Course syllabus
The course Options and Mathematics deals with the arbitragefree 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. Email: maria.roginskaya@chalmers.se
Important. I answer emails with up to 4 working days delay.
Carl Lindberg is responsible for MatLab project.
Student representatives
The following students have been appointed:
GU:
Chalmers:
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 

4 Nov KC 
10.00

1.12 (pp.15, 15) 
Basic financial concepts. Examples of financial assets. Stocks and bonds. Portfolio. Long and short positions. 
6 Nov HC3 
13.15

1.25

Returns. Historical volatility. Options. European/American financial derivatives. Calls/puts and nonstandard. Forwards, futures, and swaps. Money market. 
7 Nov HC2 
13.15

1.6, 2.1

Frictionless markets. Arbitragefree principle. Price of risk. 
11 Nov SBH4 
10.00  2.2 
Qualitative properties of options. Putcall parity. Theorem 2.3. Some of exercises 1.12.23.

13 Nov Pascal 
13.15 
2.3 
Optimal exercise of American options. Description of bonus project. Definition 2.5, Definition 2.6 Some of exercises 2.242.36 
14 Nov FB 
13.15  3.13 
Binomial markets. Logreturn and volatility. Predictable portfolio. Selffinancing portfolio. Exercise 3.41 
15 Nov Pascal 
8.30 
review so far Some of Exercises 3.3743 

18 Nov SBH4 
10.00  3.45 
Portfolio generating a cash flow. Arbitrage portfolio. Exercises 44, 45 
20 Nov Pascal 
13.15 
4.12 
Hedging/replicating portfolio Definition 4.1. Definition 4.3 Replicating portfolio of European derivatives on binomial markets. Theorem 4.10. Some of exercises 4658 
21 Nov Pascal 
13.15  5.12 
Binomial price of American derivatives. Optimal exercise of American put options. 
22 Nov Pascal 
8.30  Exercises 74 and 90  
25 Nov SBH5 
10.00  5.3 
Replicating portfolio of American derivatives. Cash flow. Exercises 6486 
27 Nov Pascal 
13.15 
3.6, 4.3, 5.4 
Computation of the binomial price of European/American derivatives with Python. Summary so far 
28 Nov Pascal 
13.15  6.1 
Finite probability spaces. Random variables. Independence Expectation and conditional expectation. 
29 Nov Euler 
8.30 
Exercises 113 and 116 

2 Dec SBH4 
10.00  6 
Stochastic processes. Martingales Applications of probability theory to the binomial model Theorems 6.36, 6.42 
4 Dec Pascal 
13.15 
6.2 (pp.137138) 8 (pp.165180) 
Riskneutral price of American derivatives Definition 6.44 General probability spaces. Brownian motion. 
5 Dec Pascal 
13.15  8.12 (pp.180188) 
Girsanov theorem. BlackScholes markets. Definition 8.19, 
6 Dec Euler 
8.30  Exercises 122 and 193  
9 Dec SBH4 
10.00  8.34 (pp.188194) 
Theorem 8.20 BlackScholes price of standard European derivatives. Definition 8.23, Theorem 8.24 
11 Dec Pascal 
13.15 
8.45 (pp.194199) 
BlackScholes price of European call and put options. Theorem 8.25 Greeks. Implied volatility. Definition 8.30. Implied volatility curves. 
12 Dec Pascal 
13.15  8.78 
The Asian option. Monte Carlo method. Standard European derivatives on a dividend paying stock. Theorem 8.34 
13 Dec Pascal 
8.30  Exercises 227 and 228  
16 Dec SBH4 
10.00  8.10 
Introduction to bonds valuation. Definition 8.40. 
18 Dec Pascal 
13.15 
Review  
19 Dec Pascal 
13.15  Old exams  
20 Dec Pascal 
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 oneperiod 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 logreturns and volatility as given by formula 3.3 (2.1), Definition of selffinancing 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 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. 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 reexamination 2023.
Suggested solution to second reexamination 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 reexamination:
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.