Mathematical Finance
Fall 2002

 

Exam
The exam has taken place on January 10 , 2003.

Grades
You can se the grades here

A day-by-day plan of what was covered in class is below.
The official course material (pensum) is

Tomas Bjørk (TB):
Arbitrage Theory in Continuous Time
Oxford University Press, 1998.

Darrell Duffie (DD):
Dynamic Asset Pricing Theory, 3.ed.
Princeton University Press, 2002,
Chapter 6, sections A-C,E-G,K,L

However, the exam will *not* use material
from the following sections of TB:
4.6, 12.2, 12.3, 13.3, 13.4, 13.5, all of chapter 14,
18.3, all of chapter 20.
Knowledge of the relevant sections of chapter 6
in DD is useful (i.e. you may refrence it)
but it is not necessary.
All material covered in the exercises is part of 'pensum' as well.

Teaching material: Tomas Bjork: Arbitrage Theory in Continuous Time.
Oxford University Press 1998.

The lecture plan below gives good details of what has been covered.
All material covered during the exercises is also part of the readings.

Exercises:

To receive credit for the course, you must sign up to present
at least one exercise in the course of the semester.
2 students may sign up to do 2 exercises together and so forth.
You sign up at the exercises one week ahead.

Sept 10 (pdf)
Sept 17 (pdf)
Sept 24 (pdf)
Oct 1 (pdf)
Oct 8 (pdf)
Oct 22 (pdf)
Oct 29 (pdf)
Nov 5 (pdf)
NO exercise session Tuesday Nov. 12.
NO exercise session Tuesday Nov. 19,
but there is an...
Assignment to be turned in on Wednesday Nov 27
NOTE the extended deadline for the assignment.
Many students (most?) have had problems with the assignment.
This might be because of trouble with some
critically important techniques. The assigment for
Nov 26 (pdf) contains repetition and training som ebasic techniques.
After doing this, I hope you will be able to do the assignment.

PLEASE NOTE CORRECTED VERSION POSTED NOV 11
A careful attempt at this problem will fulfill the requirement
to do an assignment at the exercises.
If you need a hint, stop by my office.
I don't think you really need the answer for 2
to take care of the rest.
But you need to write down solutions for the
stocks and the wealth process.
It is a good strategy to start with the case of
constant drift and constant volatility.
Then look at the determinsitic case afterwards.
You don't have to look at the case with random coefficients.

The exercises for
Dec 02 contain implementation based on a papaer that you will
see how to get using JSTOR or (if unsuccesful)
how to get from the course home page).
Dec 13 (pdf)
NOTE THAT EXERCISES HAVE BEEN MOVED TO FRIDAY.
IN THE FINAL TWO WEEKS OF THE SEMESTER
(WE HAVE INTERCHANGED FRIDAY LECTURES AND TUESDAY EXERCISES)

Dec 20 (pdf)
Exam 2001 (pdf)

What was covered in class

Wed Sept 4: Highlights from Chapter 4 and Chapter 5
in the Investments and Finance Theory Notes from 3^rd year.
This replaces Chapter 2 in Bjork.

Fri Sept 6: Chapter 3 until Proposition 3.4
Browninan motion, stochastic intergrals.

Wed Sept 11: Chapter 3 until Section 3.5
Martingales, stochastic integrals, Ito in one (space) dimension

Fri Sept 13: Chapter 3 until 3.7
Ito in one (space) dimension

Wed Sept 18: The rest of chapter 3. 4.1
Multidimensional Ito, Stochastic differential equations

Fri Sept 20: Chapter 4 until Proposition 4.5.

Wed Sept 25: The remaining parts of Section 4.5.
We skip 4.6 for now. An alternative presentation of 5.1 was given which (I think)
better motivates Definition 5.2 - and this is where we are now.

Fri Sept 27: Chapter 5 is done.
We started on section 6.1

Wed Oct 2: We finished section 6.5.
The Black-Scholes formula has been formally derived!

Fri Oct 4: Chapter 6 is done.
Chapter 7, section1 and 2 up to and including Lemma 7.4.

Wed Oct 9: We have finishged chapters 7 and 8.
We wrote down a more extensive proof of Proposition 8.3.

Fri Oct 11: We did 9.1-9.4.
Before that, we talked at length about what it means for a security price process to be of
the form F(t,S_t) when F satisfies the Black-Scholes equation.

Wed Oct 16, Fri Oct 18: FALL BREAK

Wed Oct 23: We finished chapter 9 - talking about hedging.
We then covered up to and almost including Proposition 10.2.
Note that a handout was distributed from D.Duffie: Dynamic Asset Pricing Theory.
This will be covered in class after we finish chapter 11.

Fri Oct 25: We finished chapter 10.

Wed Oct 30: Dividends, lumpy and continuous (chapter 11).
Duffie: DAPT chapter 6, sections A and B.

Fri Nov 1: Duffie 6C and 6E.
We skipped 6D for now to reach the main result quickly.

Wed Nov 6: Duffie 6E, F, Martingale representation in App. D

Fri Nov 8: Girsanov in App D, and then back to the Lemma in 6G and
the idea behind the Proposition in 6I but no formal proof.

Wed Nov 13: We will finish by looking at 6K.

Fri Nov 15: NO LECTURE

Wed Nov 20: Bjork: 12.1 (currency options), 13.1 + some of 13.2 (barrier options).

Fri Nov 22: NO LECTURE, Do the assignment and prepare for exercises Tuesday.

Wed Nov 27: We finished barrier options and
took a look at the 'reflection principle' underlying Propositions 13.3 and 13.4.
We started on Chapter 15.
(Chapter 14 is skipped).

Fri Nov 29: We did the first and second part of Proposition 15.5.
We also discussed forward rates carefully (instantaneous, simple and continuously compounded).

Wed Dec 4: We finished Proposition 15.5 (but with a
proof of the third statement which is different from
what is in the book). We also did 15.3.1-15.3.3,
lots of which we know from the third year course.
We started on - and essentially finished - Chapter 16.

Dec 6: CANCELLED.

Dec 10 (NOTE DATE): We finished Chapter 16 and most of 17.

Dec 11: We finsihed chapter 17. We didn't do Proposition 17.9.

Dec 17 (NOTE DATE): !8.1 and 18.2 was covered and we have begun
19.3 and 19.4. The introduction to the chapter is skipped and we head
almost directly to Theorem 19.8, which we then will illustrate
with examples.

Dec 18: We will do sections 19.4 and 19.5 and maybe 19.8. That'll be it!