MVE095 / MMG810 Options and mathematics
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
Maria Roginskaya. E-mail: maria.roginskaya@chalmers.se
Important. I answer e-mails with up to 4 working days delay.
Carl Lindberg is responsible for MatLab project.
Student representatives
The following students have been appointed:
(to be appointed)
News
Suggested solution to examen.
Suggested solution to first re-examination.
Suggested solution to second re-examination.
Literature
Simone Calogero: A first course in Options Pricing Theory (pdf) Errata (pdf)
Program
The schedule of the course is in TimeEdit.
Lectures
| Day | Time | Sections | Content |
|---|---|---|---|
|
31 Oct Euler |
10.15
|
1.1 (pp.1-11) |
Basic financial concepts. Long and short positions. Portfolio. |
|
2 Nov Pascal |
13.15
|
1.1 (pp.11-20)
|
Historical volatility. Options. European/American financial derivatives.
|
|
3 Nov Pascal |
13.15
|
1.1/1.2 (pp.21-26)
|
Bonds. Money market. Frictionless markets. Arbitrage-free principle
|
| 7 Nov | 8.15 | 1.2 (pp.27-30) |
Qualitative properties of options. Put-call parity. Theorem 1.2. Description of bonus project. Exercises 1.9, 1.10, 1.1, 1.14, 1.16 (1.17), 1.18, 1.19
|
|
7 Nov |
10.15 |
1.2 (pp. 31-33) |
Optimal exercise of American options. Definition 1.2, Definition 1.3 Exercise 1.22
|
| 9 Nov | 13.15 | 2.1, 2.2 (pp.37-45) |
Binomial markets. Definition 2.1. Exercise 2.2 Predictable portfolio. Self-financing portfolio. Definition 2.6. Exercise 2.4. |
| 10 Nov | 13.15 | 2.2, 2.3 (pp.45-51) |
Theorem 2.1, Theorem 2.2. Exercise 2.5 Portfolio generating a cash flow. Arbitrage portfolio. Definition 2.8. Theorem 2.4 |
| 14 Nov | 8.15 | time for questions | |
|
14 Nov |
10.15 |
3.1,3.2 |
Binomial price of European derivatives. Definition 3.1. Definition 3.2 Replicating portfolio of European derivatives on binomial markets. Theorem 3.3. |
| 16 Nov | 13.15 | EXERCISES | Exercise 3.3/3.22, 3.4, 3.6, 3.7, 3.8, 3.11 |
| 17 Nov | 13.15 | 4.1, 4.2 |
Binomial price of American derivatives. Optimal exercise of American put options.
|
| 21 Nov | 8.15 | time for questions | |
|
21 Nov |
10.15 |
4.3 |
Replicating portfolio of American derivatives. Cash flow. Exercises 4.5, 4.6 |
| 23 Nov | 13.15 | 2.4, 3.3, 4.4 |
Computation of the binomial price of European/American derivatives with Matlab. |
| 24 Nov | 13.15 | 5.1 |
Finite probability spaces. Random variables. Independence Expectation and conditional expectation. Stochastic processes. Martingales |
| 28 Nov | 8.15 |
time for questions |
|
|
28 Nov |
10.15 |
5.2 (pp.124-132) |
Applications of probability theory to the binomial model. Theorem 5.5. Definition 5.19 Exercises 5.24, 5.25, 5.38 |
| 30 Nov | 13.15 | 6.1 |
Exercise 5.26 General probability spaces. Brownian motion. Exercise 6.7 Girsanov theorem. |
| 1 Dec | 13.15 | 6.1,6.2 |
Girsanov theorem. Black-Scholes markets. Definition 6.7. Theorem 6.10 |
| 5 Dec | 8.15 | time for questons | |
|
5 Dec |
10.15 |
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
|
| 7 Dec | 13.15 | 6.4 |
Greeks. Implied volatility. Implied volatility curves. Definition 6.9
|
| 8 Dec | 13.15 | 6.6 | The Asian option. Exercises 6.22, 6.23, 6.24. Monte Carlo method. |
| 12 Dec | 8.15 | EXERCISES | time for questions, Exercises 6.12, 6.13, 6.16, 6.17 |
|
12 Dec |
10.15 |
6.7 |
Standard European derivatives on a dividend paying stock. Theorem 6.18. Exercise 6.26 |
| 15 Dec | 13.15 | 6.9 |
Introduction to bonds valuation. Definition 6.10. Exercise 6.35
|
|
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, 3.23, 4.8, 4.10, 5.39, 6.36, 6.37, 6.39, 6.40, 6.41)
|
Matlab projekt
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
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.