Portfolio theory (ST407026)
2011-2012

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.

Registration

As from the academic year 2011-2012 the Faculty of Science of the UvA has for students a procedure of registration for courses per semester. Should you haven't registered yet, please do so as soon as possible by filling out the registration form.

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! Oral exams: not in the period 15-25 January 2012. What do you have to know? The theory, i.e. all important definitions and results (lemma's, theorems, etc.). You also have to know three theorems together with their proofs. You select your favorite ones! Criteria to consider: they should be interesting, non-trivial and explainable in a reasonably short time span.

People

Lectures by Peter Spreij and homework grading by Naser Asghari.

Schedule

Fall semester, Thursdays 15.00-18.00, room D1.110 until October 19, room A1.14 from November 2. Location: Science Park 904; see the map of Science Park and the travel directions. The course will start on September 8, 2011. Last class on December 8. Changes in the schedule will appear here.

Programme
(weekly updated, last modified: )

1
Class: Sections 1.1, 1.2 up to Theorem 1.15
Homework: Read in the lecture notes also the parts before Theorem 1.15 that I skipped, read also Definition 1.16 and Proposition 1.17. Make Exercises 1.1, 1.3.
2
Class: Section 1.2 from Theorem 1,18, Section 1.3 and a very brief intro to Sections 2.1, 2.2
Homework: make Exercises 1.2, 1.4, 1.5 and read Section 2.1 (and have regularly a look at the corrections file under "Literature").
3
Class: Most of Sections 2.1, 2.2 (skip Proposition 2.10)
Homework: Exercises 2.1, 2.3, Extra exercise 1
4
Class: Section 3.1 up to Lemma 3.8
Homework: Make Exercise 3.1 and read Appendix A.4
5
Class: Section 3: Theorem 3.9, Section 4.1 and quick account of Section 4.2
Homework: Exercises 3.4 (alternatively you may finish the proof of Theorem 3.9 as a consequence of Additional exercise 2, which you have to make then), 4.1, 4.2
6
Class: Sections 5.1, 5.2
Homework: Exercises 5.2, 5.5
7
Class: Section 6.1
Homework: Exercises 6.2 (The limit in this exercise is of u(x), not of h(x) as I wrote before. I don't exclude that there is no fully satisfactory solution. Partial, incomplete, solutions are just as welcome. You may also make additional assumptions, if you think it is needed.), 6.3
8
Class: Sections 6.2, 7.1
Homework: Exercises 7.1, 7.3, 7.4
9
Class: Sections 7.2, 8.1 up to Proposition 8.7
Homework: Make exercises 7.5, 7.6, 7.7, 7.8
10
Class: Section 8.2 up to Lemma 8.11, remarks on Hahn-Banach
Homework: Read (not necessarily in depth) Section A.2 (the sets in Corollary A.8 are supposed to be disjoint!) and (in the update lecture notes) Lemma B.5. Make Exercises 8.1, 8.3.
11
Class: Appendix B
Homework: no exercises, read the parts of Appendix B that I skipped
12
Class: Sections 8.3, 8.4 (don't pay too much attention to the parts that I skipped)
Homework: Read Section 8.5, make Exercises 8.2, 8.5, 8.8
13
Class: parts of Sections 9.1, 9.2
Homework: Exercises 9.3 9.4, 9.6




Links

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