TMA285 / MMA711 Financial derivatives and partial differential equations Spring 21

Welcome to the course! 

The main topic of the course "Financial derivatives and PDE's" is the theoretical valuation of financial derivatives based on the arbitrage-free principle and using methods from stochastic calculus and partial differential equations.

This year the course is given online. The zoom link to attend the lectures will be sent to the students the week before the course starts.

Important: the lectures will be recorded and the videos will be posted on this homepage. The recording will start sometime after the beginning of the lecture and will stop sometime before the end of the lecture. During the part of the lecture which is not recorded you are free to ask questions. There will be also a Q&A session once a week (Wednesdays, 12-13, not-recorded).  Participation to the lectures and the Q&A sessions is strongly recommended, since there will be no other opportunity to ask questions.

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. 

Important: The rules for the course examination have changed substantially this year, see the section "Examination" below.


(Old News)

18th August: Important! The exam on August 27, 2021, will be an oral exam. If you want to take the oral exam, contact the examiner no later than August 25th! 

Teacher and student representatives

Teacher and examiner: Simone Calogero (

Important: I answer e-mails from Monday to Friday at office hours (9-17) and only about course administration. Questions about the course content must be asked during the lectures and the Q&A sessions.

Student representatives

Jakob Bruchhausen      
Olof Johansson    
Kubilay Muameleci     
Siddhant Som   
Elias Welander     


Basic financial concepts (PDF). Read this by yourself before the lecture on January 27th.

Stochastic Calculus, Financial Derivatives and PDE's. (PDF) Remark: Be sure to have this year version of the lecture notes!

Errata: pdf 

Additional recommended (but optional) reading:

Steven E. Shreve. Stochastic Calculus for Finance II. Continuous-Time Models (Springer)


The schedule of the course is in TimeEdit

The following schedule is approximative. 


Day Notes Content
18 Jan (15.15-17.00)
Notes  Annotes  Video1  Video2  
Probability spaces, random variables, Distribution functions (Chapters 1,2)

20 Jan (10:00-11.45)

Q&A (12-13)

Notes Annotes  Video1   Video2         
Expectation, stochastic processes, Brownian motion, quadratic variation (Chapters 2,3)
21 Jan (15.15-17.00) Notes Annotes  Video1  Video2 Conditional expectation, Martingales, Markov processes (Chapter 3)
22 Jan (15.15-17.00) Notes Annotes  Video1  Video2 Itô's integral, Itô's formula, diffusion processes (Chapter 4)
25 Jan (15.15-17.00) Notes Annotes  Video1  Video2 Exercises 2.8, 2.15, 3.3, 3.27, 4.4, 4.5, 4.6

27 Jan (10:00-11.45)

Q&A (12-13)

Notes Annotes  Video1  Video2 Diffusion processes in financial mathematics (Section 4.6) 
28 Jan (15.15-17.00) Notes Annotes  Video1  Video2 Stochastic differential equations (Section 5.1) 
29 Jan (15.15-17.00) Notes Annotes Video1  Video2 Kolmogorov PDE (Section 5.2)
1 Feb (15.15-17.00) Notes Annotes  Video1  Video2 CIR process (Section 5.3)

3 Feb (10:00-11.45)

Q&A (12-13)

Notes Annotes  Video1  Video2 Equivalent probability measures, Girsanov's Theorem (Section 4.5), arbitrage-free markets (Section 6.1)   
4 Feb (15.15-17.00) Notes Annotes  Video1  Video2 Risk-neutral pricing formula for European derivatives (Section 6.2)  
5 Feb (15.15-17.00) Notes Annotes Video1  Video2 Black-Scholes price of standard European derivatives (Section 6.3) 
8 Feb (15.15-17.00) Notes Annotes    Video1  Video2 The Asian option. Crude Monte Carlo method. Control variate Monte Carlo method (Section 6.4) 

10 Feb (10:00-11.45)

Q&A (12-13)

Notes Annotes  Video1  Video2 Local volatility models. CEV model (Section 6.6) 
11 Feb (15.15-17.00) Notes Annotes Video1  Video2 Finite different solutions of PDE's (Section 5.4) 
12 Feb (15.15-17.00) Notes Annotes Video1  Video2 Description of the projects (Appendix A) 
15 Feb (15.15-17.00) Notes Annotes  Video1 Video2 Lookback options (Section 6.5)

17 Feb (10:00-11.45)

Q&A (12-13)

