21-270
February 18, 2015
INTRO TO MATH FINANCE
Spring 2015
Solutions toTest 1
Name:
Write your answers clearly in the spaces provided. You may use the back of a page
for additional space; please indicate clearly when you do so. Show all work. No credit
wi

21-270
Intro to Math Finance
Spring 2015
Solutions to Assignments 3
2.4 The account balance at t = 1 will be
1, 000(1 + R) + 2, 000(1 + R)1/2 .
This leads to the equation
1, 000(1 + R) + 2, 000(1 + R)1/2 = 3, 128.98.
(1)
If we let x = (1 + R)1/2 then (1)

Introduction to Mathematical Finance
21—270
Solutions to Assignment 7
These exercises are taken from Chapter 4 of the lecture notes. The num-
bering of the exercises is the same as in the lecture notes.
Exercise 1. Let Q = {w1,w2,w3,w4}, and consider the

21-270
INTRO TO MATH FINANCE
Review Problems for Test 3
Spring 2015
In all problems, the present time is t = 0. In all problems except 5, 6, 7, and 11 you
should assume that arbitrage is not possible.
1. Consider a one-period binomial model with u = 1.35,

Chapter 2
Fixed-Income Securities and
Interest Rates
We now begin a systematic study of xed-income securities and interest rates. By a
xed-income security we mean a nancial instrument that promises xed (or denite)
payments at prescribed future dates. In s

Chapter 3
Forward Contracts and Put-Call
Parity
Recall that a forward contract is an agreement between two parties made at some time
concerning the sale of an asset at a future time T , called the delivery time, delivery
date, or maturity. The party taki

21-270
INTRO TO MATH FINANCE
Review Problems for Test 2
Spring 2015
In all problems, the present time is t = 0 and you should assume that arbitrage is
not possible.
1. Consider a coupon bond making 2 payments per year and having maturity 20
years, face va

21-270
Intro to Math Finance
Spring 2015
Solutions to Assignment 6
3.6 Observe that
D(1) =
1
.
1.1
By put-call parity, we have
P0 C0 = D(1)(K F).
It follows that
K=F+
P0 C 0
.
D(1)
Substituting in the numbers, we obtain
K = 296 + (9.11 6.85)(1.1) = 298.48

Chapter 3
Forward Contracts, European and
American Put and Call Options
Recall that a forward contract is an agreement between two parties made at some time
concerning the sale of an asset at a future time T , called the delivery time, delivery
date, or

Chapter 3
Forward Contracts, European and
American Put and Call Options
Recall that a forward contract is an agreement between two parties made at some time
concerning the sale of an asset at a future time T , called the delivery time, delivery
date, or

21-270
Intro to Math Finance
Spring 2015
Solutions to Assignment 9
5.13 Model (i): We seek a risk-neutral measure P with P(1 ) = p1 , P(2 ) = p2 ,
3 ) = p3 . This leads to the system of equations
and P(
251 + 202 + 53 = 15(1.05)
p
p
p
(1)
161 + 132 + 3

21-270
Intro to Math Finance
Spring 2015
Solutions to Assignment 5
3.1 To replicate the long position, we invest
1
B
(1 + R (3)3
B
units of B between t = 0 and t = 3 at the eective rate R (3) = 4% and we
borrow
B
FA
A
(1 + R (3)3
A
units of A between t =

21-270
Intro to Math Finance
Spring 2015
Review for Test 2 Selected Numerical Answers
1. (a) 5, 089.62 = 5, 00040 + 150
(1 40 ), where
1
=
1
1+
rI [2]
2
.
(b) rI [2] < .06
2. .0525
3. 3.65% with 1 point
4. F = 71, 718.87
5. F = 51.32
K
[D(N0 + .5) + D(N0

Chapter 5
Arbitrage-Free Pricing in
One-Period Finite Models
The simplest nancial models involving random evolution of prices are those in which
there are only two trading times and the prices of the basic securities are modelled
as random variables on a

CHAPTER 6
INTRODUCTION TO OPTIMAL INVESTMENT
In this chapter we provide an introduction to the theory of optimal investment
in nite one-period models. We assume throughout that a nite one-period model
with k basic risky assets and interest rate r 0 is giv

CHAPTER 6
INTRODUCTION TO OPTIMAL INVESTMENT
In this chapter we provide an introduction to the theory of optimal investment
in finite one-period models. We assume throughout that a finite one-period model
with k basic risky assets and interest rate r 0 is

21-270
Intro to Math Finance
Spring 2015
Solutions to Assignments 0 (Corrected 1/26)
1. The values of the put and call at maturity are given by
ST 50 if ST 50
CT =
0
if ST < 50
and
PT =
if ST 47.50
0
47.50 ST if ST < 47.50.
(a) If S1 = 53.47, then
CT = 5

21-270
INTRO TO MATH FINANCE
Review Problems for Test 1
Spring 2015
In all problems, the present time is t = 0. In all problems except #6, you should
assume that arbitrage is not possible.
1. Stock of the XYZ company is trading at the initial price S0 . T

21-270
Intro to Math Finance
Spring 2015
Solutions to Assignment 1
1.3 The capital received from the short sale of stock is 400 ($50) = $20, 000. The
number of puts purchased is therefore 20, 000/4 = 5, 000.
(a) If S1 = 25, the value of each put is P1 = 5

21-270
Intro to Math Finance
Spring 2015
Solutions to Assignment 2
1.19 Notice that V1 (1 ) = max cfw_$24, $24 = $24, V1 (2 ) = max cfw_$18, $20 = $24,
and V1 (3 ) = max cfw_$16, $8 = $16.
(1)
(1)
(a) Since V1 (i ) S1 (i ) for i = 1, 2, 3 and V1 (2 ) > S1

21-270
Intro to Math Finance
Spring 2015
Solutions to Assignment 2
1.19 Notice that V1 (1 ) = max cfw_$24, $24 = $24, V1 (2 ) = max cfw_$18, $20 = $24,
and V1 (3 ) = max cfw_$16, $8 = $16.
(1)
(1)
(a) Since V1 (i ) S1 (i ) for i = 1, 2, 3 and V1 (2 ) > S1

21-270
Intro to Math Finance
Spring 2015
Solutions to Unstarred Problems from Assignments 3
2.4 The account balance at t = 1 will be
1, 000(1 + R) + 2, 000(1 + R)1/2 .
This leads to the equation
1, 000(1 + R) + 2, 000(1 + R)1/2 = 3, 128.98.
(1)
If we let

Chapter 4
Finite Probability Spaces
In this chapter we briey discuss some basic concepts from the theory of nite probability. The presentation is self-contained, although it is assumed that readers have
at least some basic idea of what is meant by a state