Portfolio theory (UvA, ST407026)
2022-2023
(MINOR) CHANGES EXPECTED !!!

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). Portfolio theory is therefore typically a course in the second year of the master programmes Mathematics and Stochastics and financial mathematics.

Literature

Examination

Take home exercises (compulsory!) and oral exam. Deadlines for homework: solutions have to be handed in within one week! And you are supposed to work in pairs; this is mandatory(!), unless impossible. Hand in your solutions on the Canvas page of the course.

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. Making reservations for time slots for the exams will be possible by early December 2022.

The homework assignments count for 40% for the final grade, the result for the oral exam contributes the remaining 60%.

People

Lectures by Peter Spreij.
Assistance by Rens Kamphuis.

General schedule

Fall semester, Thursday afternoons. See datanose for details. On Canvas are some recordings (guides through the lecture notes) from 2020. In early November lectures on location will be replaced by either lectures online or by recorded lectures. More information will follow.

Programme of 2022-2023
(regularly updated, )

1
Class: Sections 1.1, 1.2 up to Proposition 1.14.
Homework: Make Exercises 1.1 (this is a lengthy exercise, involving some nasty computations; it is sufficient to be sketchy and only indicate (but rather precisely) what has to be done), 1.3, 1.8 and A.1 (appendix).
2
Class: Section 1.2 from Theorem 1.15, Section 1.3 (due to time constraints some parts only indicated), 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: 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, 2.9. [But warn me if it appears too much!]
4
Class: Section 3.1 continued.
Homework: Make Exercises 3.1, 3.4, 3.5 (optional), 3.6.
5
Class: Sections 4.1 and Section 4.2.
Homework: 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
Class: Sections 5.1, 5.2; the planned Section 6.1 up to Lemma 6.4 has not been treated yet.
Homework: Make Exercises 5.2, 5.4, 5.5; as a preparation for next week, read the definition of an u.s.c. function just above Lemma 6.3 and make Exercise 6.1.
7
Class: Sections 6.1, 6.2.
Homework: Make Exercises 6.3(b,c), 6.5, 6.7, and 6.9 (optional exercise).
8
Class (online only, see Canvas for recordings 8a, 8b, 8c; Q&A during scheduled hours): Sections 7.1, 7.2 up to Lemma 7.11.
Homework: Make Exercises 7.1, 7.3, 7.4.
9
Class (online only, see Canvas for recordings 9a--9d; Q&A during scheduled hours): Sections 8.1 and 8.2 up to Theorem 8.12 (the remainder of that section will be skipped). You may still have a look at the MTP lecture notes for some elementary properties of quantile functions, needed for Section 7.2.
Homework: Make Exercises 7.7, 7.10 (you also have to read Section 7.2, Theorem 7.12 which was not treated in class this year), 8.1, 8.3 (the last two are very easy).
10
Class: Section 8.3 (almost everything).
Homework: Make Exercises 8.11, 8.12, 8.14 (You need two sigma-algebras here to talk about arbitrage, $\mathcal{F}_1=\mathcal{F}$ the Borel sets, and $\mathcal{F}_0=\{\emptyset,\Omega\}$ the trivial sigma-algebra.).
11
Class: Sections 8.4, 8.5 (at least, most of it) and very quick mentioning only of Section 8.6.
Homework: Read the results of Section 8.6 (if you like, you can check for yourself also that the CRR model satisfies the assertion of Theorem 8.32 about atoms) and make Exercises 8.5, 8.13 (indicate only where and how you have to make changes in the proof of Proposition 8.21), 8.15. Exercise 8.10 is an optional one for the diehards. This exercise is rather new, but likely to be correct as well. Beware of potential problems! See also the added remark on Exercise 8.14 of Week 11.
12
Class: Sections 9.1, 9.2 (here and there very briefly treated).
Homework: Make Exercises 9.2, 9.4 and 9.5. Have a look at Appendix A.5 for some skipped details.
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