Portfolio theory
2016-2017
NWI-WB090, NWI-WM990B

Contents

In this course we treat the foundations of mathematical finance, its fundamental concepts like arbitrage, equivalent martingale measures, and fundamental economic notions as preference relations and utility functions and show how these are applied in portfolio optimization. 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.

Aim and objectives

The general aim is to make students familiar with the mathematical foundations of mathematical finance in discrete time and the fundamentals of portfolio selection. Specific objectives to be met at the end of the course:
1. Students are familiar with fundamental concepts of financial mathematics and know how to apply them.
2. Students know the theory behind portfolio optimization and know how to solve such optimisation problems.
3. Students are able to optimize under order restrictions.
4. Students are able to prove a number of well selected theorems.
5. Students know how to apply dynamic programming to investment-consumption problems.

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, update version of 7 June 2017, now also containing the former additional exercises.

Examination

Take home exercises and oral exam, unless the group of students is too big. This will be decided at the beginning of the course. Deadlines for homework: solutions have to be handed in within one week! Oral exams: before the start of the spring term. 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 favourite ones! Criteria to consider: they should be interesting, non-trivial and explainable in a reasonably short time span. To facilitate your planning, here are some unavailable dates: June 16 - 23 and July 11 - August 13.

People

Lectures by Peter Spreij.
Tutorials by Norbert Mikolajewski.

Schedule

Spring semester. Lectures on Wednesdays, 13:45 - 15:30, first lecture on February 1, 2017. See here for the various lecture rooms. No lectures on March 15, 29, April 5, 26. Last lecture no later than May 31. Tutorials on Tuesdays, 10:45 - 13:00, in HG02.032. First tutorial on February 7, 2017.

Programme
(regularly updated, )

1
Class: Sections 1.1, 1.2 up to Theorem 1.15.
Tutorial: Exercise 1.3 and Additional Exercise 1.
Homework: Read in the lecture notes also the parts before Theorem 1.15 that I skipped. Make Exercise 1.1 and Additional Exercise 2.
2
Class: Section 1.2 from Theorem 1.15, Section 1.3 and a very brief intro to Sections 2.1, 2.2.
Tutorial: Make Exercises 1.2 and 1.5.
Homework: Make Exercise 1.4 and Additional Exercise 5.
3
Class: Most of Sections 2.1, 2.2 (you can read Proposition 2.10 yourself, optional).
Tutorial: Make Exercises 2.2, 2.5, 2.7.
Homework: Make Exercises 2.1 and Additional Exercise 8. If you like you can consider Additional Exercise 9.
4
Class: Section 3.1 up to Example 3.7, quick mentioning of Theorem 3.9.
Tutorial: Make Exercises 2.3, 2.7 and 3.5.
Homework: Make Exercises 3.1, 3.3 and 2.6.
5
Class: Section 3: Theorem 3.9, Section 4.1 and very brief account of Section 4.2.
Tutorial: Make Exercises 3.4, 4.1 and 4.6.
Homework: Make Exercises 3.2, 4.2 and 4.3; also have a look at Section 4.2 (easy reading).
6
Class: Sections 5.1, 5.2
Tutorial: Make Exercise 5.2 and Additional exercise 12 (which replaces 5.5).
Homework: Make Exercises 5.3 and 5.4.
7
Class: Section 6.1
Tutorial: Make Exercises 6.3 (triviality), 6.6 and 6.7.
Homework: Make Exercises 6.1 and 6.4.
8
Class: Sections 6.2, 7.1
Tutorial: Make Exercises 7.1, 7.3 (and 7.4 if there is enough time).
Homework: Make Exercises 6.5 (assume that all $H(\mathbb{Q}|\mathbb{P})$ are finite, otherwise the result is not true), 7.5 and additional exercise 13.
9
Class: Sections 7.2, 8.1. You may have a look at the MTP lecture notes for some elementary properties of quantile functions (around Theorem~3.10), used in today's lecture.
Tutorial: Make Exercises 7.7, 7.9, 7.10.
Homework: Make Exercises 7.6, 7.8.
10
Class: Section 8.2 up to Lemma 8.11 (but Theorem 8.9 next week), (most of) section 8.3.
Tutorial: Make additional exercises 16, 17, 18 (just a quick look) and Exercise 8.11.
Homework: Make Exercises 8.1, 8.3 (it is sufficient to give a relation between $\mu$ and $\sigma^2$), 8.9.
11
Class: Remainder of Section 8.
Tutorial: Make Exercises 8.2, 8.4 and 8.5.
Homework: Read Section 8.6 and make Exercise 8.8.
12
Class: Most of Section 9.1 (and Appendix) and a small bit of Section 9.2.
Tutorial: Make Exercises 9.1, 9.2 and 9.5 (for which you need to read the first one and a half page of Section 9.2).
Homework: Make Exercises 9.4 and Additional exercise 19.
13
Class: Section 9.3.
Tutorial: Make Exercises 9.6, 9.7.
Homework: nothing.




Links

Institute for Mathematics, Astrophysics and Particle Physics
Korteweg-de Vries Institute for Mathematics
Master Stochastics and Financial Mathematics