Financiële Wiskunde

Financiële Wiskunde 2019-2020
NWI-WM990B and NWI-WB085B

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 regularly updated set of lecture notes will be used.

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.

Follow up material

Further reading: Tomas Björk, Arbitrage Theory in Continuous Time; Steven E. Shreve, Stochastic Calculus for Finance II.
Other, more advanced, topics in Mathematical Finance are treated in the video lectures by Lech Grzelak. Recommended!

People

Lectures: Peter Spreij
Exercise classes: Rein van Alebeek

Schedule

Due to the measures taken to prevent the corona virus from uncontrolled spreading, the schedule below will change and there may be no classroom activities anymore. In one way or another we will try to keep the week by week programme.

Lectures: Usually on Wednesdays, officially 15:30 - 17:15. First lecture (5 February 2020) in HG03.632. On 12, 19 February 2020 we try to start at 15:15!
Exercise classes: Mondays, 13:30 - 15:15 uur in HG03.085, first class on Monday 10 February 2020.
CHANGES: Starting Wednesday 26 February, the lectures will be from 13:30 to 15:15. Lecture rooms: on 26 February and on 11, 18 March in HG03.082; on 15, 22 April and on 6, 13, 20, 27 May in: HG02.028. Scheduling details also on the official website, and on this copy.
CORONA MEASURES: online screen recordings on Brightspace (login required) instead of classical lectures from 18 March 2020 on; paralles changes in the exercise classes (werkcolleges). Last "lecture" (screen recording) planned for 3 June 2020.

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 not be asked. Optional: you may prepare three/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.
Just send me a mail to make an appointment, which should in principle be before mid July. Other restrictions will appear here.

Programme

(UPDATED SCHEDULE, )

1	Lecture: Sections 1.1 (partly, until page 3 only), most of (appendix) A.1 and A.6. 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: 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.
3	Lecture: Remainder of Section 1.1, Section 1.2 (almost to the end). Exercise class: Make Exercises 1.11, 1.12, 1.13(a,b). Homework: Read the remaining part 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 until Theorem 2.5. 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 halfway page 19, Section 2.3 with Markov property of Brownian motion mentioned. Most of Section 3.1, with the exception of second half of the proof of Proposition 3.2 and of Theorem 3.3. But Proposition 3.5 on the backward heat equation mentioned in some way. 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 (in one way or another, perhaps self study): Remaining parts of Section 3.1 (i.e. Theorem 3.3), Section 4.1 Exercise class (also in a new version): Make Exercises 3.5, 4.1, 4.5, 4.6. Homework: Make Exercises 3.6, 4.3, 4.9.
7	Lecture (recordings on Brightspace): 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. (Use the 21 April 2020 version of the lecture notes.) Exercise class: Make Exercises 6.1, 6.4, 6.6(a,b) [In new version (2020 May 3, or later) this is 6.5(a,b)]. Homework: Make Exercises 6.2, 6.3, A.27.
9	Lecture: Section 6.2 from Proposition 6.7, Section 6.3 up to Example 6.17 (see lecture notes 3 May 2020, or later; numbering may have changed). Exercise class: Make Exercises (see lecture notes 3 May 2020, or later; numbering has changed) 6.5(c), 6.6, 6.8, 6.9, 6.11. Homework: Make Exercises (see lecture notes 3 May 2020, or later; numbering has changed) 6.7, 6.9, 6.19.
10	Lecture: Section 6.2 from page 53 and Section 6.3. Exercise class: Make Exercises 6.13, 6.14, 6.20; and if there is enough time, pay some attention to Exercise 6.12. Homework: Make Exercises 6.15, 6.16 and 6.18.
11	Lecture: Sections 7.1 and 7.2. Exercise class: Make Exercises 7.1, 7.5, 7.7 and 7.15. Homework: Make Exercises 7.2, 7.3, 7.4 and 7.8.
12	Lecture: Sections 7.3, 7.4, 7.5, 8.1. Exercise class: Make Exercises 7.9, 7.11, 7.14, 7.16(b,d). Homework: Make Exercises 7.10, 7.12, 7.13.
13	Lecture: Section 8.2, 8.3, 8.4. Exercise class: Make Exercises 8.1, 8.2 or 8.3 (the latter is more work), 8.5. Homework: Make Exercise 8.4.

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