Welcome to the course.
 18/03: The program and lectures notes for the first 2 weeks of the
course have been added

21/03: I listened to
this interesting radio program on P1 this morning,
which I think explains quite well how important is the role of math in
finance.
 22/03: Reference [5] on American derivatives added. Some minor
missprints in [2][4] have been corrected
 25/03:
IMP! I was asked to remind you that you can (and must) register to the course by yourself:
Som du säkert vet så skall studenterna själva registrera sig på
sina kurser – egenregistrering – det vore bra om du vid första tillfället kan
påminna dem om detta. (De har fått information i mailutskick från
mig om hur det går till).
Om det finns studenter som inte har sökt kursen – be dem att gå in
på antagning.se och gör en sen anmälan – snarast.
 26/03: The list of the course representatives has been added
 28/03: Some more exercises for the first 2 weeks added. Sample exames added. Solutions to exercises week 1 added
 31/03: Ref. [5], Introduction to probability added
Matlab codes added (see Course Literature)
Matlab project added
 17/04: New lectures notes added. Program for weeks 45 added
20/04:
IMP!
Recall that the schedule for this week is a little different: We have
class tomorrow, 21 April at 8 am in Pascal, while the lecture on
Thursday 23 has been cancelled!

21/04: The strange result that I got today with the price of the
American put computed with Matlab was not due to an error in the code!
It was due to a wrong choice of the parameters, namely u=0.01, d=0.01
and r=0.02. Since r>u, the market is not arbitrage free!
Thanks to Kristoffer for pointing this out.
 21/04: The list of the theorems for the exams has been added.
 23/04: Refs. [9], [10] added; program completed
 28/04: I posted the solutions of the exercises in Ref. [2][5], see Literature

4/05: Important information: The bonus points are valid for the exam on
June 4, 2015 and the following two reexams (August 19, 2015 and April
2016)

6/05: Important information: Note that the numbering in the list of
theorems for the exam refers to the last version of the notes! For
instance, theorem 2.4 in the older version of ref. [3] is theorem
2.6 in the new version of ref. [3] and the latter theorem is not in the
list. Be sure to study the right theorems...
 07/05: Some old exams have been added
 09/05: Some typos in ref. [6][7] have been corrected (see
literature)
 05/06: Solution of yesterday's exam posted (see old exams)
Course coordinator: Simone
Calogero (calogero@chalmers.se, tel. 317722 3562, off. L2091)
Teaching assistant: Anna Persson (peanna@chalmers.se)
lab Supervisor:
COURSE REPRESENTATIVES:
TKTEM
maber@student.chalmers.se
MARIA BERGQVIST
TKIEK
adreng@student.chalmers.se
ADRIENNE ENGVALL
TKTEM
dickh@student.chalmers.se
DICK
HEE
TKIEK
edvinma@student.chalmers.se
EDVIN MALMGÅRD
Ref. [1]: Christer Borell: Introduction to the BlackScholes theory (
pdf) (available at DC)
Ref. [2]: Basic financial concepts
Ref. [3]: The binomial asset pricing model
Version April 17. NEW:
Remark 2.3, Remark 2.5 and Example at the end of Sec.
2; missprints corrected
Ref. [4]: European derivatives
Version April 17. NEW: Remark 2.3; missprints corrected
Ref. [5]: American derivatives
Version April 17. NEW:
Exercise 6 added; missprints corrected
Ref. [6]: Introduction to probability
Version April 17. NEW:
Figure 3 improved; discussion after Th. 4.2; Eq. (27) modified; Theorem
6.3 added; missprints corrected
Version May 9: the index
"i" in eqs. (20)(21) has been changed to "k"
Ref. [7]: BlackScholes options pricing theory
Version May 9: NEW: The
payoff of butterfly options has been changed (\Delta K_1 and \Delta
K_2 are now equal)
A
factor (1a) was missing in the second last equation in the proof of
theorem 4.1
Ref. [8]: Volatilities
Version April 17.
Ref. [9]: Futures
Version April 23
Ref. [10]: The Markovitz portfolio
Version April 23
By the end of the course, the references [2][10] can be joined to
make a single text with the lectures notes
Solutions of the Exercises in refs. [2][5] (
pdf)Solutions of the Exercises in refs. [6][10] (
pdf)
Sample exams
1 (
solution),
2 (
solution),
3 (
solution),
4 (
solution)
Matlab codes:
BinomialStock,
European,
American,
ProbStock
Use the
timeedit application to see the
time and location schedule of the course
Lectures
Day

Reference

Contents

Mars
23

Ref. [2]

Introduction to the
course and to financial mathematics

Mars
25

Ref. [1], Ch. 1

The dominance
principle

Mars
26

Ref. [3]

The binomial asset
pricing model

Mars
27 (Anna)

Ref.[1]

[1]: Proof of Th.
1.1.4. Sec. 1.1: Ex. 1, 3, 4; Solutions Exercises




Mars
30

Ref.[3,4]

Arbitrage
portfolio. Implementation of the binomial model with Matlab.

