Latest news
Welcome to the course.
The schedule for the course can be found via the link to webTimeEdit top of the page.

10/24: Lecture notes added, see Course literature

10/26: Program added

10/31: Matlab project added, see here

11/05. IMP! The office hours have been moved to Mondays 15.15-17.00

11/08: The course representatives have been added, see here

11/10: Hints to recommended exercises week 1 added, see here

11/16: Some old exams added, see here

11/21: I added another slot for the office hours (Friday 13.15-15.00), see here

11/21: I added the solution to the exercise 3, Chapter 3 in Appendix D, see here

11/21: IMP On Wednesday 23 November at 3 pm I will have a short meeting with the course representatives about the first half of the course . The meeting will take place in the room HB2 just after the lectures. Anyone is welcome to participate directly or to send comments/suggestions to the course representatives before the meeting. The course evaluation is VERY important, so please take some time to think about it. Some interesting questions could be: is the teaching too fast/slow? the level too high/low? do you think to have the required math background? are the lecture notes easy to follow? etc.   

12/13: I wrote a short text with the summary of the course, see here

01/16: The solution of the exam on Friday 13th can be found here

05/23: The solution of the exam on April 18th can be found here

Course coordinator: Simone Calogero ( I answer e-mails from Monday till Friday until 5 pm
                                 Office hours: Monday 15.15-17.00, Friday 13.15-15.00. Come to my office for questions or help with the exercises

Teaching assistant: Anna Persson (

Course representatives:

MATTIAS ERIKSSON                                                             

PETTER HÄGGBERG                                                         

MARIO ALFRED INIGUEZ ORDONEZ                              

ANNIE MILDE                                     

FRIDA TIVEDAL                              
Course literature

Lecture notes: Introduction to options pricing theory

Summary of the course (pdf)


The time and place of the lectures can be found here.
IMP: the lecture on Friday, November 25th will be in the room MVH12 and not in Pascal

Day Chapter
1 Nov
Basic financial concepts. Long and short position. Portfolio.
2 Nov
Historical volatility.  Options. European/American financial derivatives.
3 Nov
Money market. Frictionless markets. 
4 Nov
Qualitative properties of option prices. Put-call parity. Optimal exercise of American put options.

8 Nov 2.1, 2.2 Binomial markets. Self-financing portfolio 
9 Nov

Proof of Th. 1.1 and Th. 1.2. Exercises 1.8, 1.9
10 Nov
Arbitrage portfolio. Arbitrage-free binomial markets.
11 Nov 3.1 Binomial price of European derivatives.
15 Nov 3.2 Hedging portfolio of European derivatives on binomial markets.
16 Nov
Proof of Th. 3.1. Exercises 3.2, 3.3
17 Nov 4.1, 4.2, 4.3 Binomial price of American derivatives. Optimal exercise time of American put options.
18 Nov 4.4
Hedging portfolio of American derivatives. Cash flow. 
22 Nov 2.4, 3.3, 4.5 Computation of the binomial price of European/American derivatives with Matlab. 
23 Nov
Exercise 3.6. Proof of Th. 4.1. Exercise 4.4
24 Nov 5.1, 5.2 1st hour: guest lecture by Carl Lindberg (Ap2 fonden)

2nd hour: Finite probability spaces. Random variables. Independence.
25 Nov 5.2, 5.3 Expectation and conditional expectation.  Stochastic processes. Martingales.
29 Nov 5.4 Applications of probability theory to the binomial model 
30 Nov
Exercises 5.7, 5.9, 5.10,  5.16
1 Dec 5.5 General probability spaces. Central limit theorem. Brownian motion. 
2 Dec 6.1 Black-Scholes markets. 
6 Dec 6.2, 6.3 Black-Scholes price of European derivatives. Hedging portfolio. Black-Scholes price and hedging portfolio of European call and put options
7 Dec
Exercises 6.6, 6.10, 6.11
8 Dec 6.4, 6.5, 6.6 Black Scholes price of binary options. Implied volatility. Standard European derivatives on a dividend paying stock
9 Dec 6.7 Optimal exercise time of American calls on a dividend paying stock.
13 Dec Review
14 Dec
Review old exams
15 Dec
Review old exams
16 Dec Forwards, Futures and Bonds. Outlook

The recommended exercises are those marked with the symbol (●) or (☆) in the lecture notes, plus the following from
Appendix D:

Chapter 1: 1, 3, 5, 6  HINTS
Chapter 3: 1, 2, 3, 4, 6   (sol. to 3, pdf)
Chapter 4: 1, 2, 3
Chapter 5: 1, 2
Chapter 6: 1, 2, 3, 4, 5, 6

Computer labs

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


Matlab project (pdf)

The deadline for the submission of the report is December, 16th. The results will be announced the week thereafter.

(1) The Matlab project is NOT compulsary.
(2) The bonus points of the Matlab project count for the exam in January and the two following re-exams
(3) 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
(4) 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
(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. 

The exam is on January 13th, 2017

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 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 in the lecture notes (max. 4 points) :

Theorem 1.2, Theorem 2.2, Theorem 2.3, Theorem 3.3, Theorem 4.3, Theorem 5.4, Theorem 5.5, Theorem 6.1, Theorem 6.2, Theorem 6.3, Theorem 6.5, Theorem 6.6, Theorem 6.9

and to provide and explain one of the following definitions in the lecture notes (max. 1 point):

Definition 1.1, Definition 2.3, Definition 2.4, Definition 3.1, Definition 3.2, Definition 3.3, Definition 4.1, Definition 4.2, Definition 4.3, Definition 5.15, Definition 6.1, Definition 6.2

(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 the lecture notes. For instance, a question asking to prove theorem 6.5 could read like "Derive the Black-Scholes price of European call options".
(iii) The explanation of the definition need not be the same as in the lecture notes. 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 Schedule 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 Student office, open hours. 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 Student office, open hours. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.
Old exams

April 2016 (pdf)    June 2016 (pdf)   August 2016 (pdf)

April 2015 (pdf)     June 2015 (pdf)     August 2015 (pdf)    

May 2014 (pdf)   August 2014 (pdf)    

Some older exams

2012 (pdf1, pdf2, pdf3)

2013 (pdf1, pdf2, pdf3)