Portfolio theory (UvA, ST407026)
2020-2021

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. In particular, students will able to prove a number of well selected theorems and demonstrate in assignments that they master the theory.

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 know results about dynamic arbitrage theory and completeness in multi-period (discrete time) models.
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. Deadlines for homework: solutions have to be handed in within one week! And you are allowed (supposed) to work in pairs.

Oral exams: before the start of the Spring semester. 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.
Register yourself on the time slot reservation page (link available on Canvas) for an appointment. The selected dates are December 23, 24, 29 and 30 and January 20 and 21. If you have a strong preference for an alternative date, let me know. Note: I may have to propose to you (by mail) an alternative time slot, but no big differences are expected.

People

Lectures by Peter Spreij.
Assistance by Peter Braunsteins.

General schedule

Fall semester. During the first half, meetings on Thursdays, 13:00-15:00. Possible changes will be announced here. The first lecture will be online, for the other ones recordings will be made available well ahead of the officially scheduled hours. The scheduled hours will be used for Q+A sessions.

Updated Programme
(regularly updated, )

1
Class (online, recordings on Canvas (Media gallery) as Pft2020-1a.mp4 etc.): Sections 1.1, 1.2 up to Proposition 1.14.
Homework: Make Exercises 1.1 (this is a lengthy exercise, involving some nasty computations; I don't mind if you are sketchy and only indicate (rather precisely) what has to be done), 1.3, 1.8 and A.1 (appendix).
2
Class (prerecorded lectures, available on Sunday 6 September 2020 on Canvas; Q+A on Thursday 10 September 2020, 13:00): 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.
Homework: Make Exercises 1.2, 1.5, 1.11, 2.1.
3
Class (prerecorded lectures, available on Sunday 13 September 2020 on Canvas; Q+A on Thursday 17 September 2020, 13:00): Most of Sections 2.1, 2.2 (you can read Proposition 2.10 yourself, optional), and an introduction to Section 3.1.
Homework: Make Exercises 2.2, 2.3, 2.6.
4
Class: Section 3.1 continued.
Homework: Make Exercises 3.1, 3.4, 3.6.
5
Class: Sections 4.1 and Section 4.2.
Homework: Make Exercises 4.1, 4.2, 4.5 (assume that the utility function $u$ (and then also $\alpha$) is defined on the whole of $\mathbb{R}$) and 4.7.
6
Class: Sections 5.1, 5.2, and 6.1 until Lemma 6.4.
Homework: Make Exercises 5.2, 5.4, 5.5, 6.1.
7
Class: Section 6.1 from Theorem 6.5, Section 6.2.
Homework: Make Exercises 6.3(b,c), 6.5 and 6.7.
8
Class: Sections 7.1, 7.2 up to Lemma 7.11.
Homework: Make Exercises 7.1, 7.3, 7.4.
9
Class: Section 7.2 from Theorem 7.12, 8.1 and 8.2 up to Theorem 8.12 (the remainder of that section will be skipped). You may have a look at the MTP lecture notes for some elementary properties of quantile functions (around Theorem 3.10), used in lecture 9 (and also in lecture 8).
Homework: Make Exercises 7.8, 7.10, 8.3.
10
Class: Sections 8.3 and 8.4 up to Proposition 8.27.
Homework: Make Exercises 8.11, 8.12, 8.14.
11
Class: Sections 8.4, 8.5 and very quick mentioning only of Section 8.6.
Homework: Read the results of Section 8.6 (and check for yourself also that the CRR model satisfies the assertion of Theorem 8.32 about atoms) and make Exercises 8.8, 8.13 (indicate only where and how you have to make changes in the proof of Proposition 8.21), 8.15.
12
Class: Sections 9.1, 9.2 (only mentioning of the existence of Appendix A.5).
Homework: Make Exercises 9.2, 9.4 and 9.5. Have a look at Appendix A.5.
13
Class: Most of Section 9.3.
Homework: Make Exercises 9.7, 9.8.




Links

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