Course syllabus
The course Options and Mathematics deals with the arbitrage-free valuation of stock options, and other financial derivatives, using both theoretical and numerical tools. It is intended as a first course in financial mathematics, for which in particular no previous knowledge in finance is required.
More information on the aim and learning outcomes of the course can be found in the course plan (Links to an external site.).
Course Responsible
Simone Calogero. E-mail: calogero@chalmers.se
Important. I answer e-mails Monday-Friday from 9 am to 5 pm and only about course administration. Math-related questions should be asked in the class.
Student representatives
The following students have been appointed:
TKTEM jasarevic.hamza@gmail.com Hamza Jasarevic
TKTEM emelielemannn@gmail.com Emelie Lemann
TKIEK saminw@student.chalmers.se William Samin
The mid-course meeting with the student representatives will be on Wednesday 24th November at 3 pm (i.e., after the lecture).
News
25/10: This course homepage has been created
4/11: An errata file for the book has been added, see Literature. If you find a typo in the book, please let me know.
08/11: Imp! Information for the students who have taken the course last year. You can submit the same Matlab project as last year, but you should address the comments that I sent you in my feedback. Moreover you have to be the single author of the report and specify that it is the re-submission of a project that you did last year.
18/11: IMP! Starting from Tuesday November 23rd, the lectures will be held in Pascal at the math department.
24/11: A discussion thread has been created that you can use to find teammates to work on the matlab project, see here
24/11: Four discussion threads have been created that you can use to shares ideais on the four different projects:
project 1 (compound options), project 2 (Lookback option), project 3 (American puts), project 4 (dividends)
14/12: The link to submit the project has been created, see Section Matlab Project below.
14/01: Solution of yesterday's exam: pdf
19/04: Solution of April 13 exam: pdf
30/08: Solution of August 23 exam: pdf
(old news)
Literature
Simone Calogero: A first course in Options Pricing Theory (pdf) Errata (pdf)
Program
The schedule of the course is in TimeEdit. In the first 3 weeks of the course, the lectures will be held in the room HC2 on Tuesday/Wednesday and in the room KC on Thursday/Friday. Starting from the 4th week, the lectures will be held in Pascal at the math department.
In the program you can find the notes that I use in the lectures. The text is taken from the book, but contains some additional annotations. Note that these notes were used last year, when the course was given online, and therefore the reference number of some theorems, definitions and exercises is different from this year version of the book!
Lectures
| Day | Time | Sections | Content |
|---|---|---|---|
|
2 Nov |
15.15
|
1.1 |
Basic financial concepts. Long and short positions. Portfolio. Notes (from last year) |
| 3 Nov | 13.15 | 1.1 |
Historical volatility. Options. European/American financial derivatives Notes (from last year) |
| 4 Nov | 13.15 | 1.1/1.2 |
Bonds. Money market. Frictionless markets. Arbitrage-free principle Notes (from last year) |
| 5 Nov | 15.15 | 1.2 |
Qualitative properties of options. Put-call parity. Theorem 1.2. Exercise 1.9. Optimal exercise of American options. Definition 1.2, Definition 1.3 Notes (from last year) |
| ******** | ******* | ************* | ******************************************************************************************* |
|
9 Nov |
15.15 |
EXERCISES |
Exercises 1.10, 1.11, 1.14, 1.16, 1.17, 1.18, 1.19, 1.22 Notes (from last year) |
| 10 Nov | 13.15 | 2.1, 2.2 |
Binomial markets. Predictable portfolio. Definition 2.1. Exercise 2.2 Notes (from last year) |
| 11 Nov | 13.15 | 2.2 |
Self-financing portfolio. Definition 2.6. Exercise 2.4. Theorem 2.1, Theorem 2.2. Exercise 2.5 Notes (from last year) |
