ASSIGNMENT 1
MATH3075 Mathematical Finance (Mainstream)
Due by 4 pm on Monday, 5 September 2016
1. [10 marks] Elementary market model. Consider the elementary two-state
market model M = (B, S) with parameters S0 > 0 and 0 < d < 1 + r < u.
b = (P(
b 1 ), P
MATH3075/3975
Financial Mathematics
School of Mathematics and Statistics
University of Sydney
Semester 2, 2016
Tutorial sheet: Week 11
Background: Chapter 5 The Black-Scholes Model.
Exercise 1 Consider the Black-Scholes model M = (B, S) with the initial
s
EXAMINATION GUIDE
MATH3075 Financial Mathematics (Normal)
You should know the basic results and computational methods of Financial Mathematics. You should be ready to answer questions regarding general singleperiod market models, the CRR model and the Bl
ASSIGNMENT 2: SOLUTIONS
MATH3075 Mathematical Finance (Mainstream)
1. [10 marks] The CRR model: European claim.
Consider the CRR model of stock price S with T periods and parameters d <
1 + r < u, where r is the one-period interest rate.
(a) Consider the
MATH3075/3975
Financial Mathematics
School of Mathematics and Statistics
University of Sydney
Semester 2, 2016
Tutorial sheet: Week 6
Background: Section 2.2 Single-Period Market Models.
Exercise 1 Consider a single-period three-state market model M = (S,
MATH3075/3975
Financial Mathematics
School of Mathematics and Statistics
University of Sydney
Semester 2, 2016
Tutorial sheet: Week 8
Background: Chapter 3 Multi-Period Market Models.
Exercise 1 We consider the two-period market model M = (B, S) with the
3: ELEMENTARY MARKET MODEL
Marek Rutkowski
School of Mathematics and Statistics
University of Sydney
Semester 2, 2016
M. Rutkowski (USydney)
Slides 3: Elementary Market Model
1 / 36
Single Period Market Model
Only one period is considered.
The beginning o
MATH3075/3975
Financial Mathematics
School of Mathematics and Statistics
University of Sydney
Semester 2, 2016
Tutorial sheet: Week 4
Background: Section 2.2 Single-Period Market Models.
Exercise 1 Verify the equality (see Section 2.2)
Vt
Vbt :=
=
Bt
(
)
MATH3075/3975
Financial Mathematics
School of Mathematics and Statistics
University of Sydney
Semester 2, 2016
Tutorial sheet: Week 9
Background: Section 4.2 European Options in the CRR Model.
Exercise 1 Consider the CRR model M = (B, S) with the horizon
MATH3075/3975
Financial Mathematics
School of Mathematical Sciences
University of Sydney
Semester 2, 2016
Tutorial sheet: Week 2
Background: Chapter 6 Probability Review.
Exercise 1 Assume that joint probability distribution of the two-dimensional
random
MATH3075/3975
FINANCIAL MATHEMATICS
Valuation and Hedging of Financial Derivatives
This course is an introduction to the mathematical theory of modern nance with the
special emphasis on stochastic modelling of asset prices and the arbitrage valuation of
h
MATH3075/3975
Financial Mathematics
Week 10: Solutions
Exercise 1 Assume the CRR model M = (B, S) with T = 3, the stock
price S0 = 100, S1u = 120, S1d = 90, and the risk-free interest rate r = 0.1.
