MVE095/MMG810, Options and mathematics, 2017/18

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 

11/08: Hints to recommended exercises for Chapter1 added (see here)

11/09: MATLAB project added (see here)

11/20: Examination information added

11/20: More past exams papers added

11/27: Modified Matlab codes added (see here)

12/07: OBS! There is an error for Friday 08/12 class on TimeEdit. It is stated there that it is a problem class (lektion), however it will be a lecture (föreläsning). Also note that the location is not Pascal, but KA. Don't get lost!


Course coordinator: Alexandr Usachev ( I reply to e-mails from Monday to Friday  9 am - 5 pm
                               Office hours: Monday 15.15-17.00, Thursday 15.15-17.00. Come to my office (MVL2126) with your questions

Teaching assistant: Anna Persson (

Course representatives:






Course literature

Lecture notes: Introduction to options pricing theory (pdf)


The time and place of the lectures can be found here.


Day Chapter
31 Oct
Basic financial concepts. Long and short position. Portfolio.
1 Nov
Historical volatility.  Options. European/American financial derivatives.
2 Nov
Money market. Frictionless markets. 

7 Nov 1.2 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.

14 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.

21 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 3.7. Proof of Th. 4.1. Exercise 4.4
24 Nov 5.1, 5.2 Finite probability spaces. Random variables. Independence.

28 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 5.9, 5.11, 5.12,  5.17
1 Dec 5.5 General probability spaces. Central limit theorem. Brownian motion. 

5 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

Proof of Theorem 5.7(ii), Exercises 5.19, 5.20, 5.27
8 Dec 6.4, 6.5, 6.6 Black-Scholes price of binary options. Implied volatility. Standard European derivatives on a dividend paying stock

12 Dec 6.7 Optimal exercise time of American calls on a dividend paying stock.
13 Dec
14 Dec

Review old exams
15 Dec

Review old exams

Recommended exercises

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 

Chapter 4: 1, 2, 3

Chapter 5: 1, 2

Chapter 6: 1, 2, 3, 4, 5, 6

Computer labs

Reference literature:

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

Modified Matlab codes:

European derivatives (see here)

American derivatives (see here)

How to use the function argument (see here)

Course requirements

The learning goals of the course can be found in the course plan.


Matlab project (pdf)

The deadline for the submission of the report is December, 15th. 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, DO NOT attach 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 12th, 2018

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 than12 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 state and 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. In addition to that, there is a schedule when exams are given for courses at University of Gothenburg.

Before the exam, it is important that you sign up for the examination. If you study at Chalmers, you will do this by the Chalmers Student Portal, and if you study at University of Gothenburg, you sign up via GU's Student Portal.

At the exam, you should be able to show valid identification.

After the exam has been graded, you can see your results in Ladok by logging on to your Student portal.

At the annual (regular) examination:
When it is practical, a separate review is arranged. The date of the review will be announced here on the course homepage. Anyone who can not participate in the review may thereafter retrieve and review their exam at the Mathematical Sciences Student office. 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 retrieved at the Mathematical Sciences Student office. 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.

Old exams

January 2017 (pdf)   April 2017 (pdf)    August 2017 (pdf)

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)