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.).
Zoom Lectures Rules
 Each lecture is divided in two sessions of 45 minutes with a break of 15 minutes between them
 Keep the microphone mute and the video off during the whole lecture (including the break)
 You can ask questions and leave comments on the chat, but I will read them only during the break. I will answer these questions in the second part of the lecture, after the break. If more questions should arise I will try to answer them in the last 510 minutes of the lecture, or just after it ends
 Ask only questions pertinent to the lecture. Questions about the exam, course administration, etc., should be sent to me by email
 The zoom lectures room will also be open for a Q&A session every Tuesday from 12 to 13, starting from the second week of the course. During this session you will be able to unmute your microphone and switch on your video
 The zoom lectures will not be recorded
Course Responsible
Simone Calogero. Email: calogero@chalmers.se
Important. I answer emails MondayFriday from 9 am to 5 pm.
Student representatives
The following students have been appointed:
Julia Brinkhagen julia.brinkhagen@gmail.com
Marco Häyhänen marco.hayhanen@live.se
Kevin Sandberg Kevinsandberg@hotmail.se
Fabian Sivengård fabsiv@student.chalmers.se
William Wester wille.wester@hotmail.se
News
(old news)
August 18th: Very important! The exam on August 24th (8.3012.30) will be carried out from home according to the following special regulation:
The examination must be conducted individually, that is, cooperation is not allowed. Due to the changed circumstances, all aids are allowed. Checks on plagiarism will be carried out.
You have 4 hours to complete the exam. Solutions are written on paper, or digitally on a digital writing pad if you have access to it. Never write more than one task on each sheet. After 4 hours, you have 30 minutes to scan your solutions and organize and submit them. After 4 hours you are not allowed to continue solving the problems. The solutions should be submitted as a single pdf file where the pages are arranged in the order of the questions. You who have a certificate for extended time will have 6 hours to complete the exam and after that 30 minutes for you to submit according to the same instructions as above. When you submit, it will be stated that the exam is submitted late and if you have not already notified the examiner that you are entitled to extended time, you should do so afterwards.
Remarks:
 It is not possible to use this exam to improve the grade from a previous already passed exam (no "plussning" is allowed).
 Only students that are signed up for the exam are allowed to take the exam.
 The exam will be posted on this home page on Tuesday August 24th at 8.30 am in the link below (or in the right column). The link is already active (but without the exam of course).
 Information on the structure of the exam and how it will be evaluated can be found in the section Examination below.
LINK TO THE EXAM ON AUGUST 24th
Solution (pdf)
Literature
Simone Calogero: A first course in Options Pricing Theory. Errata (pdf)
Other interesting readings (optional):
 C. Borell: Introduction to the BlackScholes theory (pdf). These are the lecture notes used previously in the course. They cover most of the topics discussed in the class, but with more emphasis on the mathematical side.
 A. Pascucci, W. J. Runnggaldier: Theory and Problems for Multiperiod models. Springer (2012). This book covers several topics on the binomial model, some of which are not discussed in the course, using a more advanced mathematical language.
 J. C. Hull: Futures, Options and Other Derivatives. Person Education Limited, Essex (2018). A classical book in financial engineering with plenty of discussions and examples taken from the real world. It also discusses the binomial model and the BlackScholes model, but with less mathematical rigor.
Program
The schedule of the course is in TimeEdit.
Remarks:
 All lectures (including the exercise sessions) take place on zoom.
 In addition to the lectures there will be a Q&A session every Tuesday from 12 to 13, starting from the second week of the course.
 The day of the exercises session varies from week to week. I recommend that you try to solve the exercises by yourself before these lectures. The exercises in the book not listed in the program below are optional.
 Note carefully that there is no lecture on January! The course ends on December 18th.
 Some days before each lecture I will post a pdf file with the notes that I will use in the lecture. During the lecture I will complete these notes with some annotations and after the lecture I will post the annotated version of the notes as well. I suggest that you use the same pdf file (printed on paper or on a digital notebook) to take your notes.
 The schedule reported below is only indicative.
Lectures
Day  Time  Sections  Content 

