1
Supplement to July 6 Lecture
In lecture on July 4 we showed that
Ep [Sn |Hn ] > (1 + r)Sn ,
Eq [Sn |Hn ] = (1 + r)Sn ,
where Sn denotes the period-n stock price in the multiperiod binomial model and Hn =
(S0 , S1 , . . . , Sn ). In lecture on July 6 we

MA370 - Summer 2013
Assignment #1 Solutions
1. Consider the markets defined by the payoff matrices
"
#
100 100 100
100 100 100
D1 =
, D2 = 70 110 150 .
70 110 150
10 50 90
(a) Explain why the first market is incomplete. Are there any redundant assets in t

Winter Term, 2014
Name:
Student Number:
WILFRID LAURIER UNIVERSITY
Waterloo, Ontario
MA 270 Financial Mathematics I
Midterm February 25, 2014
Time Allowed: 80 minutes
Total Value: 60 marks
Number of Pages: 6 question pages, 2 rough work pages, plus cover

1
Supplement to March 1 Lecture
In lecture we stated, but did not prove, certain properties of the function 2 (w). The purpose
of this document is to develop a detailed understanding of this function, i.e. of the relationship
between portfolio allocation

MA270 - Winter 2015 - Answers to Selected Problems on Bonds
1. Graph needs to (i) be decreasing and convex, (ii) have the horizontal axis as a horizontal asymptote, (iii) have a vertical intercept at
(0, 113.56) and (iv) pass through the point (.015, 100)

MA370 - Financial Mathematics II - Winter 2016
Assignment # 3
Due Tuesday, March 17, at the beginning of lecture.
Can be handed in any time prior to the due date by putting it under my office door (BA
546).
Please work in pairs.
1. Let fT = f (ST ) den

MA370 - Financial Mathematics II - Winter 2016
Assignment # 1
Due Thursday, January 28, at the beginning of lecture.
Can be handed in any time prior to the due date by putting it under my office door (BA
546).
Please work in pairs.
1. Consider the payo

1
Supplement to July 13 Lecture
In lecture we defined the exercize and continuation values of an American option, and argued
that the value of the option is the maximum of exercize and continuation value. In other words
if E denotes the exercize value of

MA370 - Summer 2013
Assignment #3 Solutions
1. Let = cfw_1, 2, 3, 4. Which (if any) of the following collections are
fields/algebras?
(a) F1 = cfw_, cfw_1, 2, cfw_3, 4.
Not a field since
/ F1 .
(b) F2 = cfw_, , cfw_1, cfw_2, 3, 4, cfw_1, 2, cfw_3, 4.
Not

MA370 - Financial Mathematics II - Winter 2016
Assignment # 2
Due Friday, February 5, by 2:00pm.
Can be handed in any time prior to the due date by putting it under my office door (BA
546).
Please work in pairs.
1. Suppose that portfolios 1 and 2 have

A ,4, Em; .-,Exm . Ale;
4,0,- l L Er @kag @Lwlg; VF ;A when m EMTQ 5:;
32G) Pa-M . gonad) Pa I. fsowc?)mkmhsogcgpmwi ,
143:4? Bn=Pua1e a; q ch wT-m Page wave F mgm
Haw , ,
(3n , E. .
. (J'prhn 4
'Note -00:\ Rplxtl'HiBn.

MA 370 Problem Set 1a
Let
Ax = b.
be a system of equations, where A R
M N
Question 1: Let
,xR
N
(1)
M
and b R .
2 3
1
5 6 , b = 2 , A1 = [A b].
3
8 9
1
A= 4
7
Then rank(A) = rank(A1) = 2 < 3 = N = M , so (1) is consistent and it has infinitely many soln

MA 370 Quiz [,1 (Wed, Feb. 5, 2014)
Question 1(10 marks): Consider a single period model with 2 assets and 3 states:
12
3
0
1 0 1/3
0 1 4/3
Dllll
S(0)=( 1 2 );S(1,Q)=(
If we let A = (8(1,9), b = 5(0)' and Z = [A b]. Then rref(;() = ( , 13/? > -
Determin