| 12 Nov | 15.15 | 2.2, 2.3 |
Portfolio generating a cash flow. Arbitrage portfolio. Definition 2.8. Theorem 2.4 Notes (from last year) |
| ******** | ******* | ************* | ******************************************************************************************* |
|
16 Nov |
15.15 |
3.1 |
Binomial price of European derivatives. Definition 3.1. Definition 3.2 Notes (from last year) |
| 17 Nov | 13.15 | 3.2 |
Replicating portfolio of European derivatives on binomial markets. Theorem 3.3. Exercise 3.11 Notes (from last year) |
| 18 Nov | 13.15 | EXERCISES |
Exercises 3.3/3.22, 3.4, 3.6, 3.7, 3.8 Notes (from last year) |
| 19 Nov | 15.15 | 4.1, 4.2 |
Binomial price of American derivatives. Optimal exercise of American put options. Notes (from last year) |
| ******** | ******* | ************* | ******************************************************************************************* |
|
23 Nov |
15.15 |
4.3 |
Replicating portfolio of American derivatives. Cash flow. Exercises 4.5, 4.6 Notes (from last year) |
| 24 Nov | 13.15 | 2.4, 3.3, 4.4 |
Computation of the binomial price of European/American derivatives with Matlab. Description of the projects. Notes (from last year) |
| 25 Nov | 13.15 | 5.1 |
Finite probability spaces. Random variables. Independence Notes (from last year) |
| 26 Nov | 15.15 | 5.1 |
Expectation and conditional expectation. Stochastic processes. Martingales Notes (from last year) |
| ******** | ******* | ************* | ******************************************************************************************* |
|
30 Nov |
15.15 |
5.2 |
Applications of probability theory to the binomial model. Theorem 5.5. Definition 5.19 Notes (from last year)
|
| 1 Dec | 13.15 | EXERCISES |
Exercises 5.24, 5.25, 5.26, 5.38 Notes (from last year) |
| 2 Dec | 13.15 | 6.1 |
General probability spaces. Brownian motion. Exercise 6.7 Notes (from last year) |
| 3 Dec | 15.15 | 6.1, 6.2 |
Girsanov theorem. Black-Scholes markets. Definition 6.7. Theorem 6.10 Notes (from last year) |
| ******** | ******* | ************* | ******************************************************************************************* |
|
7 Dec |
15.15 |
6.3, 6.4 |
Black-Scholes price of standard European derivatives. Definition 6.8 Black-Scholes price of European call and put options. Theorem 6.14 Notes (from last year) |
| 8 Dec | 13.15 | 6.4 |
Greeks. Implied volatility. Implied volatility curves. Definition 6.9 Notes from last year: same as previous lecture |
| 9 Dec | 13.15 | EXERCISES |
Exercises 6.12, 6.13, 6.16, 6.17 Notes (from last year) |
| 10 Dec | 15.15 | 6.6 |
The Asian option. Exercises 6.22, 6.23, 6.24. Monte Carlo method. Notes (from last year) |
| ******** | ******* | ************* | ******************************************************************************************* |
|
14 Dec |
15.15 |
6.7 |
Standard European derivatives on a dividend paying stock. Theorem 6.18. Exercise 6.26 Notes (from last year) |
| 15 Dec | 13.15 | 6.9 |
Introduction to bonds valuation. Definition 6.10. Exercise 6.35 Notes (from last year) |
|
16 Dec |
13.15 |
Review and exercises (1.24, 1.25, 1.26, 1.28, 1.29, 1.31, 1.32, 1.33, 1.34) Notes (from last year) |
|
|
17 Dec |
15.15 |
Review and exercises (3.23, 4.8, 4.10, 5.39, 6.36, 6.37, 6.39, 6.40, 6.41) Notes (from last year) |
|
|
|
Matlab Project
Choose one of the following projects in the book:
- Compound options (Project 1, page 76) Discuss the project with other groups here
- Loopback options (Project 2, page 81) Discuss the project with other groups here
- American put options (Project 3, page 95) Discuss the project with other groups here
- Call options on dividend paying stocks (Project 4, page 96) Discuss the project with other groups here
Use this discussion thread if you are looking after teammates to work on the project.
Remarks:
- The project is not compulsary and can be worked out in groups of max 4 students.
