TMA285/MMA711 Financial derivatives and partial differential equations
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.
News
2020-01-16: The text "Basic financial concepts" has been added. You should go through this text by yourself before the beginning of the second week of the course.
2020-01-16: Lecture notes added. Note that the first 4 chapters are part of the pre-requisites for this course. These chapters will be reviewed in the first two weeks of the course, but it is up to you to read the details in the notes (which you must do if you don't have the required background in stochastic calculus)
2020-02-12: New project on Lookback options added, see Literature
2020-02-28: Project submission is now changed to ``e-mail submission"
2020-03-06: IMP! The following theorems have been removed from the list of the theorems that you have to know for the exam, as we did not have time to go through them in the class:
Theorem 6.8, Theorem 6.29, Theorem 6.30
2020-03-06: There are some typos in the notes, see here
2020-03-06: IMP! We will have an extra lecture on Monday 9th March, 15.15-17 in MVF21. During this lecture I will solve a few exercise from Chapter 6.
2020-03-13: VERY IMP! As you probably heard by now you will take the exam next Tuesday from home. This examination is subject to the following special regulation. See below for an English translation AND for some additional instructions that will apply to the exam of this specific course.
” Tentamen ska genomföras enskilt, det vill säga samarbete är inte tillåtet. På grund av de förändrade omständigheterna är alla hjälpmedel tillåtna oavsett vad som står på tentamenstesen. Plagiatkontroll kommer att genomföras.
Du har 4 timmar på dig att genomföra tentamen. Lösningar skrivs på papper, alternativt digitalt på en skrivplatta om du har tillgång till det. Behandla aldrig mer än en uppgift på varje blad. Efter 4 timmar har du 30 minuter på dig att scanna/fotografera dina lösningar och organisera och lämna in dina lösningar enligt ett av följande alternativ i prioritetsordning:
- En enda pdf-fil där sidorna är ordnade så att uppgifterna kommer i nummerordning
- Bildfiler (jpg eller png) eller pdf-filer där varje fil innehåller lösning till bara en uppgift och som är namngivna enligt "Uppgift 1", alternativt "Uppgift 1 sida 1", Uppgift 1 sida 2" etc om det finns flera sidor till en uppgift.
Du som har intyg för förlängd tid kommer att ha 6 timmar på dig att genomföra tentamen och efter det 30 minuter på dig för inlämning enligt samma instruktioner som ovan. När du lämnar in kommer det att stå att tentan är inlämnat sent och om du inte redan har anmält till examinator att du har rätt till förlängd tid ska du göra det i efterhand.”
ENGLISH:
The exam is to be carried out individually, ie., collaboration is not allowed. Due to the current circumstances, all examination aids are allowed regardless of what is written on the exam. Control for plagiarism will be carried out.
You have 4 hours to write the exam. The solutions should be written on paper or on a digital notebook if you have access to one. No more that one task or exercise should appear in the same page. After 4 hours you have 30 minutes to scan/take a picture of your solutions and organise them according to one of the following ways:
1) (preferred) A single pdf file where the pages are ordered so that all solutions appear in numerical order
2) Image files (jpg or png) or pdf files where each file contains the solution of just one exercise and which are named according to "Task 1", or "Task 1 page 1", "Task 1 page 2", etc., if the solution requires several pages
If you have a certificate for extended time then you will have 6 hours to write the exam and 30 minutes after that to organise your solutions as described above. When you submit your exam it will be marked as "late submission"and if you do not have already informed the examiner that you have right to extended time then you should do it upon submission
SOME EXTRA INFORMATION:
The exam will be posted on this home page on Tuesday March 17 at 8.30 am, see the link below.
The exam has to be submitted, in the way described previously, using the link in the right column. YOU MUST BE LOGGED IN TO CANVAS TO SEE THE LINK!
Since the first task of the exam requires to prove theorems that you can find in the lecture notes, the number of points assigned to this task 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 by writing "just" what is written in the lecture notes. See this example (pdf)
LINK TO THE EXAM ON MARCH 17th: PDF (posted on March 17th, h. 8.29)
For questions on the exam you can reach me at +46 (0)31 772 35 62
2020-05-28: VERY IMP! Oral exam on June 9th
2020-06-03: An e-mail has been sent to the students registered to the exam on June 9th. In the e-mail there is a link to book a time slot for the oral exam.
2020-08-21: VERY IMP! Oral Exam on August 28th! The re-exam in August will be an online oral exam. Register to the course as usual and after the registration period is closed I will send an e-mail to the registered students to schedule a Zoom meeting for the exam on August the 28th. You should have the lecture notes with you during the exam, as questions will target precise results discussed in the notes.
Teacher and student representatives
Teacher and examiner: Simone Calogero (calogero@chalmers.se)
Student representatives
John Hang hang@student.chalmers.se
Filip Johansson Gunnarsson gufilip@student.chalmers.se
Astrid Liljenberg astridl@student.chalmers.se
Sara Nordin Hällgren sarhal@student.chalmers.se
Aditya Singh adityasingh.asm@gmail.com
Literature
Basic financial concepts (PDF).
Stochastic Calculus, Financial Derivatives and PDE's (pdf). Remark: Be sure to have this year version of the lecture notes!
Lookback options (pdf)
Program
The schedule of the course is in TimeEdit.
The following schedule is approximative. Check in TimeEdit for possible room changes.
