Portfolio theory 2008-2009

Portfolio theory
2008-2009

Aim

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

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. Typos and other errors in the lecture notes will be corrected throughout the course. They will always communicated to you during the classes. Last corrections made on 30-09-2008.

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!

People

Lectures by Bert van Es and by Peter Spreij.

Schedule

Fall semester, Mondays 13.00-15.45, room JK 3.85, Universiteit van Amsterdam (Valckenierstraat 65, see map). The course will start on September 1. Note that classes start at 13.00 sharp (new UvA rules)! No classes on October 20. Changes in the schedule will also appear here.

Programme (weekly updated!)

1
Class: Section 1.1 up to Example 1.7.
Homework: Exercises 1.1, 1.4.

2
Class: Definition 1.8 - Theorem 1.18
Homework: Exercises 1.3, 1.5

3
Class: Remark 1.19 - Theorem 1.24
Homework: Exercise 1.6

4
Class: Sections 2.1 - 2.2 (up to Proposition 2.10)
Homework: Exercises 2.1, 2.5, 2.6

5
Class: Section 2.2, Proposition 2.11 - Section 3.1, Lemma 3.5 (without the proof)
Homework: 2.2, 2.3, 3.1

6
Class: Remainder of Chapter 3 (except Corollary 3.9, but you can easily understand this by yourself)
Homework: 3.1, 3.2, 3.3

7
Class: Section 4.1
Homework: 4.1, 4.2, 4.6; something to think about: from Proposition 4.6, we get 0 < \lambda^* <1 implies E[Xu'(X)]\leq EX Eu'(X). If you have nothing better to do, you can prove this as an optional exercise.

8
Class: Sections 4.2 and 5.1 up to Remark 5.4.
Homework: 4.3, 5.2

9
Class: Proposition 5.5 - Lemma 6.4
Homework: 5.3, 5.4, 6.1

10
Class: Up to Theorem 6.5 - Proposition 6.7
Homework: 6.2, 6.5, 6.6

11
Class: Sections 7.1, 8.1
Homework: Read definition 6.12 and make Exercises 7.1, 7.2, 7.3 (I don't think that the questions concerning relative entropy causes you difficulties, otherwise you may skip them)

12
Class: Section 8.2 up to Lemma 8.11
Homework: Exercises 8.1, 8.3

13
Class: remainder of Section 8.2 except Proposition 8.13, Section 8.3 up to Theorem 8.20
Homework: Exercises 8.2, 8.5

14
Class: Theorem 8.20 and Section 8.4
Homework: Read Section 8.5 and make Exercises 8.4, 8.8

Links

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

1	Class: Section 1.1 up to Example 1.7. Homework: Exercises 1.1, 1.4.
2	Class: Definition 1.8 - Theorem 1.18 Homework: Exercises 1.3, 1.5
3	Class: Remark 1.19 - Theorem 1.24 Homework: Exercise 1.6
4	Class: Sections 2.1 - 2.2 (up to Proposition 2.10) Homework: Exercises 2.1, 2.5, 2.6
5	Class: Section 2.2, Proposition 2.11 - Section 3.1, Lemma 3.5 (without the proof) Homework: 2.2, 2.3, 3.1
6	Class: Remainder of Chapter 3 (except Corollary 3.9, but you can easily understand this by yourself) Homework: 3.1, 3.2, 3.3
7	Class: Section 4.1 Homework: 4.1, 4.2, 4.6; something to think about: from Proposition 4.6, we get 0 < \lambda^* <1 implies E[Xu'(X)]\leq EX Eu'(X). If you have nothing better to do, you can prove this as an optional exercise.
8	Class: Sections 4.2 and 5.1 up to Remark 5.4. Homework: 4.3, 5.2
9	Class: Proposition 5.5 - Lemma 6.4 Homework: 5.3, 5.4, 6.1
10	Class: Up to Theorem 6.5 - Proposition 6.7 Homework: 6.2, 6.5, 6.6
11	Class: Sections 7.1, 8.1 Homework: Read definition 6.12 and make Exercises 7.1, 7.2, 7.3 (I don't think that the questions concerning relative entropy causes you difficulties, otherwise you may skip them)
12	Class: Section 8.2 up to Lemma 8.11 Homework: Exercises 8.1, 8.3
13	Class: remainder of Section 8.2 except Proposition 8.13, Section 8.3 up to Theorem 8.20 Homework: Exercises 8.2, 8.5
14	Class: Theorem 8.20 and Section 8.4 Homework: Read Section 8.5 and make Exercises 8.4, 8.8

Aim

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