Stochastic integration 2007-2008

Contents

Stochastic calculus is an indispensable tool in modern financial mathematics. In this course we present this mathematical theory and apply it to the problem of pricing and hedging of financial derivatives. We treat the following topics from martingale theory and stochastic calculus: martingales in discrete and continuous time, construction and properties of the stochastic integral, Ito's formula, Girsanov's theorem, stochastic differential equations. As an application, we explain how stochastic differential equations are typically used to model financial markets and we discuss the problem of the pricing of derivatives such as stock options.

Prerequisites

Measure theory, stochastic processes at the level of the course Measure Theoretic Probability

Literature

Recommended background reading: I. Karatzas and S.E. Shreve, Brownian motions and stochastic calculus and D. Revuz and M. Yor, Continuous martingales and Brownian motion. The contents of the course are described in the lecture notes (based on these books).

Until 31 July 2008 Springer Verlag holds its Yellow Sale with many mathematics books at low prices. The above two reference books are both available, see Brownian motions and stochastic calculus (€ 28,84) and Continuous martingales and Brownian motion (€ 53,45), as well as many more.

On 5 June 2008 the symposium `30 years of option excellence' takes place. Information and registration via the website of The Derivatives Technology Foundation.

Companion course

Students are recommended to take also the course on Stochastic Processes, see the Spring Courses of the Dutch Master Program in Mathematics.

Follow up courses

Courses that heavily rely on stochastic calculus are Financial Stochastics and Control of Stochastic Systems in Continuous Time.

Lecturer

P.J.C. Spreij

Homework

Grading by Enno Veerman

Schedule

Spring semester: Tuesdays, 11.00-14.00, Room I.103. First lecture on 5 February 2008. No classes on March 4, March 25, May 6.

Examination

The final grade is a combination of the results of the take home assignments and the oral exam. Make an appointment for the oral exam by email, taking into account the following impossibilities: June 23 - July 1, July 7 - 10, July 14 and about 4 weeks starting sometime in late July.


Programme

Please, regularly check the programme below.
It is not definitive and adjustments may be made throughout the course

1 Lecture notes: section 1, section 2.1
Homework: 4 exercises from section 1.3
2 Lecture notes: sections 2.1 (what remained), 2.2, 2.3 (uniqueness of DM decomposition, proposition 2.17)
Homework: choose 3 exercises from 2.1, 2.3, 2.5, 2.8, 2.9
3 Lecture notes: section 2.3 (existence part of the proof of theorem 2.14, theorem 2.15 (weaker statement)), section 3.1, definition of quadratic variation
Homework: select four exercises from 2.6, 2.7, 2.12, 2.14, 3.1
4 Lecture notes: most of section 3.2; section 4.1 (definition of local martingales under the restriction that they start in zero) and section 4.2 up to proposition 4.6.
Homework: One exercise from 3.2, 3.3 (part (c) has been extended with a hint, read the changed version in the lecture notes), and two from 3.4, 3.5, 3.6, 3.9
5 Lecture notes: sections 5 and the beginning of section 6
Homework: Two exercises from 4.1 - 4.4 and two from 4.5 - 4.8, 5.1-5.2
6 Lecture notes: section 6.1 (except proposition 6.7) and proposition 6.8 (partly).
Homework: read sections 3 and 6 of "The Radon-Nikodym theorem", make three exercises from 6.1, 6.3, 6.6, 6.8, 6.9
7 Lecture notes: sections 6.2 (remaining parts), 6.3, section 7.1 up to Proposition 7.7
Homework: three exercises from the set 6.7, 6.8, 6.10 - 6.14
8 Lecture notes: remainder of section 7.1, sections 7.2, 7.3, 7.4
Homework: three exercises from section 7, read the first example of a local martingale that is not a martingale
9 Lecture notes: section 8 until theorem 8.6 (with only a sketch of the proof)
Homework: four exercises from section 8
10 Lecture notes: sections 9.1-9.3 (with the exception of Propositions 9.5 and 9.13)
Homework: make three exercises from 9.2-9.8
11 Lecture notes: remainder of section 9.3, section 9.4 and definition 10.1 plus examples of ODEs.
Homework: none
12 Lecture notes: (most of) section 10.1, parts of section 10.2
Homework: three exercises from 10.3, 10.4, 10.6, 10.8, 10.16
13 Lecture notes: sections 10.3 and 11
Homework: one exercise from 10.2, 10.14, 10.15, two from 11.1, 11.2, 11.3, 11.4, 11.5 and read the second example of a local martingale that is not a martingale.


To Korteweg-de Vries Instituut voor Wiskunde or to the homepage of the master's programme in Stochastics and Financial Mathematics.

Email: spreij@science.uva.nl