Portfolio theory (ST407026)
2009-2010

Aim

To make students familiar with the mathematical fundamentals of portfolio selection.

Contents

In this course we treat fundamental economic notions as preference relations and utility functions and show how these are applied in portfolio optimization and measure of risk. This will be done first for static markets and extended later on to a dynamic setting, where time is discrete. Finally we will show how stochastic control theory and dynamic programming can be applied to problems of portfolio optimization.

Prerequisites

Basic concepts of probability and measure theory, for instance at the level of Measure theoretic probability.

Literature

The course is mainly based on H. Föllmer and A. Schied, Stochastic Finance, An Introduction in Discrete Time and on S.R. Pliska, Introduction to Mathematical Finance: Discrete Time Models. A set of lecture notes partly based on these sources contains the contents of the course.

Related course

Students are advised to also take the course Financial Stochastics on derivative pricing in continuous time.

Examination

Take home exercises (you are strongly encouraged to work in pairs) and oral exam. Deadlines for homework: solutions have to be handed in within one week!

Planning for the oral exam: You are welcome any time, taking the exceptions below into account. You can make an appointment for a date that suits you and on which you think that you will be well prepared. Made appointments can always be shifted to a later date in the future, should you wish so (it doesn't make sense to have the exam, while you think that you need more time for preparation). I will not be available in the periods 22 November - 2 December and 13 - 20 December. Appointments between Christmas and New Year are in principle not excluded.

People

Lectures by Bert van Es and by Peter Spreij.

Schedule

Fall semester, Mondays 13.00-15.45, room P.015B, Universiteit van Amsterdam (Plantage Muidergracht 24, see map). The course will start on August 31. No classes on October 19, November 23 and 30; on December 7 class from 13:00 to 16:00.

Programme
(last modified: )

1
Class: Sections 1.1, 1.2 up to Theorem 1.15
Homework: Exercises 1.1, 1.3.
2
Class: Section 1.2 from Definition 1.16, Section 1.3
Homework: Exercises 1.2, 1.4, 1.5.
3
Class: Most of Sections 2.1, 2.2 (skip Proposition 2.10)
Homework: Exercises 2.1, 2.3, 2.5.
4
Class: Section 3.1 up to Lemma 3.7
Homework: Exercise 3.1
5
Class: Section 3: Theorem 3.9, Section 4.1
Homework: Exercises 3.3, 4.1, 4.2
6
Class: Sections 4.2, 5.1
Homework: Exercises 5.2, 5.5
7
Class: Section 6.1
Homework: Exercises 6.2, 6.3
8
Class: Section 7.1
Homework: Exercises 7.1, 7.3 (look at Definition 6.12 of the relative entropy), 7.4
9
Class: Sections 8.1, 8.2
Homework: Read the end of Section 8.2 and make exercises 8.1, 8.2, 8.3
10
Class: Section 8.3 and part of Theorem 8.22
Homework: Exercises 8.5, 8.6
11
Class: Sections 8.4, parts of Section 9.1
Homework: Read Section 8.5, make Exercises 8.8, 9.2
12
Class: Remainder of Section 9.1, parts of Sections 9.2, 9.3
Homework: Exercises 9.4, 9.7
Remark: The problem that I encountered in the text on the bottom of page 80 can indeed easily be solved. The condition $E^\star C_1+C_0=w$ implies that $C_1$ has a unique price (under any risk neutral measure). By Proposition 1.21 this implies that $C_1$ is attainable, from which the assertion should follow. Something similar applies to Proposition 9.16, where you could use Theorem 8.20 to conclude that the $\gamma_t$ are attainable.




Links

Korteweg-de Vries Institute for Mathematics
Master Stochastics and Financial Mathematics