3 Nov 
15.15

1.1 
Basic financial concepts. Long and short positions. Portfolio 
4 Nov  13.15  1.1 
Historical volatility. Options. European/American financial derivatives 
5 Nov  13.15  1.1/1.2 
Bonds. Money market. Frictionless markets. Arbitragefree principle 
6 Nov  15.15  1.2 
Qualitative properties of options. Putcall parity (Theorem 1.2(v)). Optimal exercise of American options 
********  *******  *************  ******************************************************************************************* 
10 Nov (Q&A 
15.15 1213) 
EXERCISES 
Exercises 1.9, 1.12, 1.14, 1.16, 1.17, 1.28, 1.30 
11 Nov  13.15  2.1, 2.2 
Binomial markets. Exercise 2.2. Predictable portfolio 
12 Nov  13.15  2.2 
Selffinancing portfolio. Exercise 2.4. Theorem 2.1. Exercise 2.5 
13 Nov  15.15  2.2, 2.3 
Portfolio generating a cash flow. Arbitrage portfolio. Theorem 2.4(step 1) 
********  *******  *************  ******************************************************************************************* 
17 Nov (Q&A 
15.15 1213) 
3.1 
Binomial price of European derivatives 
18 Nov  13.15  3.2 
Replicating portfolio of European derivatives on binomial markets. Theorem 3.3. Exercise 3.11 
19 Nov  13.15  EXERCISES 
Exercises 3.3, 3.23, 3.6, 3.7, 3.8 
20 Nov  15.15  4.1, 4.2 
Binomial price of American derivatives. Optimal exercise of American put options 
********  *******  *************  ******************************************************************************************* 
24 Nov (Q&A 
15.15 1213) 
4.3 
Hedging portfolio of American derivatives. Cash flow. Exercises 4.4, 4.5, 4.12 
25 Nov  13.15  2.4, 3.3, 4.4 
Computation of the binomial price of European/American derivatives with Matlab 
26 Nov  13.15  5.1 
Finite probability spaces. Random variables. Independence 
27 Nov  15.15  5.1 
Expectation and conditional expectation. Stochastic processes. Martingales 
********  *******  *************  ******************************************************************************************* 
1 Dec (Q&A 
15.15 1213) 
5.2 
Applications of probability theory to the binomial model. Theorem 5.4 
2 Dec  13.15  EXERCISES 
Exercises 5.22, 5.23, 5.32, 5.33, 5.34 
3 Dec  13.15  6.1 
General probability spaces. Brownian motion. Girsanov theorem 
4 Dec  15.15  6.2, 6.3 
BlackScholes markets. BlackScholes price of standard European derivatives 
********  *******  *************  ******************************************************************************************* 
8 Dec (Q&A 
15.15 1213) 
6.4 
BlackScholes price of European call and put options. Theorem 6.14. Implied volatility. Implied volatility curves 
9 Dec  13.15  EXERCISES 
Exercises 6.6, 6.7, 6.9, 6.12, 6.13, 6.15, 6.17 
10 Dec  13.15  6.6 
The Asian option. Exercises 6.23, 6.24. Monte Carlo method 
11 Dec  15.15  6.7 
Standard European derivatives on a dividend paying stock. Exercise 6.29 
********  *******  *************  ******************************************************************************************* 
15 Dec (Q&A 
15.15 1213) 
6.9 
Introduction to bonds valuation. Exercise 6.38 
16 Dec  13.15 
Review and exercises (1.10, 1.13, 1.15, 1.19, 1.25, 1.26, 1.27) (I will present the solution of the underlined exercises and help you to do the others) Notes Annotes (complete) 

17 Dec 
13.15 
Review and exercises (3.4, 3.24, 4.6, 6.14, 6.16, 6.41, 6.42) (I will present the solution of the underlined exercises and help you to do the others) Notes Annotes (complete) 

18 Dec 
15.15 
Project Assistance  
5 Jan 
1213 
Q&A session 
Matlab Project
Choose one of the following projects:
 American put options (pdf)
 Call options on dividend paying stocks (pdf)
 Compound options (pdf)
 Loopback options (pdf)
Remarks:
 The projects can be worked out in groups of max 4 students. If you are looking for teammates to create a group post a message in the discussion thread here
 The report has to be submitted in PDF form by email to calogero@chalmers.se. Use OPTIONS 2021 as subject and include all members of the group as recipients.
 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 3rd, 2021 at 23.59. The grade on the project (max 2 points) will be communicated on January 7th. The points of the Matlab project are valid only for the exams in 2021. If you take the exam later you will have to redo the project
 The grade of the project is mostly based on the ``aesthetical'' quality of the report, e.g., on having nice and informative plots (remember to specify labels for the axes). 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 ebook 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
This year the examination of the course will be slightly different than it was last year. In particular the 2 points for the Matlab project will be counted as normal points, and not as bonus points, at the exam. The Matlab project is notcompulsory (as last year), but be aware that if you do not submit the Matlab project the maximum total number of points that you can get at the final exam is 18/20.
The final test comprises 18 points and the Matlab project gives max 2 more points. The least number of points to pass the course is 10.
 at GU a result greater than or equal to 15 points is graded VG;
 at Chalmers a result greater than or equal to 13 points and smaller than 16 points is graded 4 and a result greater than or equal to 16 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 one of the following theorems in the lecture notes (max. 2 points):
 Theorem 1.2 (v) (put call parity), Theorem 2.1, Theorem 2.4 (Step 1: the oneperiod model), Theorem 3.3, Theorem 5.4, Theorem 6.14
to provide and explain one of the following definitions in the lecture notes (max. 2 points):
 Definition 1.2, Definition 2.1, Definition 2.5, Definition 2.7, Definition 3.1, Definition 3.2, Definition 6.8, Definition 6.9, Definition 6.10
and to answer a critical thinking question similar (perhaps even identical) to one of those marked with the symbol (?) in the lecture notes (max. 2 points).
The second part of the exam consists of 3 exercises (max. 12 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. (6.18) in Definition 6.8, then you have to explain the meaning of all mathematical symbols in this formula. You must also explain the financial meaning of the definition.
(iv) The exam in January will be (most likely) carried out from home. Some additional special rules apply to this exam, which will be reported on the news section in due time (at least 10 days before the exam). In particular, when the exam is carried out from home the number of points assigned to the proof of the theorem will be based on the amount of details provided in the solution. Use your own words and explain each step in the proof, in more details than in the lecture notes. No point will be awarded for writing "just" what is written in the lecture notes.
The dates and times for the exams can be found in the student portal (Links to an external site.).
Old exams
Last year exams: January 2020, April 2020, August 2020
Remarks:
 all exercises from older exams can be found in the lecture notes.
 IMP! This year the proof of the theorems gives max 2 points (and not 4 as in 2020). The maximum number of points that you can get in the final exam is 18, to which the points on the Matlab project will be added (max 2 points)
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.