Lectures
| Day | Room | Content |
|---|---|---|
| 20 Jan (15.15-17.00) | MVF21 | Probability spaces, random variables, stochastic processes (Chapters 1,2) |
| 22 Jan (10:00-11.45) | Pascal | Expectation, quadratic variation, conditional expectation, martingales, Markov processes (Chapter 3) |
| 23 Jan (15.15-17.00) | Euler | Itô's integral, Itô's formula, diffusion processes, Girsanov's theorem (Chapter 4) |
| 24 Jan (15.15-17.00) | MVF26 | Exercises |
| 27 Jan (15.15-17.00) | MVF21 | Diffusion processes in financial mathematics (Section 4.6) |
| 29 Jan (10:00-11.45) | Pascal | Stochastic differential equations (Section 5.1) |
| 30 Jan (15.15-17.00) | Euler | Kolmogorov PDE (Section 5.2) |
| 31 Jan (15.15-17.00) | MVF26 | CIR process (Section 5.3) |
| 3 Feb (15.15-17.00) | MVF21 | Arbitrage-free market. Risk-neutral pricing formula for European derivatives (Sections 6.1, 6.2) |
| 5 Feb (10:00-11.45) | Pascal | Black-Scholes price of standard European derivatives (Section 6.3) |
| 6 Feb (15.15-17.00) | Euler | The Asian option. Crude Monte Carlo method. Control variate Monte Carlo method (Section 6.4) |
| 7 Feb (15.15-17.00) | MVF26 | Local volatility models. CEV model (Section 6.5) |
| 10 Feb (15.15-17.00) | MVF21 | Finite different solutions of PDE's (Section 5.1) |
| 12 Feb (10:00-11.45) | Pascal | Description of the projects (Appendix A) |
| 13 Feb (15.15-17.00) | Euler | Stochastic volatility models. Variance swaps (Section 6.5) |
| 14 Feb (15.15-17.00) | MVF26 | Lookback options |
| 17 Feb (15.15-17.00) | My office | Project assistance |
| 19 Feb (10:00-11.45) | My office | Project assistance |
| 20 Feb (15.15-17.00) | My office | Project assistance |
| 21 Feb (15.15-17.00) | My office | Project assistance |
| 24 Feb (15.15-17.00) | MVF21 | Bonds (Section 6.6) |
| 26 Feb (10:00-11.45) | Pascal | Classical approach to ZCB pricing. Interest rate swaps, caps and floors (Section 6.6) |
| 27 Feb (15.15-17.00) | Euler | HJM approach to ZCB pricing (Section 6.6) |
| 28 Feb (15.15-17.00) | MVF26 | Forward contracts. Forward measure (Section 6.7) |
| 2 Mar (15.15-17.00) | MVF21 | Futures. Multi-dimensional markets (Sections 6.7, 6.8) |
| 4 Mar (15.15-17.00) | Pascal | Multi-dimensional markets (Section 6.8) |
| 5 Mar (15.15-17.00) | Euler | American derivatives. Perpetual American put option (Section 6.9) |
| 6 Mar (15.15-17.00) | MVF26 | American calls on dividend-paying stocks (Section 6.9) |
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 exercises session.
Assignments
The are two types of assignments:
- Exercises: There are 10 exercises in the lecture notes which are marked with the symbol (☆). The assignment consists in finding these exercises and solve them. Bonus points: Max. 2 points. Deadline for submission: February 7th, 5pm. The written exercises must be handed in on paper (no e-mail submission!)
- Matlab project: One of the two projects in Appendix A, namely the Asian Option (app. A.1) or the CEV model (app. A.2), or the project on Lookback options (see Literature). Bonus points: Max. 3 points. Deadline for submission: March 2nd. The project must be submitted by e-mail to the examiner. Add all members of the group as recipients. One submission per group.
Remarks:
(1) The assignments are not compulsory, although they are strongly recommended
(2) The assignments can be carried out in groups of max. 3 students
(3) On week 5th of the course there will be no lecture, so that you can focus on writing your project. I will be in my office during the lectures hours to answer your questions and help you with the project
Information about the exam
The first exam is on March 17th, 2020
The test comprises 15 points and to pass at least 6 points are required
- at GU a result greater than or equal to 11 points is graded VG;
- at Chalmers a result greater than or equal to 9 points and smaller than 12 points is graded 4 and a result greater than or equal to 12 points is graded 5.
The assigments give max. 5 points
The test is divided in two parts. The first part will be of theoretical nature and will require to prove one or more of the following theorems (max. 4 points) :
Theorem 6.1, Theorem 6.2, Theorem 6.3, Theorem 6.5, Theorem 6.6, Theorem 6.9, Theorem 6.11, Theorem 6.14, Theorem 6.15, Theorem 6.16, Theorem 6.17, Theorem 6.18, Theorem 6.26 (i.e., 6.27)
and to provide and explain one of the following definitions (max. 1 point):
Definition 6.1, Definition 6.2, Definition 6.3, Definition 6.5, Definition 6.6, Definition 6.9, Definition 6.11,
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 fair swap rate of interest rate swaps".
(iii) The explanation of the definition need not be the same as in the lecture notes. You can use your own intuition.
The second part consists of exercises; one of the exercises will be very similar (perhaps even identical) to one of those in Chapter 6 of the lecture notes.
Some old exams can be found here: Old_Exams.zip
Course summary:
| Date | Details | Due |
|---|---|---|