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Portfolio theory 2020-2021 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 objectivesThe 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:
PrerequisitesBasic 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
ExaminationTake 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! Oral exams: before the start of the summer holidays. 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. Send a mail with a request for an appointment on a day and time that suits you well, preferably not much later than the end of June.PeopleLectures by Peter Spreij.Tutorials by Rein van Alebeek. General scheduleSpring semester. Initial, "zero" lecture on Monday 25 January 2021 at 10:30. Next lectures will be available as prerecorded sessions on Zoom. The contents of these lectures can be discussed during the Q+A sessions on Thursdays, starting on 28 January 2021, 10:30. Be sure you have then seen the first recordings, available on Brightspace, and/or that you have studied the corresponding pages in the lecture notes. There will be exercise classes (tutorials) on Mondays, starting on Monday 1 February 2021, 10:30. See Brightspace for the Zoom links.
Updated Programme |
| 0 |
Online class: Background in probability and measure theory, Appendix B in the lecture notes.
Homework: none |
| 1 |
Class (recordings on Brightspace 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, but rather precisely, what has to be done), 1.3, 1.8 and A.1 (appendix). |
| 2 |
Class (recordings available on Brightspace from 28/01/2021 on as Pft2020-2a.mp4 etc.): 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 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.
Homework: Make Exercises 2.2, 2.3, 2.6, 2.9. |
| 4 |
Class: Section 3.1 continued.
Homework: Make Exercises 3.1, 3.4, 3.5, 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 the Arrow-Pratt $\alpha$ 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, 6.7, 6.9 and 6.4 optional. |
| 8 |
Class: Sections 7.1, 7.2 up to Lemma 7.11. You may have a look at the MTP lecture notes for some elementary properties of quantile functions (around Theorem 3.10).
Homework: Make Exercises 7.1, 7.2, 7.3, 7.4 (and 7.5 as optional exercise if there is enough time). |
| 8bis |
In the Q&A session of Week 8 there will be a quick treatment of conditional expectation and martingales, based on Section B.4 of the Appendix. |
| 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).
Homework: Make Exercises 7.7, 7.8, 7.9 (optional exercise), 7.10, 8.3. |
| 10 |
Class: Sections 8.3 and 8.4 up to Proposition 8.27.
Homework: Make Exercises 8.1 (should be simple), and the more demanding 8.11, 8.12; if you want you also make 8.3 (it is sufficient to give a relation between $\mu$ and $\sigma^2$). |
| 11 |
Class: Sections 8.4, 8.5 and very quick mentioning only of Section 8.6.
Homework: Read Section 8.6 if you don't follow the presentation in Q&A 11bis. Make Exercises 8.5, 8.8, 8.13, 8.15. Exercise 8.10 (optional) is for the diehards. [This exercise has never been tested before, beware of problems!] |
| 11bis |
In the Q&A session of Week 11 there will be a quick treatment of Section 8.6. |
| 12 |
Class: Sections 9.1, 9.2 (only mentioning of the existence of Appendix A.5).
Homework: Have a look at Appendix A.5. Make Exercises 9.1, 9.2, 9.4 and 9.5. |
| 12bis |
Extra session with Rein (on 17 May 2021), a potpourri of exercises across the various chapters of the lecture notes: 2.11, 3.3, 4.10(a,b and certainty equivalents), 5.6, 6.6, 8.10. |
| 13 |
Class: Most of Section 9.3.
Homework: Make Exercises 9.6, 9.7, 9.8 (use the new version, not the one in the lecture notes of January 25, 2021, or earlier), and 9.10 when there is time left. |