Portfolio theory
2022-2023
NWI-WB090, NWI-WM990B

THERE WILL BE REGULAR UPDATES !!!

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 (used for the Mastermath course with the same name).

Literature

Examination

Take home exercises (compulsory!) and oral exam. The take home assignments count for 25%. Deadlines for homework: solutions have to be handed in within one week! Details concerning homework assignment will be communicated by Rein van Alebeek.

Oral exams: preferably in the first three weeks of June. What do you have to know? The theory, i.e. all important definitions and results (lemma's, theorems, etc.). You also have to know four 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.

People

Lectures by Peter Spreij.
Tutorials by Rein van Alebeek.

General schedule

Spring semester. Lectures on Wednesdays, 10:30 - 12:15 in HG01.029, first lecture on 1 February 2023. Exercise classes (tutorials) on Fridays, 13:30 - 15:15 in Transitorium 00.008, first exercise class on 3 February 2023. See also the official schedule page.

Updated Programme
(regularly updated, )

1
Class: Sections 1.1, 1.2 up to Proposition 1.14.
Tutorial: Make Exercises 1.3, 1.5, 1.6 and A.1 (for this you have to read section A.1).
Homework: Read in the lecture notes also the parts before Theorem 1.12 that I skipped. Make Exercises 1.1 (this is a lengthy exercise, involving some nasty computations; I don't mind if you are sketchy and only indicate, but rather precisely, what has to be done) and 1.8.
2
Class: Section 1.2 from Theorem 1.15, Section 1.3, Section 2.1, Section 2.2 up to the first part of the proof of Theorem 2.6.
Tutorial: Make Exercises 1.2, 1.10, 1.13, 1.14.
Homework: Make Exercises 1.11, 1.12 (perhaps 1.15) and 2.1.
3
Class: Most of Sections 2.1, 2.2 (you can read Proposition 2.10 yourself, optional), and an introduction to Section 3.1.
From this week on, exercises for the tutorial session and homework will be merged and will simply be called `Exercises'. Note that there may still be weekly updates of the schedule.
Make Exercises: 2.2, 2.3, 2.6, 2.9, 2.11.
4
Class: Section 3.1 continued.
Make Exercises 3.1, 3.4, 3.5, 3.6.
5
Class: Sections 4.1 and Section 4.2.
Make Exercises 4.1, 4.2, 4.5 and 4.7. [If you have nothing better to do, you may also try 4.10.]
6
No classroom lecture this week, instead you may want to view the three videos Pft2020-6(a,b,c) on Brightspace, or simply read in the lecture notes Sections 5.1, 5.2, and 6.1 until Lemma 6.4.
Make Exercises 5.2, 5.4, 5.5, 6.1.
7
Class: Section 6.1 from Theorem 6.5, Section 6.2.
Make Exercises 6.3(b,c), 6.5, 6.7, 6.9 and 6.4 optional.
8
Class (ambitious plan): Sections 7.1, 7.2 (with superficial treatment of Theorem 7.12). You may have a look at the MTP lecture notes for some elementary properties of quantile functions (around Theorem 3.10).
Make Exercises 7.1, 7.2, 7.3, 7.5, 7.7, 7.8 (or a selection of them if this turns out to be too demanding).
9
Class: Sections 8.1 and 8.2 up to the proof of Theorem 8.12 (the remainder of that section will be skipped).
Make the remains of Week 8, and Exercises 7.9 (only if there is enough time), 8.1, 8.3 (forget about the gamma distribution). Have a look at Section B.4 if you are not familiar with conditional expectation and martingales.
10
Class: Sections 8.3 and 8.4 up to Proposition 8.27.
Make Exercises 8.1 (if not made in Week 9), and the more demanding 8.11, 8.12, 8.14.
11
Class: Sections 8.4, 8.5 and very quick mentioning only of Section 8.6.
Read the parts of Section 8.6 that have been skipped during claaa. Make Exercises 8.5, 8.8, 8.13, 8.15. Exercise 8.10 (optional) is for the diehards.
12
Class: Sections 9.1, 9.2 (up to the first financial application, expected utility from terminal wealth).
Make Exercises 9.1, 9.2 (do these quickly), and 9.4. These will be merged with the exercises of Week 13.
13
Class: Section 9.2 from Definition 9.11 and most of Section 9.3.
Make Exercises 9.6, 9.7, and 9.8 (bonus exercise). These will be merged with the exercises of Week 12.




Links

Institute for Mathematics, Astrophysics and Particle Physics (Radboud University)
Korteweg-de Vries Institute for Mathematics (University of Amsterdam)
Master Stochastics and Financial Mathematics (University of Amsterdam)