We consider the American put option on the stock S with t
7: The CRR Market Model
Ben Goldys and Marek Rutkowski
School of Mathematics and Statistics
University of Sydney
MATH3075/3975 Financial Mathematics
Semester 2, 2015
7: The CRR Market Model
Outline
We will examine the following issues:
1
The Cox-Ross-Rubi
8: The Black-Scholes Model
Ben Goldys and Marek Rutkowski
School of Mathematics and Statistics
University of Sydney
MATH3075/3975 Financial Mathematics
Semester 2, 2015
8: The Black-Scholes Model
Outline
We will examine the following issues:
1
The Wiener
6: MULTI-PERIOD MARKET MODELS
Ben Goldys and Marek Rutkowski
School of Mathematics and Statistics
University of Sydney
Semester 2, 2015
B. Goldys and M. Rutkowski (USydney)
6: Multi-Period Market Models
1 / 55
Outline
We will examine the following issues:
5: FILTRATIONS AND CONDITIONING
Ben Goldys and Marek Rutkowski
School of Mathematics and Statistics
University of Sydney
Semester 2, 2015
B. Goldys and M. Rutkowski (USydney)
Slides 5: Filtrations and Conditioning
1 / 37
New Features
In the case of multi-
ASSIGNMENT 1: SOLUTIONS
MATH3075 Mathematical Finance (Normal)
1. [10 marks] Elementary market model.
Consider a single-period two-state market model M = (B, S) with the two dates:
0 and 1. Assume that the stock price S0 at time 0 is equal to $27 per shar
4: SINGLE-PERIOD MARKET MODELS
Marek Rutkowski
School of Mathematics and Statistics
University of Sydney
Semester 2, 2016
M. Rutkowski (USydney)
Slides 4: Single-Period Market Models
1 / 87
General Single-Period Market Models
The main differences between
MATH3075/3975
Financial Mathematics
Week 11: Solutions
Exercise 1 We consider the Black-Scholes model M = (B, S) with the
initial stock price S0 = 9, the continuously compounded interest rate r = 0.01
per annum and the stock price volatility = 0.1 per ann
MATH3075/3975 FINANCIAL MATHEMATICS
Valuation and Hedging of Financial Derivatives
Marek Rutkowski
School of Mathematics and Statistics
University of Sydney
Semester 2, 2016
M. Rutkowski (USydney)
MATH3075/3975 Financial Mathematics
1/4
Information
Lectur
ASSIGNMENT 1: SOLUTIONS
MATH3075 Mathematical Finance (Mainstream)
1. [10 marks] Elementary market model. Consider the elementary two-state
market model M = (B, S) with parameters S0 > 0 and 0 < d < 1 + r < u.
b = (P(
b 1 ), P(
b 2 ) = (b
(a) Find the uni
MATH3075/3975
Financial Mathematics
School of Mathematics and Statistics
University of Sydney
Semester 2, 2016
Tutorial sheet: Week 10
Background: Section 4.4 American Options in the CRR Model.
Exercise 1 Assume the CRR model M = (B, S) with T = 3, the st
MATH3075/3975
Financial Mathematics
Week 6: Solutions
Exercise 1 (a) To show that the model is arbitrage-free, we need to show
that there is no strategy (x, ) satisfying conditions of Denition 2.2.3 of an
arbitrage opportunity:
(i) x = V0 (x, ) = 0, (ii)
MATH3075/3975
Financial Mathematics
School of Mathematics and Statistics
University of Sydney
Semester 2, 2015
Tutorial sheet: Week 4
Background: Section 2.2 Single-Period Market Models.
Exercise 1 Verify the equality (see Section 2.2)
Vt
Vt :=
=
Bt
(
)
MATH3075/3975
Financial Mathematics
Week 3: Solutions
Exercise 1 Let (x, ) be the replicating strategy for the contingent claim
X = h(S1 ) = S1 , that is,
V1 (x, ) := (x S0 )(1 + r) + S1 = S1 .
More explicitly, for every i = cfw_1 , 2 ,
V1 (x, )(i ) = (x
MATH3075/3975
Financial Mathematics
School of Mathematics and Statistics
University of Sydney
Semester 2, 2015
Tutorial sheet: Week 3
Background: Section 2.1 Elementary Market Model.
Exercise 1 What is the price at time 0 of a contingent claim represented
MATH3075/3975
Financial Mathematics
Week 2: Solutions
Exercise 1 (a) The conditional distributions of X given Y = j are:
for j = 1 : (1/5, 3/5, 1/5),
for j = 2 : (2/3, 0, 1/3),
for j = 3 : (0, 3/5, 2/5),
and thus the conditional expectations EP (X| Y = j)