MATH508 Exam solutions.
2011
1. (a) Write down the general principle used for valuing forwards.
Using this, calculate the 3-month future on Oil given the following information:
(i) Spot Oil is trading at 90USD per barrel.
(ii) USD interest rates are at 12
MATH508 Exam solutions.
2010
1. (a) Write down the general principle used for valuing forwards. Using
this, calculate the one year future on Morgan Stanley (MS) shares given
that:
(i) MS is currently trading at 30 USD per share
(ii) USD interest rates are
M3508 Exercises 6
November 2013
1. Conditional expectation
Consider the following ltration (Pt )T=1 , where r = 0:
t
1
2
3
4
S (0, ) S (1, )
100
120
100
120
100
90
100
90
S (2, )
150
110
110
70
(i) Calculate the conditional probabilities for each edge of
Math 2101
Homework 6
Due: November 21, 2012
1. Let C be the circle |z i| = 2 traversed anticlockwise. Compute
C
cos z
dz,
z (z 2 + 8)
C
cosh z
dz,
z4
C
(z 2
1
dz.
+ 4)2
Do not use the residue theorem.
(a) Using partial fractions we easily see that
1
z
1
=
Math 2101
Homework 6
Due: November 21, 2012
1. Let C be the circle |z i| = 2 traversed anticlockwise. Compute
C
cos z
dz,
z (z 2 + 8)
C
cosh z
dz,
z4
C
(z 2
1
dz.
+ 4)2
Do not use the residue theorem.
2. By evaluating the integral
1
2i
C
1
dz
(z a)(z a1 )
Math 2101
Homework 7
Due: November 28, 2012
1. Let f be entire.
(a) Show that, if ef is bounded, then f is constant.
Since ef is entire, as composition of ew and f , and bounded, by assumption, we can
apply Liouvilles theorem and deduce that ef is constan
Math 2101
Homework 7
Due: November 28, 2012
1. Let f be entire.
(a) Show that, if ef is bounded, then f is constant.
(b) Show that, if
(f ) is bounded above, then f is constant.
2. Let a, b R. Let f be entire and satisfy
f (z + a) = f (z ),
f (z + ib) = f
Math 2101
Homework 5
Due: November 14, 2012
1. In the following you are not allowed to use the residue theorem or Cauchys integral
formula, as they have not been discussed yet.
(a) Compute the integral
x dz
|z |= r
for the positive sense of the circle, in
Math 2101
Homework 5
Due: November 14, 2012
1. In the following you are not allowed to use the residue theorem or Cauchys integral
formula, as they have not been discussed yet.
(a) Compute the integral
x dz
|z |= r
for the positive sense of the circle, in
Math 2101
Homework 9
NOT DUE
While this assignment will not be marked, please study Part A for the nal exam in
May.
PART A
1. (a) How many roots of the equation z 4 6z + 3 = 0 have their modulus between 1
and 2?
Let f (z ) = z 4 and g (z ) = 6z + 3. On th
Math 2101
Homework 2
Due: October 17, 2012
1. Verify the Cauchy-Riemann equations for f (z ) = z 3 and f (z ) = ex (cos y + i sin y ).
For the function f (z ) = z 3 we have u = x3 3xy 2 and v = 3x2 y y 3 and
u
v
= 3x2 3y 2 =
,
x
y
u
v
= 6xy = .
y
x
For th
Math 2101
Homework 1: Solutions
Due: October 10, 2012
The homework consists mostly of a selection of problems from the suggested books.
1. (a) Find the value of (1 + i)n + (1 i)n for every n N.
We will use the polar form of 1 + i = 2(cos(/4) + i sin(/4) a
Math 2101
Homework 1
Due: October 10, 2012
Problems marked with will be marked. The students are advised to work over ALL
problems in the assignments, so as to keep pace with the material.
1. (a) Find the value of (1 + i)n + (1 i)n for every n N.
(b) Show
Math 2101
Homework 4
Due: October 31, 2012
1. (a) Solve the equation ez = 1 + 3 i, i.e. nd all complex numbers z satisfying it.
