1: PROBABILITY REVIEW
Ben Goldys and Marek Rutkowski
School of Mathematics and Statistics
University of Sydney
Semester 2, 2015
B. Goldys and M. Rutkowski (USydney)
Slides 1: Probability Review
1 / 56
Outline
We will review the following notions:
1
Discre
MATH3075/3975
Financial Mathematics
Week 5: Solutions
Exercise 1 For a trading strategy (x, ) we have
V1 (x, ) = (x S0 )(1 + r) + S1 .
Hence the class of all attainable contingent claims is the two-dimensional
subspace of R3 spanned by the vectors (1, 1,
1: PROBABILITY REVIEW
Marek Rutkowski
School of Mathematics and Statistics
University of Sydney
Semester 2, 2016
M. Rutkowski (USydney)
Slides 1: Probability Review
1 / 56
Outline
We will review the following notions:
1
Discrete Random Variables
2
Expecta
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 9: Solutions
Exercise 1 Consider the CRR model M = (B, S) with the horizon date
T = 2, the risk-free rate r = 0.1, and the following values of the stock price
S at times t = 0 and t = 1:
S0 = 10,
S1u = 13.2,
S1d =
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, 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
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
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 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
(
)
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 8
Background: Chapter 3 Multi-Period Market Models.
Exercise 1 We consider the two-period market model M = (B, S) with 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,
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-
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:
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
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
MATH3075/3975
Financial Mathematics
School of Mathematics and Statistics
University of Sydney
Semester 2, 2016
Tutorial sheet: Week 7
Background: Chapter 3 Multi-Period Market Models.
Exercise 1 We consider the conditional expectation EP (X | G) where G i
2: SECURITIES MARKETS
Marek Rutkowski
School of Mathematics and Statistics
University of Sydney
Semester 2, 2016
M. Rutkowski (USydney)
Slides 2: Securities Markets
1 / 19
Books
Recommended readings:
Hull, J. C. (2002). Options, Futures, and Other Derivat
2: SECURITIES MARKETS
Ben Goldys and Marek Rutkowski
School of Mathematics and Statistics
University of Sydney
Semester 2, 2015
B. Goldys and M. Rutkowski (USydney)
Slides 2: Securities Markets
1 / 19
Books
Recommended readings:
Hull, J. C. (2002). Option
3: ELEMENTARY MARKET MODEL
Ben Goldys and Marek Rutkowski
School of Mathematics and Statistics
University of Sydney
Semester 2, 2015
B. Goldys and M. Rutkowski (USydney)
Slides 3: Elementary Market Model
1 / 36
Single Period Market Model
Only one period i
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 4: Single-Period Market Models
1 / 37
New Features
In the case of multi-p
4: SINGLE-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)
Slides 4: Single-Period Market Models
1 / 87
General Single-Period Market Models
MATH3075/3975
Financial Mathematics
School of Mathematical Sciences
University of Sydney
Semester 2, 2015
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
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)
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 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 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 7: Solutions
Exercise 1 (a) The cumulative distribution function of X reads
0,
x < 1,
0.1, 1 x < 2,
0.2, 2 x < 3,
FX (x) =
0.5, 3 x < 4,
0.7, 4 x < 5,
1,
x 5.
Equivalently, we may represent the probability distribu
MATH3075/3975
Financial Mathematics
School of Mathematics and Statistics
University of Sydney
Semester 2, 2016
Tutorial sheet: Week 5
Background: Section 2.2 Single-Period Market Models.
Exercise 1 Consider the market model M = (B, S) introduced in Exerci
MATH3075/3975
Financial Mathematics
School of Mathematics and Statistics
University of Sydney
Semester 2, 2016
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 8: Solutions
Exercise 1 (a) We rst focus on the conditional risk-neutral probabilities.
To compute them, we consider the three embedded single-period two-state
models. In each of these models, we may use the single
6: MULTI-PERIOD MARKET MODELS
Marek Rutkowski
School of Mathematics and Statistics
University of Sydney
Semester 2, 2016
M. Rutkowski (USydney)
6: Multi-Period Market Models
1 / 55
Outline
We will examine the following issues:
1
Trading Strategies and Arb
5: FILTRATIONS AND CONDITIONING
Marek Rutkowski
School of Mathematics and Statistics
University of Sydney
Semester 2, 2016
M. Rutkowski (USydney)
Slides 5: Filtrations and Conditioning
1 / 37
New Features
In the case of multi-period market models, we need