Financiële Wiskunde 2017-2018
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:

Literature

A set of lecture notes (as of December 18 a revised version with less typos, but also slightly different numbering, is available!) will be used, which might be extended in the last weeks of the course. Further reading: Tomas Björk, Arbitrage Theory in Continuous Time; Steven E. Shreve, Stochastic Calculus for Finance II.

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.

People

Lectures: Peter Spreij
Exercise classes: Stijn Cambie

Schedule

Lectures: Wednesdays, 15:45 - 17:30. First lecture (6 September) in HG00.071. For other lecture rooms, have a look here for up to date information.
Exercise classes: Mondays, 10:45 - 12:30 uur in HG03.054, first class on Monday 11 September.
After the autumn break, there will be a completely rescheduled programme. Days, hours and lecture rooms as in the original schedule, but lectures and exercise classes mixed up. See the table below.

Adjusted schedule

November Monday 13: Tutorial 8
Wednesday 15: Lecture 9
Monday 20: Lecture 10
Wednesday 22: Lecture 11
Monday 27: Tutorial 9
Wednesday 29: Tutorial 10
December Monday 04: Tutorial 11
Wednesday 06: None
Monday 11: Lecture 12
Wednesday 13: Tutorial 12
Monday 18: Lecture 13
Wednesday 20: Tutorial 13

Examination

Oral exam. Homework assignments (compulsory, every week) count for 25%. What you have to know: The theory, i.e. all important definitions and results (lemma's, theorems, etc.), but proofs will not be asked. Optional: you may prepare two 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). You will be asked to present one of them. Selected dates: Thu 4 Jan, Tue 30 Jan, Wed 31 Jan. You make an appointment for your favourite day, cancellations are always allowed. Absolutely impossible are the periods 25 Dec - Jan 2, Jan 6 - Jan 17, Jan 22 - Jan 25, Feb 5 - 25, but on request there should be more possibilities. To be continued...


Programme

(UPDATED SCHEDULE, )

1 Lecture: sections 1.1 (partly, until p.7 half way), a bit of A.5.
Exercise class: 1.1, 1.2(a), A.12.
Homework: read what has been treated during class (even if it was superficially) and make Exercises 1.3, 1.4, A.13. See this too as experimenting a bit with what is doable (or not).
2 Lecture: Review of the foundations of probability theory, conditional expectations in the finite and general case.
Exercise class: Read Proposition A.11, Section A.6 up to Definition A.12. Make Exercises A.1, and A.15 if time permits. Prove Proposition A.11(iii). Show that the conditional expectation of Definition A.11 is a.s. unique. Show also that $X+Y$ is a random variable, if $X$ and $Y$ are random variables. What about $XY$?
Homework: None
3 Lecture: Remainder of Section 1.1, Section 1.2.
Exercise class: Make Exercises 1.11, 1.12, 1.13(a,b).
Homework: Make Exercise 1.5, 1.8, 1.9 and read the main results of Section A.4 (unless you already know this).
4 Lecture: Sections 2.1, 2.2 until the first part of the proof of Theorem 2.5.
Exercise class: Make Exercises 2.1, 2.3, 2.4, 2.5.
Homework: Make Exercises 2.2, 2.13 (needed for next time) 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: Some essentials of characteristic functions and Hilbert spaces, remainder of Section 2.1, starting with the second part of the proof of Theorem 2.5. Markov (Proposition 2.10) and martingale property of Brownian motion
Exercise class: Make Exercises 2.6, 2.8 and Exercise 6 of the additional exercises. Show also that $E\exp(uZ)=\exp(\frac{1}{2}u^2)$ if $Z$ is standard normal and $u$ is real, simply by integration.
Homework: Make Exercise 2.9, Exercise 5 of the additional exercises, and then Exercise 2.11.
6 Lecture: Section 2.3, the beginning of Section 3.1, and the backward heat equation.
Exercise class: Make Exercises 3.1, 3.2 (for this exercise you quickly glance at the proof of Proposition 3.2, in particular Eq (3.4) and what is around it), 3.4.
Homework: Make Exercises 2.10(b), 2.14.
7 Lecture: Remainder of Section 3.1 (Proposition 3.2 + proof, mentioning of Thm 3.3) and Section 4.1 up to Proposition 4.2.
Exercise class: none
Homework: Make Exercises 3.5, 3.6 and 3.9.
8 Lecture: Section 4.1 from Corollary 4.3, Sections 5.1, 5.2.
Exercise class: Make Exercises 4.5, 4.7, 4.8, and 4.1 if time permits.
Homework: Make Exercises 4.3, 4.6, 4.9, 5.1.
9 Lecture: Section 5.3, Section 6.1 up to Proposition 6.1.
Exercise class: Make Exercises 5.3, 5.4(c,e) [the answer to (e) should be familiar to you], 6.1.
Homework: Make Exercises 5.2, 5.4(a,b), 6.2.
10 Lecture: Section 6.1 from Proposition 6.1, proof of Proposition 6.5 only for simple processes (planned recap of Section A.6 skipped).
Exercise class: Make Exercises 6.6, 6.7 and 6.12.
Homework: Make Exercises 6.4, 6.5 and 6.8 (think of using quadratic variation).
11 Lecture: Section 6.2.
Exercise class: Make Exercises 6.9, 6.13 and 6.16.
Homework: Make Exercises 6.10, 6.11 and 6.19.
12 Lecture: Sections 6.3, 7.1, Section 7.2 up to Proposition 7.4 / Corollary 7.5.
Exercise class: Make Exercises 4.12, 6.14, 7.5.
Homework: Make Exercises 6.15 and 7.4.
13 Lecture: Section 7.2 from Corollary 7.5, Sections 7.3, 7.4.
Exercise class: Make exercises 7.2, 7.6 (if enough time), 7.7 (don't give an explicit expression (unless you really want), but express the price of the straddle in terms of the prices of the constituents of the constant portfolio), 7.8, 7.11.
Homework: None.