Notes Annotes Video1 Video2 Stochastic volatility models. Variance swaps (Section 6.6)
18 Feb (15.15-17.00) Notes Annotes  Video1 Video2  Bonds (Section 6.7)
19 Feb (15.15-17.00) Notes Annotes Video1 Video2 Classical approach to ZCB pricing. Interest rate swaps, caps and floors (Section 6.7)
22 Feb (15.15-17.00) Notes Annotes Video1 Video2 HJM approach to ZCB pricing (Section 6.7)

24 Feb (10:00-11.45)

Q&A (12-13)

Notes Annotes Video1 Video2 Forward contracts. Forward measure (Section 6.8)
25 Feb (15.15-17.00) Notes Annotes  Video1 Video2 Futures (Section 6.8)
26 Feb (15.15-17.00) Notes Annotes  Video1 Video2  Multi-dimensional markets (Sections 6.9)
1 Mar (15.15-17.00) Notes Annotes  Video1 Video2 Multi-dimensional markets (Section 6.9)

3 Mar (10-11.45)

Q&A (12-13)

Notes Annotes  Video1 Video2 American derivatives. Perpetual American put option I (Section 6.10)
4 Mar (15.15-17.00) Annotes Video1 Video2 Perpetrual American put II (Section 6.10)
5 Mar (15.15-17.00) Review Project_hints (PDF) Review (if time permits)
8 Mar (15.15-17.00) Written exercise for the exam

Recommended exercises

All exercises whose solutions are found in Appendix B of the lecture notes are recommended. Some of these exercises will be solved in the class, but there is no specific exercise session (except on January 25th)


This year the course will be examined through a series of assignments and an oral examination. Each assignment gives a certain number of points to the overall score and the total maximum number of points to be earned is 30. The minimum number of points to pass the course is 18 and

- at GU a result greater than or equal to 25 points is graded VG;
- at Chalmers a result greater than or equal to 23 points and smaller than 27 points is graded 4 and a result greater than or equal to 27 points is graded 5.


1) There are 10 exercises in Chapters 1 through 5 of the lecture notes which are marked with the symbol (☆).    The assignment consists in finding these exercises and solve them. Awarded points: Max. 5 points. Deadline for submission: February 7th, h. 23.59. You can submit a picture of handwritten solutions, but be sure that they are clearly readable! This assignment has to be worked out individually.

2) The two projects in appendix A of the lecture notes on the Asian option and the CEV model. Awarded points: Max. 5 points for each project. Deadline for submission: March 14th, h. 23.59. This assignment can be worked out on groups of up to 3 students.  If you are looking for teammates to create a group post a message in the discussion thread here.

3) Written exam on March 8th, h. 15.15-16.45. In the last lecture of the course I will assign one exercise based on Chapter 6 of the lecture notes. Awarded points: Max. 5 points. You have to be connected via zoom with the camera on while you are working on this exercise. All aids are permitted, but you have to work individually. At 16.45 you have to stop writing the solution, take a picture of it and submit it through Canvas (the link will be posted on the news section). Submissions received after 17.00 will not be accepted.

4) Oral examination in the week 15th-19th of March. The exact schedule will be agreed upon with the students. During the oral exam, which will be done via zoom, I will ask to explain one definition and one theorem proof in the following list

Definitions: 2.17, 4.4, 4.5, 6.1, 6.2, 6.3, 6.5, 6.6, 6.9, 6.11

Theorems: 6.1, 6.2, 6.3, 6.5, 6.6, 6.9, 6.11, 6.13, 6.15, 6.17, 6.18, 6.19, 6.20, 6.25, 6.27, 6.28

Awarded points: Max. 5 points for each question.

Important: Be sure to have a copy of the lecture notes with you during the exam. You will use it to explain the steps in the proof that I ask (only the argument, not the detailed computations). Be prepared in particular to explain the financial interpretation of the definition/theorem. Active participation to the course (attending the lectures, discussions, etc.) will reduce the extension of the oral exam 


- The assignments are submitted to me by e-mail 

- None of the assignments is compulsory (not even the oral examination), but be aware that skipping assignments will reduce the total number of points that you can get

- The points awarded by the assignments 1) and 2) are valid for all exams in 2021, while those awarded by 3) and 4)  are only valid for the exam in March. For the re-exams in 2021 the grade is decided by the points in assignments 1) and 2) and by a new oral examination, which will be more extensive than the one in March.  

- Together with the projects in assignment 2) 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
















Course summary:

Date Details Due