We write 1 + 3i in polar form. Since |1 + i 3| = 2 and the argument can be taken
to be /3 = arctan 3, we have 1 + i 3 = 2ei/3 .
Math 2101
Homework 4
Due: October 31, 2012
1. (a) Solve the equation ez = 1 + 3 i, i.e. nd all complex numbers z satisfying it.
(b) Show that ez = ez .
(c) Find the image of the semi-innite strip x 0 and 0 y under the transformation w = ez . Exhibit corre
Math 2101
Homework 8
Due: December 5, 2012
1. Explain why the following integrals have the given value using the residue theorem.
Complete explanations are required.
/2
(a)
0
1
dt =
, a > 0, (b)
a + cos2 t
2 a2 + a
2
cos2 (3t)
3
(c)
dt =
.
5 4 cos(2t)
8
Math 2101
Homework 8
Due: December 5, 2012
1. Explain why the following integrals have the given value using the residue theorem.
Complete explanations are required.
/2
(a)
0
1
,
dt =
2t
a + cos
2 a2 + a
2
(c)
0
a > 0,
(b)
0
log x
dx = 0,
x2 + 1
2
cos (
Math 2101
Homework 9
NOT DUE
While this assignment will not be marked, please study the Part A for the nal exam
in May.
PART A
1. (a) How many roots of the equation z 4 6z + 3 = 0 have their modulus between 1
and 2?
(b) Find the number of the roots of the
ECON 2601: Profit Maximisation
Suppose that a firm operating in a perfectly competitive industry in the long run
1
1
has the production function f ( x1 , x 2 ) = x1 4 x 2 4 and faces prices p , w1 , and w2
for the output and the two input factors, respect
ECONOMICS 2601: Demo 3
Suppose that a consumer has utility function u(f, s) = ln f + ln s, faces
prices pf = 1 and ps = 1, and has exogenous income m = 200. We
would like to calculate compensating and equivalent variations along
with the change in consum
ECON2601: Demo 1
1. Suppose that your utility function is u(t, Q) = t + 10 Q, where t is the
number of hours spent online and Q is expenditure on all other goods.
Assume that a two-part tari exists for broadband such that there is a
xed charge of f = 19 a
M3508 Exercises 4
October 2013
1. In a one-period model, the share price starts at S and in one months
time is either SU or S/U where U > 1. Assuming rates are zero,
show that the risk-neutral probability p of the upmove is given by
p = 1/(U + 1). Hence,
M3508 Exercises 3
October 2013
1. An economist writes a 1-period expectation model for valuing options.
The model assumes that the stock starts at S and moves to 2S or 1 S
2
in 1 years time with equal probability. Assume rates are zero.
(a) Using this exp
M3508 Exercises 2
October 2013
1. On January 1st 2013 Morgan Stanley shares were trading at 20USD per
share. The shares pay a quarterly dividend of 0.40c per share if the
shareholder has held the share for at least 10 days before the dividend
payment. Cal
STAT2001/STAT3101:
Probability and Inference
Lecturer: Yvo Pokern,
Room 144 Dept. of Statistical Science
Lectures:
Christopher Ingold XLG1 on Thursdays 1pm-2pm
Anatomy G29 J. Z. Young Lecture Theatre on Fridays
2pm-4pm
Ofce Hour: Fridays 4pm-5pm
Email:
YP: STAT2001/3101, 2013 2014
1
STAT2001: Probability and Inference (0.5 unit)
STAT3101: Probability and Statistics II (0.5 unit)
Lecturer: Yvo Pokern
Aims of course
To continue the study of probability and statistics beyond the basic concepts introduced
i
STAT2001/STAT3101:
Probability and Inference
Lecturer: Yvo Pokern,
Room 144 Dept. of Statistical Science
Lectures:
Christopher Ingold XLG1 on Thursdays 1pm-2pm
Anatomy G29 J. Z. Young Lecture Theatre on Fridays
2pm-4pm
Ofce Hour: Fridays 4pm-5pm
Email: