2x+

1

2y � 30

�,

the Kuhn-Tucker First Order Conditions are:8>>>>>>>>>>><>>>>>>>>>>>:

Lx = 0

Ly = 0

l � 0Ll � 0

l � Ll = 0

,

8>>>>>>>>>>><>>>>>>>>>>>:

13y � x� 1

2l + 30 = 0

13x� 1

2l � y + 40 = 0

l � 030� 1

2y � 1

2x � 0

l�30� 1

2y � 1

2x�= 0 .

For l = 0 (we assume a painless constraint), we have:8><>:13y � x+ 30 = 0

13x� y + 40 = 0

,

8><>:x = 195

4= 48: 75

y = 2254= 56: 25 .


The unconstrained maximum-pro�t point is such that P�1954; 2254

�= 7145

4= 1786: 25. It turns out to be

unfeasible as the constraint is violated:

1

248: 75 +

1

256: 25 = 52: 5 � 30 :

For l > 0 (we assume a painful constraint), we have:8>>>>>><>>>>>>:

13y � x� 1

2l + 30 = 0

13x� 1

2l � y + 40 = 0

30� 12y � 1

2x = 0 (the constraint is binding)

,

8>>>>>><>>>>>>:

x = 1054= 26: 25

y = 1354= 33: 75

l = 30 .

The constrained maximum pro�t is

P

�105

4;135

4

�=

5795

4= 1448: 75 .


SOLUTIONS

4 The correct answer is c).

By the First Fundamental Theorem of Asset Pricing, any arbitrage opportunity is ruled out if the

market M supports a risk-neutral probability measure Q (recall that the riskfree rate is r = 0):264 1:0

3: 2

4: 8

375 =1

1 + 0

264 1 + 0 3 5

1 + 0 5 7

1 + 0 2 3

375T 264 Q (!1)Q (!2)

Q (!3)

375 .

Since

det

0B@264 1 3 5

1 5 7

1 2 3

3751CA = � 2 ,

the unique measure Q is:

264 Q (!1)Q (!2)

Q (!3)

375 =

0BB@264 1 3 5

1 5 7

1 2 3

375T1CCA�10B@(1 + 0)

264 1:0

3: 2

4: 8

3751CA =

264 0:30:30:4

375

with

0BB@264 1 3 5

1 5 7

1 2 3

375T1CCA�1

=

0B@264 1 1 1

3 5 2

5 7 3

3751CA�1

=1

�2

264 1 1 �44 �2 �2�3 1 2

375| {z }matrix of cofactors

T

.


The payo¤ to be priced is

eX (1) = max� eS2 (1)� 3 ; 0 �

m

264 X (1) (!1)X (1) (!2)

X (1) (!3)

375 =

264 max ( 5� 3 ; 0 )max ( 7� 3 ; 0 )max ( 3� 3 ; 0 )

375 =

264 240

375 .

Its no-arbitrage price is

X (0) =1

1 + 0

264 240

375T 264 0:30:3

0:4

375 = 1: 8 .

An alternative would be the calculation of the initial cost of the unique replicating strategy #X :

264 #X0

#X1#X2

375 =

264 1 3 5

1 5 7

1 2 3

375�1 264 24

0

375 =1

�2

264 1 4 �31 �2 1

�4 �2 2

375| {z }matrix of cofactors

T 264 240

375 =

264 �301

375

and

V#X (0) =

264 �301

375T 264 1:0

3: 2

4: 8

375 = 1: 8 .


SOLUTIONS

5 The correct answer is a).

8><>:#�0 +

2920#�1 +

1110#�2 = �0:10

#�0 + #�1 + #

�2 = 0

#�0 + 2#�1 + 2#

�2 = 0

()

264 1145100

1110

1 1 1

1 2 2

375264 #

�0

#�1#�2

375 =264 �

10100

0

0

375 .

Since

det

0B@264 1

145100

1110

1 1 1

1 2 2

3751CA = � 7

20,

we have

264 #�0

#�1#�2

375 =

264 1145100

1110

1 1 1

1 2 2

375�1 264 �

10100

0

0

375 =

264 0

�2727

375 ,

where

264 1145100

1110

1 1 1

1 2 2

375�1

=1

� 720

264 0 � 710

720

�1 910

110

1 �1120

� 920

375 =

264 0 2 �1207

�187�27

�207

117

97

375 .

Hence,

V#� (1) (!2) =h1 1 0

i264 #�0

#�1#�2

375 = � 27.


Exam 3

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