- The report has to be submitted through Canvas at this link: Project Submission
- Together with the report you have to submit a statement, signed by all members of the group, certifying that this report is your own work and that all members of the group have equally contributed to the assignment. All reports will be subjected to a plagiarism check. Information on how to avoid plagiarism and on Chalmers policy on plagiarism can be found here.
- The deadline for submission is January 2nd, 2022 at 23.59. The grade on the project (max 2 points) will be communicated the following week. The bonus points of the Matlab project are valid only for the exams in 2022.
- Some help and tips on how to carry out the project are given in the lecture on November 24th. Due to the high number of students in the course, no further assistance on the project can be provided.
- The grade of the project is mostly based on the "aesthetical'' quality of the report, e.g., on having nice and informative plots. You should also write a short text explaining how the Matlab codes work, e.g., as comments within the codes themselves. An incorrect interpretation of the results will not affect the grade, so be bold in your arguments!
- You may carry out the project using another program language instead of Matlab, but remember to explain how the codes work!
Reference literature:
- Learning MATLAB, Tobin A. Driscoll. Provides a brief introduction to Matlab to the one who already knows computer programming. Available as e-book from Chalmers library.
- Physical Modeling in MATLAB 3/E, Allen B. Downey
The book is free to download from the web. The book gives an introduction for those who have not programmed before. It covers basic MATLAB programming with a focus on modeling and simulation of physical systems.
Examination
The final test comprises 30 points and the Matlab project gives max 2 more bonus points. The least number of points to pass the course is 15.
- at GU a result greater than or equal to 23 points is graded VG;
- at Chalmers a result greater than or equal to 20 points and smaller than 25 points is graded 4 and a result greater than or equal to 25 points is graded 5.
The test is divided in two parts:
The first part will be of theoretical nature and will require to state and prove two of the following theorems in the lecture notes (max. 3 points for each theorem):
- Theorem 1.2 , Theorem 2.1, Theorem 2.2, Theorem 2.4 (Step 1: the one-period model), Theorem 3.3, Theorem 5.5, Theorem 6.10, Theorem 6.14, Theorem 6.18
to provide and explain one of the following definitions in the lecture notes (max. 3 points):
- Definition 1.2, Definition 1.3, Definition 2.1, Definition 2.6, Definition 2.8, Definition 3.1, Definition 3.2, Definition 5.19, Definition 6.7, Definition 6.8, Definition 6.9, Definition 6.10
and to answer a critical thinking question similar to one of those marked with the symbol (?) in the lecture notes (max. 3 points).
The second part of the exam consists of 3 exercises (max. 18 points).
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 6.14 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. If the definition involves a mathematical formula, e.g., eq. (6.13) in Definition 6.7, then you have to explain the meaning of all mathematical symbols in this formula. You must also explain the financial meaning of the definition.
The dates and times for the exams can be found in the student portal (Links to an external site.).
Old exams
2020: January 2020, April 2020, August 2020
2021: January 2021, April 2021, August 2021
IMP! The evaluation rules for the exams in 2022 has changed. In particular each exercise now gives more points and the maximum number of points has passed from 20 to 30, see the section Examination above. Moreover in 2021 the exam was a bit different, since it was carried out from home. Specifically, in 2021 it was required to prove only one theorem, the Matlab project was considered part of the exam itself (and not a bonus assignment) and all aids were permitted. For the exam in 2022 no aids are permitted.
Examination procedures
In Chalmers Student Portal (Links to an external site.) you can read about when exams are given and what rules apply to exams at Chalmers. In addition to that, there is a schedule (Links to an external site.) when exams are given for courses at the University of Gothenburg.
Before the exam, it is important that you sign up for the examination. You sign up through Ladok (Links to an external site.).
At the exam, you should be able to show valid identification.
After the exam has been graded, you can see your results in Ladok.
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 (Links to an external site.). 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 (Links to an external site.). 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.
Course evaluation
At the beginning of the course, at least two student representatives should have been appointed to carry out the course evaluation together with the teachers. The evaluation takes place through conversations between teachers and student representatives during the course and at a meeting after the end of the course when the survey result is discussed and a report is written.
Guidelines for Course evaluation (Links to an external site.) in Chalmers student portal.