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
here23-08: The solution of the exam on August 17 can be found
here
Course coordinator: Simone Calogero (calogero@chalmers.se, tel. 317722 3562, off. L2091)
Teaching assistant: Anna Persson (peanna@chalmers.se)
COURSE REPRESENTATIVES:
TKIEK
hovi@student.chalmers.se
VI HOANG
MPENM
ottedag@student.chalmers.se HARALD
OTTEDAG
MPCAS
stolf@student.chalmers.se GUSTAVO
STOLF JEUKEN
TKTEM
johanul@student.chalmers.se JOHAN
ULANDER
TKIEK
oskarwa@student.chalmers.se
OSKAR
WAHLBÄCK
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
Lectures
Day |
Reference
|
Contents
|
WEEK 1 | | |
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.
|
WEEK 2
|
|
|
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)
|
WEEK 3 | | |
April 18 | Ref. [1], Ch. 3 | Binomial price of European derivatives. |
April 20 | Ref. [1], Ch. 3 | Hedging 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 22 | Ref. [1], Ch. 4 | Binomial price of American derivatives. |
WEEK 4 | | |
April 25 | Ref. [1], Ch. 4 | Replicating
portfolio of standard American derivatives. Computation of
the binomial price of American derivatives with Matlab. |
April 27 | Ref. [1], Ch. 5 | Introduction to probability theory. Finite probability spaces. |
April 29 (Anna) | | Ref. [1], Ex. 4.3, 4.6, 4.7. (solutions) |
WEEK 5 | | |
May 2 | Ref. [1], Ch. 5 | Random variables. Expectation. Variance. Independence. Joint distribution. |
May 4 | Ref. [1], Ch. 5 | Conditional expectation. Stochastic processes. Martingales. |
WEEK 6 | | |
May 9 | Ref. [1], Ch. 5 | Probabillistic formulation of the binomial options pricing model. |
May 11 | Ref. [1], Ch. 5 | Infinite probability spaces. Geometric Brownian motion. |
May 12 | Ref. [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) |
WEEK 7 | | |
May 16 | Ref. [1], Ch. 6 | Black-Scholes price of call and put options. The greeks. Implied volatility. Volatiility smile. |
May 18 | Ref. [1], Ch. 6 | Standard 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 20 | Guest lecture | Carl 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
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)
Remarks:
(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.
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
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 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.
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.
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)
...