MA370 - Summer 2013
Assignment #1
Work in groups of two.
Due Friday, May 31 by 4:00 PM. You may hand in your assignment
at any time prior to the due date. My office is BA-546 - slip your
assignment under the door if I am not there.
You are strongly enc

MA370 - Financial Mathematics II - Winter 2016
Assignment # 2 Solutions
1. To say they have the same payoffs in all states of nature is to say that 1 D = 2 D.
Now if they had different costs then 1 S0 6= 2 S0 , in particular either 1 S0 > 2 S0 or
1 S0 < 2

MA370 - Financial Mathematics II - Winter 2016
Assignment # 4 Solutions
1. Stock tree is
93.04
75.64
61.50
50.00
65.81
53.51
43.50
46.55
37.84
32.93
Risk-neutral probability is q = 0.5. Tree for (a) is, subject to rounding error,
0.00
0.00
0.78
2.96
0.00

MA 370 - Summer 2013
Assignment #4 (Optional) Solutions
1. In this problem we investigate continuously compounded stock returns
in the binomial model. To this end consider the N -period binomial
model with parameters u, d and p.
(a) Determine the mean and

MA370 - Financial Mathematics II - Winter 2016
Assignment # 4 (Fixed)
Due by 4:00pm on Monday, April 4.
Can be handed in any time prior to the due date by putting it under my office door (BA
546).
Please work in pairs.
1. Consider a three-period binomi

MA 370 First Midterm Test
[10 marks]
Page 1 of 2
1. Consider the payoff matrix
100 100 100
D=
.
80 110 140
It can be shown (you do not need to show this) that a basis for the set of attainable payoffs
in this market is
cfw_ 1 0 1 , 0 1 2 .
(a) If a por

MA370 - Financial Mathematics II - Winter 2016
Assignment # 1 Solutions
1. Determinant of D1 is 4, 000, therefore D1 is invertible, therefore the first market is complete. For the second market we have two assets but three states of nature, so the market

1
Duration and Sensitivity to Changes in Interest Rates
In lecture we showed that
P V 0 (r) = P V (r) D(r) ,
where P V (r) and D(r) represent the present value and duration, respectively, of a stream of
cash flows. The formula assume that all cash flows a

MA 270 Midterm Test Winter 2014
[10 marks]
Page 1 of 8
1. Bell Canada has an outstanding bond that matures in exactly 4.5 years and pays a semi-annual
coupon of 3% on a face value of $100.
(a) Sketch the pricing function P (j) for this bond. Be sure to la

MA 370 Second Midterm Test
[10 marks]
Page 1 of 2
1. Consider an Arrow-Debreu market that is free from arbitrage, in which a risk-free asset is
available.
P
1
(a) Briefly explain (intuitively or otherwise) why M
j=1 j = 1+r for any state price vector,
whe

MA370 - Summer 2013
Assignment #2 Solutions
1. In lecture on June 5 we considered an instance of the binomial model
where the expected return from buying a put option was negative. In
this problem we prove that this will be the case for any derivative tha

1
Supplement to May 25 Lecture
In lecture we introduced the notion of state prices and Arrow-Debreu securities. We briefly
discussed state prices in complete markets without redundant assets, but not incomplete markets.
1.1
State Prices and Arrow-Debreu S

The St. Petersburg Paradox
Adam Metzler
Mathematics Department
Wilfrid Laurier University
MA270 Lecture
February 2, 2016
Motivating Question
I
I
Suppose we play the following game.
I
Two fair dice are rolled.
I
If the sum of the two faces is 7, I pay you

Winter Term, 2015
Name:
Student Number:
WILFRID LAURIER UNIVERSITY
Waterloo, Ontario
MA 270 Financial Mathematics I
Midterm February 25, 2015
Time Allowed: 80 minutes
Total Value: 60 marks
Number of Pages: 6 question pages, 2 rough work pages, plus cover

1
March 24 Lecture Notes
This document summarizes the lecture notes that I would have used on March 24 (the day the
university was shut down because of freezing rain).
2
Value of a Call Option at Maturity
In lecture on March 22 we were considering a Europ