Portfolio theory 2013-2014

Portfolio theory (ST407026)
2013-2014

Aim

To make students familiar with the mathematical foundations of mathematical finance in discrete time and the fundamentals of portfolio selection.

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.

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.

Registration

As from the academic year 2011-2012 the Faculty of Science of the UvA has for students a procedure of registration for courses per semester. Should you haven't registered yet, please do so as soon as possible by filling out the registration form, UvAnetID required. Alternatively, registration via Mastermath should be also possible.

Reimbursement of travel costs

Students who are registered in a master program in Mathematics at one of the Dutch universities can claim their travel expenses, see the rules.

Examination

Take home exercises (you are strongly encouraged to work in pairs) 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: January 20-29, February 8-12, March 3-8.

People

Lectures by Peter Spreij.

Schedule

Fall semester, Thursdays 14:00-16:00 (with the exception of September 26: 15:00-17:00, October 10, 13:00-15:00), room A1.04. Location: Science Park 904; see the map of Science Park and the travel directions. No lecture on October 24, 2013 (autumn break). Classes from November 7 on in G4.15; count on a 5 minutes walk from the main hall (indoor connection on the 1st floor). No lecture on November 21. On November 28 and December 5 (last lecture), 13:00 - 16:00.

Programme
(weekly updated, last modified: )

1
Class: Sections 1.1, 1.2 up to Theorem 1.15
Homework: Read in the lecture notes also the parts before Theorem 1.15 that I skipped, read also Definition 1.16 and Proposition 1.17. Make Exercises 1.1, 1.3.

2
Class: Section 1.2 from Theorem 1.15, Section 1.3 and a very brief intro to Sections 2.1, 2.2
Homework: make Exercises 1.2, 1.4, 1.5 (this exercise is new, be aware of mistakes; should this happen you are free to adjust the questions to get something sensible) and read Section 2.1

3
Class: Most of Sections 2.1, 2.2 (skip Proposition 2.10)
Homework: Exercises 2.1, 2.3

4
Class: Section 3.1 up to Example 3.7, quick mentioning of Theorem 3.9
Homework: Make Exercises 3.1, 3.3 and read Appendix A.4

5
Class: Section 3: Theorem 3.9, Section 4.1 and quick account of Section 4.2
Homework: Exercises 3.4, 4.1, 4.2. Note: I found in the lecture notes many times a funny looking misplaced (iii) instead of (c); please be aware of this; (iii) should only used for the third item in a theorem or so.

6
Class: Sections 5.1, 5.2
Homework: Exercises 5.2, 5.5; please make sure that I have your solutions on Oct 17 by 18:00.

7
Class: Section 6.1
Homework: Exercises 6.4, 6.5, 6.7; optional: Exercise 6.2 (I don't exclude that there is no fully satisfactory solution. Partial, incomplete, solutions are just as welcome. You may also make additional assumptions, if you think it is needed.)

8
Class: Sections 6.2, 7.1
Homework: Make Exercises 7.1, 7.3, 7.4 and read Section 8.1 to make yourself familiar with the terminology.

9
Class: Sections 7.2, 8.1 up to Proposition 8.7 (without proof)
Homework: Make exercises 7.5, 7.6, 7.7, 7.8. The numbering refers to the version of the lecture notes that you probably use. In the newest version these are 7.7, 7.8, 7.9, 7.10.

10
Class: Section 8.2 from Definition 8.8 to Theorem 8.12, (most of) section 8.4
Homework: Read section A.2 (at least take notice of the results) and make Exercises 8.1, 8.2.

11
Class: Sections 8.3 and (the essentials of) 8.5
Homework: Quickly digest Section 8.6 and make Exercises 8.5, 8.8

12
Class: (Parts of) Sections 9.1, 9.2
Homework: Exercises 9.3 9.4, 9.6 (if the totality of the exercises is beyond a reasonable limit, you may drop one exercise, but please inform me in that situation on your sheet/file with solutions)

Links

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

1	Class: Sections 1.1, 1.2 up to Theorem 1.15 Homework: Read in the lecture notes also the parts before Theorem 1.15 that I skipped, read also Definition 1.16 and Proposition 1.17. Make Exercises 1.1, 1.3.
2	Class: Section 1.2 from Theorem 1.15, Section 1.3 and a very brief intro to Sections 2.1, 2.2 Homework: make Exercises 1.2, 1.4, 1.5 (this exercise is new, be aware of mistakes; should this happen you are free to adjust the questions to get something sensible) and read Section 2.1
3	Class: Most of Sections 2.1, 2.2 (skip Proposition 2.10) Homework: Exercises 2.1, 2.3
4	Class: Section 3.1 up to Example 3.7, quick mentioning of Theorem 3.9 Homework: Make Exercises 3.1, 3.3 and read Appendix A.4
5	Class: Section 3: Theorem 3.9, Section 4.1 and quick account of Section 4.2 Homework: Exercises 3.4, 4.1, 4.2. Note: I found in the lecture notes many times a funny looking misplaced (iii) instead of (c); please be aware of this; (iii) should only used for the third item in a theorem or so.
6	Class: Sections 5.1, 5.2 Homework: Exercises 5.2, 5.5; please make sure that I have your solutions on Oct 17 by 18:00.
7	Class: Section 6.1 Homework: Exercises 6.4, 6.5, 6.7; optional: Exercise 6.2 (I don't exclude that there is no fully satisfactory solution. Partial, incomplete, solutions are just as welcome. You may also make additional assumptions, if you think it is needed.)
8	Class: Sections 6.2, 7.1 Homework: Make Exercises 7.1, 7.3, 7.4 and read Section 8.1 to make yourself familiar with the terminology.
9	Class: Sections 7.2, 8.1 up to Proposition 8.7 (without proof) Homework: Make exercises 7.5, 7.6, 7.7, 7.8. The numbering refers to the version of the lecture notes that you probably use. In the newest version these are 7.7, 7.8, 7.9, 7.10.
10	Class: Section 8.2 from Definition 8.8 to Theorem 8.12, (most of) section 8.4 Homework: Read section A.2 (at least take notice of the results) and make Exercises 8.1, 8.2.
11	Class: Sections 8.3 and (the essentials of) 8.5 Homework: Quickly digest Section 8.6 and make Exercises 8.5, 8.8
12	Class: (Parts of) Sections 9.1, 9.2 Homework: Exercises 9.3 9.4, 9.6 (if the totality of the exercises is beyond a reasonable limit, you may drop one exercise, but please inform me in that situation on your sheet/file with solutions)

Aim

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