Latest news
Welcome to the course.

15-03: The program of the course and the lecture notes have been added

16-03 IMP: The lectures on the first week of the course will be held in the room K2 in the chemistry building, and NOT in Pascal. More information will follow concerning the remaining lectures

16-03: Info about the exam added

05-04 IMP: From next week the lectures will take place in Pascal on Monday, Wednesday and Thursday, and in MVF31 on Friday

05-04: Matlab project added (see here)

07-04: The barrier and compound options in the matlab project are intended to be on European style options! This previously missing information has been added to the project 

11-04: More information on the Matlab project added (see here). 

18-04: Solutions of the exercises for week 1 have been added (including some hints on how to solve the other recommended exercises)

20-04: IMP: I will be in the office every thursday from 10 am to 1 pm to answer your questions and revise your solutions of the recommended exercises. 

21-04: Hints for the recommended exercises on week 3 added

22-04: Solutions to Exercises week 3 added

26-04: More information on the Matlab project added (point (5))

27-04: As we have just started today the discussion on probability theory, I will post next week the hints to the exercises of week 4

27-04: After the discussion with the course representatives I decided to remove the topic "Markowitz portfolio" from the program, so that we can spend more time on probability theory.

28-04: I posted a list of errata in Ref. [1], see literature. If you find more, please let me know and I will update the list.

02-05: Hints and solutions for week 4 exercises added

04-05: List of errata updated. The example at the end of Section 5.2.1 is formulated incorrectly... the correct formulation is given in the list of errata below

09-05: The bonus points of the Matlab project count until the re-exam in April 2017 included!

13-05: IMP: I will give an extra lecture on Tuesday 24 May, 13.15-15.00. In this lecture we shall do some more exercises together, taken from old exams.

13-05: IMP: I noticed that many students have decided to work together on the first task of the project (barrier option). Remember however that groups of more than 1 student must carry out the first n tasks of the project, where n is the number of students in the group. Hence a group of 2 students should work on barrier options AND compound options, while a group of 3 students should work on barrier, compound AND bermuda options. If the same project on just barrier options is presented by more than one student, the bonus point will be divided by all students that present the project.

16-05: Info on the guest lectures added.

18-05: Errata list updated. IMP: Tomorrow I will upload a new version of the note with the errata corrected

18-05: As I did not have time to review the proof if Theorem 6.5 in the class, then this theorem has been removed from the exam list

20-05: I posted a new version of the lecture notes in which the typos of the previous one have been corrected

20-05: Solutions to exercises week 7 added.

20-05: IMP: in the list of "further exercises" I underlined those which are most relevant for the exam

24-05: IMP: The lecture today will be at Euler

03-06: The solution of yesterday exam can be found here

23-08: The solution of the exam on August 17 can be found here

Course coordinator: Simone Calogero (, tel. 317722 3562, off. L2091)
Teaching assistant: Anna Persson (


TKIEK                                     VI HOANG

MPENM                             HARALD OTTEDAG

MPCAS                                  GUSTAVO STOLF JEUKEN

TKTEM                               JOHAN ULANDER

TKIEK                               OSKAR WAHLBÄCK

Course litterature
Ref. [1] Introduction to options pricing theory

Ref. [2] Christer Borell: Introduction to the Black-Scholes theory (pdf)   (available ad DC)

Further reading: Steven E. Shreve: Stochastic Calculus for Finance I: The Binomial Asset Pricing Model. Springer Finance New York (2004)

Use the timeedit application to see the time and location schedule of the course

Day Reference
Mars 21          
Ref. [1], Ch. 1
Introduction to the course. Basic financial concepts. Portfolio.
Mars 23
Ref. [1], Ch. 1
Basic financial concepts (cont.). Historical volatility. Options. Money market. Frictionless market.
Mars 24
Ref. [1], Ch. 1
Qualitative properties of option prices. Dominance principle. Optimal exercise time of American put options.

April 11
Ref. [1], Ch. 2
Binomial stock price. Binomial markets. Self-financing portfolio.
April 13
Ref. [1], Ch. 2
Arbitrage portfolio. Absence of arbitrage in binomial markets.
April 14
Ref. [1], Ch. 3
Computation of the binomial stock price with Matlab.
April 15 (Anna)

