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.).
Solutions for reexamination on 28/8.
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:
GU:
Sebastian Persson
Joel Dalebjörk
Chalmers:
Jacob Burman
Tharinrath Jatupattrapiboorn
Maria Jidah
Danai Kiewwan
Meixi Lin
News
List of formulas useful to know by heart to the exam.
Literature
Simone Calogero: A first course in Options Pricing Theory
Program
The schedule of the course is in TimeEdit.
Preliminary program of lectures
| Day | Time | Sections | Content |
|---|---|---|---|
|
30 Oct KC |
10.15
|
1.1-2 (pp.1-5, 15) |
Basic financial concepts. Examples of financial assets. Stocks and bonds. Portfolio. Long and short positions. |
|
1 Nov HC3 |
13.15
|
1.2-5
|
Returns. Historical volatility. Options. European/American financial derivatives. Calls/puts and non-standard. Forwards, futures, and swaps. Money market. |
|
2 Nov Pascal |
13.15
|
1.6, 2.1
|
Frictionless markets. Arbitrage-free principle. Price of risk. |
|
6 Nov SB-H4 |
10.00 | 2.2 |
Qualitative properties of options. Put-call parity. Theorem 2.3. Some of exercises 1.1-2.23.
|
|
8 Nov Pascal |
13.15 |
2.3 |
Optimal exercise of American options. Description of bonus project. Definition 2.5, Definition 2.6 Some of exercises 2.24-2.36 |
|
9 Nov Pascal |
13.15 | 3.1-3 |
Binomial markets. Log-return and volatility. Predictable portfolio. Self-financing portfolio. Exercise 3.41 |
|
10 Nov Pascal |
8.30 |
review so far Some of Exercises 3.37-43 |
|
|
13 Nov SB-H4 |
10.00 | 3.4-5 |
Portfolio generating a cash flow. Arbitrage portfolio. Exercises 44, 45 |
|
15 Nov Pascal |
13.15 |
4.1-2 |
Hedging/replicating portfolio Definition 4.1. Definition 4.3 Replicating portfolio of European derivatives on binomial markets. Theorem 4.10. Some of exercises 46-58 |
|
16 Nov Pascal |
13.15 | 5.1-2 |
Binomial price of American derivatives. Optimal exercise of American put options. |
|
17 Nov Pascal |
8.30 | Exercises 74 and 90 | |
|
20 Nov SB-H5 |
10.00 | 5.3 |
Replicating portfolio of American derivatives. Cash flow. Exercises 64-86 |
|
22 Nov Pascal |
13.15 |
3.6, 4.3, 5.4 |
Computation of the binomial price of European/American derivatives with Python. Summary so far |
|
23 Nov Pascal |
13.15 | 6.1 |
Finite probability spaces. Random variables. Independence Expectation and conditional expectation. |
|
24 Nov Pascal |
8.30 |
Exercises 113 and 116 |
|
|
27 Nov SB-H4 |
10.00 | 6 |
Stochastic processes. Martingales Applications of probability theory to the binomial model Theorems 6.36, 6.42 |
|
29 Nov Pascal |
13.15 |
6.2 (pp.137-138) 8 (pp.165-180) |
Risk-neutral price of American derivatives Definition 6.44 General probability spaces. Brownian motion. |
|
30 Nov Pascal |
13.15 | 8.1-2 (pp.180-188) |
Girsanov theorem. Black-Scholes markets. Definition 8.19, |
|
1 Dec Pascal |
8.30 | Exercises 122 and 193 | |
|
4 Dec SB-H4 |
10.00 | 8.3-4 (pp.188-194) |
Theorem 8.20 Black-Scholes price of standard European derivatives. Definition 8.23, Theorem 8.24 |
|
6 Dec Pascal |
13.15 |
8.4-5 (pp.194-199) |
Black-Scholes price of European call and put options. Theorem 8.25 Greeks. Implied volatility. Definition 8.30. Implied volatility curves. |
|
7 Dec Pascal |
13.15 | 8.7-8 |
The Asian option. Monte Carlo method. Standard European derivatives on a dividend paying stock. Theorem 8.34 |
|
8 Dec Pascal |
8.30 | Exercises 227 and 228 | |
|
11 Dec SB-H4 |
10.00 | 8.10 |
Introduction to bonds valuation. Definition 8.40. |
|
13 Dec Pascal |
13.15 |
Review | |
|
14 Dec Pascal |
13.15 | Old exams | |
|
15 Dec |
8.30 |
Old exams
|
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 2.3(1.2) , Theorem 3.16 (2.1), Theorem 3.17 (2.2), Theorem 3.23 (2.4) (Step 1: the one-period model), Theorem 4.10 (3.3), Theorem 6.36 (5.5), Theorem 8.20 (6.10), Theorem 8.25 (6.14), Theorem 8.34 (6.18)
to provide and explain one of the following definitions (max. 3 points):
- Definition of arbitrage: either 2.1 (1.2) or the one given at lecture, Definition 2.6 (1.3), Definition of annualized min of log-returns and volatility as given by formula 3.3 (2.1), Definition of self-financing portfolio process 3.13 (2.6), Definition 3.22 (2.8), Definition 4.1 (3.1), Definition 4.3 (3.2), Definition 6.44 (5.19), Definition 8.19 (6.7), Definition 8.23 (6.8), Definition 8.30 (6.9), Definition 8.40 (6.10)
and to answer a critical thinking question similar to one of those marked with the symbol (?) in the book (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
Suggested solution to examen 2023.
Suggested solution to first re-examination 2023.
Suggested solution to second re-examination 2023.
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.