April
1

Ref. [4]

Fair price of European derivatives Computation of the price of European derivatives with Matlab

April
2 (Anna)

Ref. [3,4]

[3]: Proof of Th. 2.4 [4]: Sec. 2.1, 2.2 



April 20 
Ref. [4,5] 
Hedging portfolio
of European derivatives. Fair price of American derivatives

April 21 
Ref. [5,6] 
Hedging American derivatives. Introduction to probability theory. 
April 22 
Ref. [6] 
Random variables. Stochastic processes 
April 24 (Anna) 
Ref. [1,4,5] 
[4]: Proof of Th.
3.1, Ex. 1; [1] Ex. 5, Sec. 2.2, [5]: Exercise 6 Solutions Exercises 



April 27 
Ref. [6] 
The binomial model rivisited. Computing probabilities with Matlab 
April 29 
Ref. [6] 
Infinite probability spaces. The geometric Brownian motion 
April 30 (Anna) 
Ref. [6] 
[6]: Ex. 6, 7, 8, 9, 19, 22 Solutions Exercises 



May 4 
Ref. [7] 
The BlackScholes formula. BlackScholes price of call and put options. The greeks 
May 6 
Ref. [7] 
Binary call, butterfly and chooser options. Options on dividendpaying stocks 
May 7 
Ref. [8] 
Volatilities 
May 8 (Anna) 

[1]: Th. 5.1.1; Example 5.3.1. [7]: Ex. 4, 6 Solutions Exercises 



May 11 
Ref. [10] 
The Markowitz portfolio 
May 13 
Ref. [9] 
Forward and Futures 



May 18 

Lecture by Joakim Björnander (Front Arena, Sungard) 
May 20 (Anna) 
Ref. [9,10] 
[10]: Exercises 1,2,3 Solutions Exercises 
May 21 
Ref. [1], Sec 7.27.3 
Properties of American options on dividend paying stocks. 
May 22 (Anna) 

Revision of some old exams 
Furher Exercises
Week 1: Ref. [1], Sec. 1.1, Ex. 2, 5, 6, 7, 8, 9; Sec 1.2 (problem with solution)
Ref. [2], Ex. 1
Week 2: Ref. [1], Sec 2.1, Ex. 1, 2, 5; Sec. 2.2, Ex. 2, 3, 4, 6
Sample Exam 1, Ex. 1
Week 3: Ref. [5], Ex. 1, 2, 3, 4, 5
Ref. [6], Ex. 1, 2, 3, 5, 10, 11, 12
Week 4: Ref. [6], Ex. 17, 18
Ref. [1]: Sec. 3.1, Ex. 2, 4, 8, 12. Sec. 4.1, Ex. 1
Week 5: Ref. [7] Ex. 2, 3, 5, 7
Ref. [1]: Sec. 5.2, Ex. 5; Sec. 5.3, Ex. 2,3,4
Reference literature:
Tobin A. Driscoll, Learning MATLAB, ISBN:
9780898716832 (The book is published by SIAM)
This year I will assign a Matlab
project which gives max 2 bonus points.
The bonus points will be assigned only if the exam reaches the minimum
number of points to pass (6).
Each student can work on the project with at most 2 other students.
Matlab projectThe
deadline to submit the project is May, 22nd. Send the project to me
(calogero@chalmers.se) using MVE095PROJECT in the subject and add to
the list of the recipients all the students who contributed to the
project. The grade of the project will be announced to the students the
week after (hence, before the exam).
The bonus points are valid for the exam on June 4, 2015 and the following two reexams (August 19, 2015 and April 2016)
The exam is on June 4th
The first reexam in on August 19th
The test comprises 15 points and to pass at least 6 points are required
 at GU a result greater than or equal to 11 points is graded VG;

at Chalmers a result greater than or equal to 9 points and smaller than
12 points is graded 4 and a result greater than or equal to 12 points
is graded 5.
The bonus points from the Matlab project (max 2) cannot be used to
pass the exam, i.e., they will be assigned only if the result of the
test is higher or equal to 6.
The test is divided in three parts, each one valid for a maximum of 5
points
One part will be of theoretical nature and
will require to prove one or more of the following theorems:
Ref. [1]: Theorems 1.1.2, 1.1.3, 1.1.4, 5.1.1
Ref. [3]: Theorems 2.4, 3.2 (only for the single period model)
Ref. [4] : Theorem 3.1 (for N=2)
Ref. [5]: Theorem 4.4
Ref. [6]: Theorems 4.1, 6.3
Ref. [7]: Theorems 2.1, 2.3 (only for Delta, which is computed on pag. 6), 3.1 (which is 5.1.1 in Ref. [1]), 4.1
Ref. [8]: Theorem 1.1
Remark:
If in the exam it is asked to prove theorem X and the proof requires
the result of theorem Y, you don't need to prove also Y
In
Chalmers Student Portal you can read about when exams are
given and what rules apply on exams at Chalmers.
At the link
Scedule
you can find when exams are given for courses at University of
Gothenburg.
At the exam, you should be able to show valid identification.
Before the exam, it is important that you report that you want to take
the examination. If you study at Chalmers, you will do this by the
Chalmers Student Portal, and if you study at University of
Gothenburg, so sign up via
GU's
Student Portal.
You can see your results in Ladok by logging on to the Student portal.
At the annual
examination:
When it is practical a separate review is arranged. The date of the
review will be announced here on the course website. Anyone who can not
participate in the review may thereafter retrieve and review their exam
on Mathematical sciences study expedition, Monday through Friday, from
9:00 to 13:00. Check that you have the right grades and score. Any
complaints about the marking must be submitted in writing at the
office, where there is a form to fill out.
At reexamination:
Exams are reviewed and picked up at the Mathematical sciences study
expedition, Monday through Friday, from 9:00 to 13:00. Any complaints
about the marking must be submitted in writing at the office, where
there is a form to fill out.
May 2014 (
pdf), August 2014 (
pdf), April 2015 (
pdf,
solution) June 2015 (
solution)
Some older exams
2012 (
pdf1,
pdf2,
pdf3)
2013 (
pdf1,
pdf2,
pdf3)
...