Financiële Wiskunde 2021-2022
NWI-WM990B (???), NWI-WB085

Contents

We treat a number of basic but fundamental issues in financial mathematics, in particular pricing of financial derivatives and hedging by self-financing portfolios. The course starts with financial models in discrete time, treats convergence of these models to models in continuous time. Equally important are some relevant mathematical concepts and techniques that are central in the field: Brownian motion, heat equation and related PDEs, stochastic calculus and measure transformation.

Aims and objectives

The general aim is to make students familiar with the fundamental mathematical techniques and topics, and the diversity of them, that play a role in mathematical finance, as well as specific topics in the field. Specific objectives to be met at the end of the course:
  • Students are familiar with fundamental concepts like no arbitrage, self financing portfolios, financial derivatives, hedging.
  • Students are familiar with core concepts from the diverse necessary techniques, like the role of the heat equation and related PDEs, Brownian motion and other martingales, stochastic integration and the Ito rule, and now how to apply these.
  • Students know limits of discrete time models result in continuous time models.
  • Students know the theory behind hedging portfolios and how to derive these.
  • Students are able to prove a number of well selected theorems.

Literature

  • A set of lecture notes (version July 15, 2020) will be used. They also contain the exercises. A revised and extended version has become available after the course.
    We will sometimes use a bit of measure theory, partly covered in the lecture notes. For basic knowledge (definitions and theorems) of sigma-algebras, random variables and integrals (expectation), you may look at the MTP lecture notes, sections 1.1, 1.2, 1.5, 3.1, 3.2, 4.1, 4.2, 4.4. Other concepts will be treated during the course.
  • Further reading: Tomas Björk, Arbitrage Theory in Continuous Time; Steven E. Shreve, Stochastic Calculus for Finance II.
  • Completely different topics in Mathematical Finance are treated in the video lectures by Lech Grzelak. Recommended!

People

Lectures: Peter Spreij
Exercise classes: Rein van Alebeek

Schedule

Lectures: Usually on Wednesdays, officially 15:30 - 17:15. First lecture (2 February 2022) in Transitorium 00.010. From 22 Feb 2022 up to 16 Mar 2022 in HG03.082; from 13 Apr 2022 up to 1 Jun 2022 in HG00.086. We will try to start at 15:20 from 22 Feb 2022 on. Plan is to have the last lecture on 1 June 2022.
Exercise classes: Thursdays, 13:30 - 15:15 uur in E 3.20 (first weeks, later in HG01.029), first class on Thursday 3 February 2022.
CHANGES in the schedule will appear here. Scheduling details of course also on the official website.

Examination

Oral examination. Homework assignments (compulsory, every week) count for 25%. What you have to know at the exam: The theory, i.e. all important definitions and results (lemma's, theorems, etc.), but proofs will usually not be asked. Optional: you may prepare four theorems or so 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 (at most 15 minutes). If you exercise the option, you will be asked to present one or two of them.
Available remaining dates: Fri 17 June (afternoon), Mon 20 June (afternoon), Fri 24 June (afternoon).


Programme

(UPDATED SCHEDULE, )

1 Lecture: Sections 1.1 (partly, until page 3 only), most of (appendix) A.1 and A.6 (at high speed, no details).
Exercise class: Make Exercises 1.1, 1.3, A.1, A.22.
Homework: read what has been treated during class and make Exercises A.13, A.14, A.23 [Hint: take $G=\{\hat{X}> \hat{X}'\}$ and use the definitions of $\hat{X}$ and $\hat{X}'$].
2 Lecture: (most of) Section 1.1.
Exercise class: Make Exercises 1.2, 1.7, A.16, A.17 and prove Proposition A.11(iii).
Homework: Make Exercises 1.4, 1.5, 1.6, A.15.
Note: There may be a swap between the exercises for class and for homework, also in upcoming weeks.
3 Lecture: Section 1.2 (but not all details of the later results).
Exercise class: Make Exercises 1.11, 1.12, 1.13(a,b).
Homework: Read the remaining parts of Section 1.2; make Exercise 1.8, 1.9, 1.14 and read the main results of Section A.4 (unless you already know this).
4 Lecture: Sections 2.1, 2.2 up to Theorem 2.5 (with complete proof).
Exercise class: Make Exercises 2.3, 2.5, 2.6.
Homework: Make Exercises 2.1, 2.2, 2.4, 2.7 and show that Equation (2.7) holds true. Make yourself familiar with the results of Section A.2 (and ask questions next time if necessary).
5 Lecture: Remainder of Section 2.2, starting on page 20, Section 2.3 (proof of Proposition 2.10 skipped). Most of Section 3.1, only first half of the proof of Proposition 3.2, Proposition 3.5; Theorem 3.3 perhaps next week.
Exercise class: Make Exercises 2.8, 2.10, 2.14, 3.2 (look at the proof of Proposition 3.2).
Homework: Make Exercises A.3, 2.9, 2.11, 3.4.
6 Lecture: Remaining parts of Section 3.1 (i.e. Theorem 3.3), Section 4.1.
Exercise class: Make Exercises 3.5, 4.1, 4.5, 4.12.
Homework: Make Exercises 3.6, 4.3, 4.6, 4.9.
7 Lecture: Sections 5.1-5.3.
Exercise class: Make Exercises 4.8, 5.2, 5.3, 5.4 (c,e).
Homework: Make Exercises 5.1, 5.4 (a,b,f).
8 Lecture: Section 6.1 up to Example 6.6.
Exercise class: Make Exercises 6.1, 6.4, 6.5(a,b).
Homework: Make Exercises 6.2, 6.3, A.27.
9 Lecture (plan): Section 6.1 from Proposition 6.7, Section 6.2 up to Example 6.17.
Exercise class: Make Exercises 6.5(c), 6.6, 6.8, 6.11.
Homework: Make Exercises 6.7, 6.9, 6.19.
10 Lecture: Section 6.2 from page 53 and Section 6.3; perhaps introduction to Section 7.1.
Exercise class: Make Exercises 6.13, 6.14, 6.20; have a quick look at Exercise 6.12.
Homework: Make Exercises 6.15, 6.16 and 6.18.
11 Lecture: Sections 7.1, 7.2, 7.3.
Exercise class: Make Exercises 7.1, 7.5, 7.7 and 7.15.
Homework: Make Exercises 7.2, 7.4 and 7.8.
12 Lecture: Sections 7.4, 7.5, 8.1, and a part of Section 8.2 (up to Equation (8.7)). Due to Radboud Rocks there are only very few students expected at the regular lecture hour. Therefore there will be no lecture on campus. Instead you are invited to look at the video recordings from 2020 on Brightspace: fw2020Week12aSection7374: from 27:50 (circa 13 minutes), fw2020Week12bSection7581: all (circa 30 minutes), fw2020Week13aSection82: up to 16:13 (i.e. up to Equation (8.7)). See also the announcement by Rein.
Exercise class: Make Exercises 7.9, 7.11, 7.14; if there is enough time: 7.16(b,d).
Homework: Make Exercises 7.10, 7.12, 7.13.
13 Lecture: Quick review of Section 8.1 and of the first part of Section 8.2; remainder of Section 8.2, Sections 8.3, 8.4.
Exercise class: Make Exercises 8.1, 8.3, 8.5.
Homework: Make Exercises 8.2, 8.4.