Financial Derivatives and Stochastic Analysis

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

Lecturer and examinator: Torbjörn Lundh, tel. 772 3503

TA:  Ibidem

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
Wednesdays, 8-12
andFridays, 13-15
Week no 44-50 (i.e. lv 1-7).
21 lectures and 6 exercise sessions.
Place: Seminar room 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.

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. And here a translation.
Here are a couple of older finals (in Swedish): December '02,
April '03.

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

Date, time
25/10, 13.15
Introduction of the course, General Probability Theory Chapter 1 in Shreve.
27/10, 8.00
Continuation of Chapter 1.
27/10, 10.00
 	- " -
Information and Conditioning, Chapter 2.
3/11, 8.00
Lesson 1, some exercises from Chapter 1.
3/11, 10.00
Continuation with some exercises and the end of Chapter 2.
5/11, 13.15
Introduction to Brownian Motion, Chapter 3. Definition of BM and its Martingale property. 
8/11, 13.15
Quadratic Variation.  Delivery of home-assignment no. 1. <- Uppdated!
10/11, 8.00
2.1, 2.3, 2.6, 2.11, 3.1, 3.2.
10/11, 10.00
Realized volatility of the geometric Brownian motion. First passage time. Reflection principle. Introduction to Ito's Integral, Chapter 4.1
12/11, 13.15
Ito's formula, Section 4.2-4.3.
15/11, 13.15
Ito-Doeblin's Formula, Section 4.4.
A few problems from chapter 3 and 4.
The Black-Scholes-Merton's Formula. 
Continuation of The Black-Scholes-Merton's Formula. Read multidimensional version yourself.
The Risk-Neutral Measure, Chapter 5. Girsanov's Theorem.  Dead-line for home-assignment no. 1. Delivery of home-assignment no. 2.
Ericsson B, stock.
Ericsson B, call options.

Furthermore, the Proof list is complete!
3.1, 4.4, 4.9, 4.18 (i).
The Martingale Representation Theorem, Sec. 5.3. Girsanov's Thm, Sec. 5.4.
Dividend-Paying Stocks, Sec. 5.5.
Forwards and Futures, Sec. 5.6 
4.18 (ii) and (iii), 4.20, ...
Connections between SDE:s and PDE:s, i.e. the Feynman-Kac Formula, Chapter 6.
Exotic Options, Chapter 7.
Continuation of Chapter 7, i.e. Asian Options. Dead-line for home-assignment no. 2.
6/12, 15.15
Extra lecture given by Patrik Albin as a preparation to Shirayev's talk on Wednesday.

 "Hörsalen" (basement)
Albert Shiryaev, Moscow, Title: 
Stochastic integral representations of functionals of maximum type for Brownian motion (with some applications to financial mathematics), see also this.
American Options, Chapter 8.
16/12, V-house,
14.00 - 18.00
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.
1/4 (no joke!),
8.30-12.30, V-house
Written re-examination I.
26 augusti,
8.30-12.30, V-house
Written re-examination II.

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

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

The page will be  updated during the course. Lates update by
T. Lundh: 050311