Welcome to the course!
The schedule for the course can be found in TimeEdit.
10/25: Lecture notes added, see Course literature
10/25: MATLAB project added (see here)
10/25: Examination information added
11/08: OBS! The location for the next Friday 11/16 class is not Pascal, but EA. Don't get lost!
12/04: Modified Matlab codes added (see here)
20190828 Solution of yesterday's exam (pdf)
Course coordinator/Lecturer/Teaching assistant: Alexandr Usachev (usachev@chalmers.se) I reply to emails from Monday to Friday 9 am  5 pm.
Course representatives:
TKMAS brauns@student.chalmers.se Simon Braun
TKIEK mattc@student.chalmers.se Mattias Carlsson
MPENM hnavid@student.chalmers.se Navid Haddad
TKIEK zingmark@student.chalmers.se William Zingmark
TKIEK zingmark@student.chalmers.se William Zingmark
Day 
Sections  Contents 

6 Nov 
1.1 
Basic financial concepts. Long and
short position. Portfolio. 
7 Nov 
1.1 
Historical volatility.
Options. European/American financial derivatives. 
8 Nov 
1.1 
Money market. Frictionless
markets. 
9 Nov  1.2  Qualitative properties of option prices. Putcall parity. Optimal exercise of American put options. 
13 Nov  2.1, 2.2  Binomial markets. Selffinancing portfolio 
14 Nov 
2.3  Arbitrage portfolio. Arbitragefree binomial markets. 
15 Nov lektion 
Proof of Th. 1.1 and Th. 1.2. Exercises 1.8, 1.9  
16 Nov  3.1  Binomial price of European derivatives. 



20 Nov  3.2  Hedging portfolio of European derivatives on binomial markets. 
21 Nov 
4.1, 4.2, 4.3  Binomial price of American derivatives. Optimal exercise time of American put options. 
22 Nov lektion 
Proof of Th. 3.1. Exercises 3.2, 3.3  
23 Nov  4.4  Hedging portfolio of American derivatives. Cash flow. 
27 Nov  2.4, 3.3, 4.5  Computation of the binomial price of European/American derivatives with Matlab. 
28 Nov 
5.1, 5.2  Finite probability spaces. Random variables. Independence. 
29 Nov lektion 
Exercise 3.7. Proof of Th. 4.1. Exercise 4.4  
30 Nov  5.2, 5.3  Expectation and conditional expectation. Stochastic processes. Martingales. 


4 Dec  5.4  Applications of probability theory to the binomial model 
5 Dec 
5.5  General probability spaces. Central limit theorem. Brownian motion. 
6 Dec lektion 
Exercises 5.9, 5.11, 5.12, 5.17  
7 Dec  6.1  BlackScholes markets. 


11 Dec  6.2, 6.3  BlackScholes price of European derivatives. Hedging portfolio. BlackScholes price and hedging portfolio of European call and put options 
12 Dec 
6.4, 6.5, 6.6  BlackScholes price of binary options. Implied volatility. Standard European derivatives on a dividend paying stock 
13 Dec lektion 
Proof of Theorem 5.7(ii), Exercises 5.19, 5.20, 5.27  
14 Dec  6.7  Optimal exercise time of American calls on a dividend paying stock. 


8 Jan  Review  
9 Jan lektion 
Review old exams  
10 Jan lektion 
Review old exams  
11 Jan lektion 
Review old exams 
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
Chapter 3: 1, 2, 3, 4, 6
Chapter 4: 1, 2, 3
Chapter 5: 1, 2
Chapter 6: 1, 2, 3, 4, 5, 6
Learning MATLAB, Tobin A. Driscoll ISBN: 9780898716832
(The book is published by SIAM).
Modified Matlab codes:
European derivatives (see here)
American derivatives (see here)
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, 14th. The
results will be announced the week thereafter.
Remarks:
(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 reexams.
(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 18th, 2019
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
Remarks:
(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 BlackScholes 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.
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 can do this from 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 reexamination:
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.
January 2018 (pdf) April 2018 (pdf)
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)