Financial Derivatives and Stochastic Analysis

Code: TMA 285 [Chalmers] or MAM 695 [GU] 

Lecturer and examinator: Torbjörn Lundh, tel. 031 - 772 3503

Literature: S. Shreve, Stochastic Calculus for Finance II, Continuous-Time Models, Springer, 2004.

Suitable background: The course Optioner och matematik or something similar.

Some of the concepts that will be discussed in the course are
Measure theory, probability theory; Brownian motion an Stochastic Integration. Ito's, and Feynman-Kac's formulas. The connection between Brownian motion and PDEs. Stochastic differential equations. Self financed portfolio strategies and arbitrage. Black-Scholes' model, and stability analysis of that model. Wiener measures and Cameron-Martin's Theorem. Martingale representation. Complete capital markets. Financial derivative depending on multiple stocks. Currency depending options. Exotic options. Bonds- and interest models. Option price in Gaussian interest models.
The course has 56 hours of lecturing or teaching.

Times and dates:
Mondays, 13-15 (except Monday Oct. 31, when it is 10-12)
Wednesdays, 10-15
and Fridays, 13-15
Week no 44-50 (i.e. lv 1-7).
21 lectures and 6 exercise sessions.
Place: Seminar room S1 (Wednesdays), and S4 in the mathematics center.

There will be two home-assignments, which can add up to 1.5 bounus-points each to your final exam.
You may work together in pairs, but no larger groups are allowed. Note, that points earned this way, might only be used within one year. (I.e. only for the ordnary exam, and the re-examination.)

In the written exam, you will be given two theoretical tasks that are more or less picked from the following proof list.

Here you can find last year's final exam.
Here are a couple of older finals (in Swedish): December '02,
April '03.

Here you can find '03:s final exam. And here a translation.

Preliminary lecture and teaching plan, which will be continuously updated:

Date, time
24/10, 13.15
Introduction of the course, General Probability Theory Chapter 1 in Shreve.
26/10, 10.00, S1
Continuation of Chapter 1.
26/10, 13.15, S1
 	- " -
28/10,13.15, S4
Information and Conditioning, Chapter 2 .
31/11, 10.00 <-Obs the time!
Continuation with some exercises, 1.1, 1.2, and the end of Chapter 2.
2/11, 10.00
Lesson 1, some exercises from Chapter 1. 1.3, 1.4, 1.5, 1.7, 1.12.
2/11, 13.15
Introduction to Brownian Motion, Chapter 3. Definition of BM and its Martingale property. 
7/11, 13.15
Quadratic Variation.  Delivery of home-assignment no. 1. 
9/11, 10.00
Exercises 2.1, 2.3, 2.6, 2.11, 3.1, 3.2.
9/11, 13.15
Realized volatility of the geometric Brownian motion. First passage time. Reflection principle. Introduction to Ito's Integral, Chapter 4.1
11/11, 13.15
Ito's formula, Section 4.2-4.3.
14/11, 13.15
Ito-Doeblin's Formula, Section 4.4.
16/11, 10.00
A few problems from chapter 3 and 4.
16/11, 13.15
The Black-Scholes-Merton's Formula. 
18/11, 13.15
OBS! No lecture. Study eagerly for your self!
Continuation of The Black-Scholes-Merton's Formula. Read multidimensional version yourself.
3.1, 4.4, 4.9, 4.18 (i).
The Risk-Neutral Measure, Chapter 5. Girsanov's Theorem.  
Dead-line for home-assignment no. 1.

Delivery of home-assignment no. 2. <- corrected version Nov. 29
Volvo B, 21 Nov. '04 to 21 Nov '05.
Volvo B, optioner 22 november 2005.
The Martingale Representation Theorem, Sec. 5.3. Girsanov's Thm, 
The Martingale Representation Theorem, Sec. 5.3. Girsanov's Thm, 
5.2, 5.5, 5.10,
some remarks on the first home assignment.
End of section 5.2.
Section 5.3.
Sec. 5.4
Dividend-Paying Stocks, Sec. 5.5. Forwards and Futures, Sec. 5.6 
Connections between SDE:s and PDE:s, i.e. the Feynman-Kac Formula, Chapter 6.
Dead-line for home-assignment no. 2. 
Introduction to exotic options.
Continuation of Exotic Options, Chapter 7.
American Options, Chapter 8.
G4, 9.00-10.00
Extra office-hour for last minute questions, as I promised.
Since I do not have an office then, we meet at G4, also known as
16/12, V-building, 8.30-12.30
Written final exam.
As on previous exams: Note that no calculators are allowed, but on the other hand you may bring a copy of the book Beta, i.e. Mathematics Handbook for Science and Engineering, Råde and Westergren. 
Short solutions.
New: The exam is now graded.  031 772 3500 for information 8.30-13.00.
September 1,

Written re-examination.
Note, that there will be only one re-examination this year. This is due to the fact that there have been "too few" student doing the re-examination over the last years.  (Short solutions.)

This plan is just preliminary and will be upgraded during the course.

Here is the link to our own, private, course-page. You'll need a user-name and a password.

A few links that might be of some interest:
Steven Shreve's home-page
Olle Häggström and Torgny Lindvall's  lecture notes from the course Sannolikhetens Grunder.
Some advice Jones, Mardon och Cook to someone who wants to become a "quant". 
See also
Here are some trading sites:

The page will be  updated during the course. Lates update by
T. Lundh: 051221