Ref. [2], Theorem 1.1.1; Ref. [2], Sec. 1.1: Ex. 1, 3, 4;. Ref. [1], Ex. 1.7  (solutions)
April 18Ref. [1], Ch. 3Binomial price of European  derivatives.
April 20Ref. [1], Ch. 3Hedging portfolio of standard Euroepan derivatives. Computation of the binomial price of European derivatives with Matlab.
April 21 (Anna)Ref. [2], Sec. 2.2: Ex. 5. Ref. [1]: Ex. 3.1, 3.2, Ex. 3.3.   (solutions)
April 22Ref. [1], Ch. 4Binomial price of American derivatives.
April 25Ref. [1], Ch. 4Replicating portfolio of standard American derivatives.  Computation of the binomial price of American derivatives with Matlab.
April 27Ref. [1], Ch. 5Introduction to probability theory. Finite probability spaces. 
April 29 (Anna)Ref. [1], Ex. 4.3, 4.6, 4.7. (solutions)   
May 2Ref. [1], Ch. 5 Random variables. Expectation. Variance. Independence. Joint distribution. 
May 4Ref. [1], Ch. 5Conditional expectation. Stochastic processes. Martingales.
May 9Ref. [1], Ch. 5Probabillistic formulation of the binomial options pricing model. 
May 11Ref. [1], Ch. 5Infinite probability spaces. Geometric Brownian motion.
May 12Ref. [1], Ch. 6 Black-Scholes options pricining model. Black-Scholes formula.
May 13 (Anna)Ref. [1], Ex. 5.6, 5.7, 5.8, 5.15, 5.25.   (solutions)
May 16Ref. [1], Ch. 6Black-Scholes price of call and put options. The greeks. Implied volatility. Volatiility smile.
May 18Ref. [1], Ch. 6Standard European derivatives on a dividend paying stock.
May 19 (Anna)Ref. [2], Example 5.3.1 , Ref. [1], Sec. 6.3.1, Ex. 6.5, 6.6.  (solutions)
May 20Guest lectureCarl Lindeberg (adjunt professor and porfolio manager):
Derivatives-Or, why math is essential in finance

Joakim Björnander (Software developer at FIS Front Arena):
Working as a (financial) software developer

May 23 (Anna)Review old exams.

Further exercises

Week 1: Ref. [2], Sec. 1.1, Ex. 2, 5, 6, 7, 8, 9; Sec 1.2 (problem with solution) Ref. [1], Ex. 1.6  (hints)
Week 3: Ref. [2], Sec 2.1, Ex. 1, 2, 5; Sec. 2.2, Ex. 2, 3, 4, 6  (hints)

Week 4: Ref. [1], Ex. 4.4, 5.1, 5.2, 5.3, 5.4, 5.5   (hints)

Week 5: Ref. [1], Ex. 5.9, 5.10, 5.11, 5.12 ,5.17, 5.18, 5.19 (hints)

Week 6: Ref. [1], Ex. 5.22, 5.26, 6.1 (hints)

Week 7: Ref. [1], Ex. 6.2, 6.3, 6.4, Ref. [2], Sec. 5.3, Ex. 5   (hints)

Remark: While all exercises above are important for a full understanding of the content of the course, the underlined exercises are most relevant for the examination

Computer labs and project assignment

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

Matlab project (deadline for submission: 20th of May)

(1) The Matlab project is NOT compulsary. The bonus points of the Matlab project count until the re-exam in April 2017 included
(2) Note that each option in the project has many different variants (e.g., compound options exist as call on call, call on put, etc.). While you should describe all possible variants, for the numerical part you can focus on one example for each option
(3) Attach only the most relevant Matlab codes (e.g., the Matlab functions to compute the price). In particular, avoid attaching the Matlab scripts used to create the plots
(4) For the Matlab code of European barrier options you can assume that the barrier is European, i.e., it is only checked at maturity whether the barrier is crossed or not (while for an American barrier, the option expires immediately when the barrier is crossed, regardless of whether in the future the stock price will return below the barrier)
(5) The main part of the project is the matlab code and the pictures. As far as the theoretical evaluation you can simply list the different approaches used in the literature and give some reference to where details can be found. If there exists an exact formula, you can write it in your report. Moreover you could give an example with the binomial model for N=2 or 3... this is really up to you! As I ask you to write no more than 5 pages for each option (including the matlab code and the figures), you really can't write much. 

Course requirements


The exam is on June 2nd

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 Matlab project gives max 1 point (the project can be found here)

The test is divided in three parts, each one giving a maximum of 5 points.

One part will be of theoretical nature and will require to prove one or more of the following theorems form Ref. [1] (max. 4 points) :

Theorem 1.1, Theorem 2.1, Theorem 2.2, Theorem 3.2, Theorem 4.1, Theorem 5.3, Theorem 5.4, Theorem 5.10, Theorem 6.2,

and to provide and explain one of the following definitions from Ref. [1] (max. 1 point):

Definition 1.1, Definition 2.2, Definition 2.3, Definition 3.1, Definition 3.2, Definition 3.3, Definition 4.1,Definition 4.2, Definition 4.3, Definition 4.4, Definition 5.4, Definition 5.15, Definition 5.19, Definition 6.1

(i) 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
(ii) When asked to prove one of the above theorems, the question does not necessarily contain the exact statement as it appears in Ref. [1]. For instance, a question asking to prove theorem 6.2 could read like "Derive the Black-Scholes price of European call and put options".
(iii) The explanation of the definition need not be the same as in Ref. [1]. You can use your own intuition.
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
Remark: I changed slightly the notation in the text of the exams to make it consistent with the one in the lecture notes

May 2014 (pdf),   August 2014 (pdf),     April 2015 (pdf)     June 2015 (pdf)     August 2015 (pdf)    April 2016 (pdf)

Some older exams

2012 (pdf1, pdf2, pdf3)

2013 (pdf1, pdf2, pdf3)