SEMESTER 2 EXAMINATION 2012/2013
MST 30030
Financial Mathematics
Dr. P. Murphy
Dr. H. Render
Time Allowed: 2 hours
Instructions for Candidates
Full marks will be awarded for complete answers to all 8 questions.
Instructions for Invigilators
Non-programmab
SOLUTIONS TO SEMESTER 1 EXAMINATION 2013/14
MST 30030 Financial Mathematics
1. a) What are European and American call options?
A European call option is an agreement between two parties giving the holder
of the option the right to buy a certain asset at a
Financial Mathematics, MST 30030
Lecturer: Masha Vlasenko
Introduction: nancial markets, derivatives and traders
Interest rates
Properties of stock options
Some probability theory
Binomial model for options pricing
The Black-Scholes Model
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Literatur
Lecture 3: Interest Rates
Reading: J.C.Hull, Chapter 4
Example (interest compounded annually)
Suppose we have a principal of 50 Euro and an interest rate of
10% compounded annually. Then after one year the future value is
50 1 +
10
100
= 50 1.1 = 55.
A ge
SOLUTIONS TO SEMESTER 2 EXAMINATION 2012/13
MST 30030 Financial Mathematics
1. Solution: a) A European call option is an agreement between two parties
giving the holder of the option the right to buy a certain asset at a certain time T
to a certain price
Lecture 2. Traders
We will consider the following types of traders depending on the
way they use nancial derivatives:
hedgers to reduce the risk they face from potential future
movements in a market variable
speculators to bet on the future direction of a
Lecture 5: Put - Call Parity
Reading: J.C.Hull, Chapter 9
Reminder: basic assumptions
1. There are no arbitrage opportunities, i.e. no party can get a
riskless prot.
2. Borrowing and lending are possible at the risk-free interest
rate r > 0 which is const
Lecture 4: Properties of stock options
Reading: J.C.Hull, Chapter 9
An European call option is an agreement between two parties
giving the holder the right to buy a certain asset (e.g. one stock
unit) for a price K (strike price) at time T (maturity).
An
Lecture 16: Delta Hedging
We are now going to look at the construction of binomial trees as
a rst technique for pricing options in an approximative way.
These techniques were rst proposed in:
J.C. Cox, S.A. Ross, M. Rubinstein, Option Pricing: A Simplied
Chapter 9: Two-step binomial trees
Example
Suppose we have a 6 month European call option with K = A 21.
C
Suppose S0 = A 20 and in two time steps of 3 months the stock
C
can go up or down by 10% (u = 1.1, d = 0.9). Let r = 0.12.
D
p
24.2
B
p
22
1-p
A
20
SEMESTER 1 EXAMINATION 2013/2014
MST 30030
Financial Mathematics
Dr. P. Murphy
Dr. M. Vlasenko
Time Allowed: 2 hours
Instructions for Candidates
Full marks will be awarded for complete answers to seven out of eight questions.
Instructions for Invigilators
Lectures 6-10: Axioms of Probability Theory
Let X be a set.
We denote by 2X the set of all subsets of X .
Denition
A collection of subsets F 2X is called an algebra if
, X F
A F Ac F
A, B F A B F
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Algebras of sets
Exercises:
1. Check that
AB
c
= Ac