Latest news
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 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 4-5 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 re-exams (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 (, tel. 317722 3562, off. L2091)
Teaching assistant: Anna Persson (
lab Supervisor:


TKTEM                   MARIA BERGQVIST
TKIEK                     ADRIENNE ENGVALL
TKTEM                     DICK HEE
TKIEK                  EDVIN MALMGÅRD

Course litterature

Ref. [1]: Christer Borell: Introduction to the Black-Scholes 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]: Black-Scholes options pricing theory 
             Version May 9: NEW: The pay-off of butterfly options has been changed (\Delta K_1 and \Delta K_2 are now equal)
                                              A factor (1-a) 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

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

Mars 30
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 Black-Scholes formula. Black-Scholes price of call and put options. The greeks
May 6 Ref. [7] Binary call, butterfly and chooser options. Options on dividend-paying 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 7.2-7.3 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

Computer labs

Reference literature:
Tobin A. Driscoll, Learning MATLAB, ISBN: 978-0-898716-83-2 (The book is published by SIAM)
Course requirements

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 project

The deadline to submit the project is May, 22nd. Send the project to me ( 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 re-exams (August 19, 2015 and April 2016)

The exam is on June 4th

The first re-exam 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

Examination procedures
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 re-examination:
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.
Old exams
May 2014 (pdf),   August 2014 (pdf),     April 2015 (pdf, solution)     June 2015 (solution)

Some older exams

2012 (pdf1, pdf2, pdf3)

2013 (pdf1, pdf2, pdf3)