Kursöversikt
The main topic of the course "Financial derivatves 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 page contains the program of the course: lectures, exercise sessions and computer labs. Other information, such as learning outcomes, teachers, literature and examination, are in a separate course PM.
Teacher and student representatives
Teacher and examiner: Moritz Schauer (smoritz@chalmers.se)
Student representatives
Literature
Basic financial concepts (PDF Download PDF ). Read this by yourself during the first week.
Stochastic Calculus, Financial Derivatives and PDE's. (PDF Download PDF)
Additional recommended (optional) reading:
Steven E. Shreve. Stochastic Calculus for Finance II. Continuous-Time Models (Springer Links to an external site.).
Errata: Errata6-1.pdf Download Errata6-1.pdf
Program
The schedule of the course is in TimeEdit Links to an external site.
The week plan is preliminary and might be changed during the course.
Lectures
| Week | Notes | Sections | Old Slides |
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| 3 |
tma285recap1.pdf Download tma285recap1.pdf | Introduction. Probability spaces, random variables, distribution functions (Ch 1, 2) | TMA285_MMA711_230117.pdf Download TMA285_MMA711_230117.pdf |
| tma285recap23.pdf Download tma285recap23.pdf |
Lebesgue integral, expectation. Change of measure. | TMA285_MMA711_230118.pdf Download TMA285_MMA711_230118.pdf | |
| Conditional expectation. Stochastic processes. Brownian motion, quadratic variation (Ch 2, 3) | TMA285_MMA711_230119.pdf Download TMA285_MMA711_230119.pdf | ||
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Brownian motion. Martingales. Markov processes (Ch 3) |
TMA285_MMA711_230120.pdf Download TMA285_MMA711_230120.pdf | ||
| 4 |
tma285recap-6.pdf Download tma285recap-6.pdf |
Itô's integral (Ch 4) |
TMA285_MMA711_230124.pdf Download TMA285_MMA711_230124.pdf |
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Itô's formula. Diffusion processes(Sec 4.6) |
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Diffusion processes. Girsanov's Theorem (Sec 4.5) |
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Exercises 2.8, 2.15, 3.3, 3.27, 4.4, 4.5, 4.6 |
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| 5 | tma285recap-10.pdf Download tma285recap-10.pdf |
Stochastic differential equations Kolmogorov PDE (Sec 5.2) |
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| tma285recap-12.pdf Download tma285recap-12.pdf |
Kolmogorov PDE, Markov property, and transition density (Sec 5.2) Exercise 3.33, 5.7 |
TMA285_MMA711_230201.pdf Download TMA285_MMA711_230201.pdf | |
| tma285recap-14.pdf Download tma285recap-14.pdf |
Arbitrage-free markets (Sec 6.1) Risk-neutral formula in discrete case |
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| tma285recap-17.pdf Download tma285recap-17.pdf | Exercises |
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| 6 | tma285recap-18.pdf Download tma285recap-18.pdf |
Risk-neutral pricing formula for European derivatives (Sec 6.2) |
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tma285recap-21-1.pdf Download tma285recap-21-1.pdf Gist: Julia program for Black-Scholes Links to an external site. |
Black-Scholes price of standard European derivatives (Sec 6.3) |
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Black-Scholes price of standard European derivatives (Sec 6.3) |
No notes because I followed Sec 6.3 | ||
| tma285recap-27.pdf Download tma285recap-27.pdf |
Exercises: (Instead of The Asian option. (Sec 6.4). Finite difference solutions of PDE's (Sec 5.4)) |
TMA285_MMA711_230210 Download TMA285_MMA711_230210.pdf Download .pdf |
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| 7 | Julia Monte Carlo programs Links to an external site. |
Finite difference for heat equation. Crude Monte Carlo method. Control variate Monte Carlo method (Sec 6.4) |
Compendium page 134-137 |
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Local volatility models. CEV model (Sec 6.6) |
TMA285_MMA711_230215 Download TMA285_MMA711_230215.pdf Download .pdf |
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Local volatility models (Sec 6.6) |
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Stochastic volatility models. |
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| 8 |
Work on projects |
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Work on projects |
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Work on projects |
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Variance swaps (Sec 6.6) |
See compendium, Sec 6.6 |
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| 9 |
Yield curve. Classical approach to ZCB pricing. |
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| tma285recap-31.pdf | Classical approach to ZCB pricing. HJM model (Sec 6.7). | ||
| Interest rate swaps (Sec 6.7) (Sec 6.8) |
TMA285_MMA711_230302 Download TMA285_MMA711_230302.pdf Download .pdf |
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| tma285recap-35.pdf | Forwards (Sec 6.8) | TMA285_MMA711_forwards.pdf Download TMA285_MMA711_forwards.pdf | |
| 10 |
Futures (Sec 6.8). |
TMA285_MMA711_futures.pdf Download TMA285_MMA711_futures.pdf |
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| Repetition/questions |
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AI Tools / Användning av AI-verktyg
Under studierna är det fritt fram att använda AI-verktyg som stöd för inlärningen. Betygen fastställs genom att bedöma deltagarnas individuella prestationer. På tentamen är det inte tillåtet att använda någon form av AI-verktyg. / During the course, students are free to use AI tools to support their learning. Grades are determined by assessing the individual performance of each participant. The use of any form of AI tool is not permitted during the exam.
Examination
This course will be examined through a series of assignments and a written exam.
The minimum number of points to pass the course is 15p, of which 12p come from the written exam and 3p from the assignments.
- at GU a result greater than or equal to 24 points is graded VG;
- at Chalmers a result greater than or equal to 23 points and smaller than 25 points is graded 4 and a result greater than or equal to 25 points is graded 5.
Note that the written exam gives at most 20 points, and so is not sufficient to get VG or 5.
Assignments:
1) There are 10 exercises in Chapters 1 through 5 of the lecture notes which are marked with the symbol (☆). Awarded points: Max. 5 points. Deadline for submission: February 12th, 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 and submitted via canvas. This assignment is not compulsory.
2) The two projects in appendix A of the lecture notes on the Asian option and the CEV model. You can use either Python (preferable) or Matlab for the computer part. Awarded points: Max. 5 points for each project. Deadline for submission: March 6th, 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. The assignments should be submitted via canvas. This assignment is not compulsory. Note: Projektarbetet ska visa på ett genuint engagemang i ämnet. God kommunikation bygger på precisa argument, korrekt användning av facktermer och ett tydligt sammanhang. Fokusera på tydlighet och djup, och se till att din text speglar din förståelse och ditt kritiska tänkande snarare än ytlig finess / The project work should demonstrate a genuine commitment to the subject. Good communication is based on precise arguments, correct use of technical terms, and a clear context. Focus on clarity and depth, and ensure that your text reflects your understanding and critical thinking rather than superficial finesse.
Together with the projects, 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
3) Written exam in MARCH. Awarded points: Max. 20 points. The written exam is compulsory and no aids are allowed. The exam will include practical exercises as well as theoretical questions. The list of definitions and theorems:
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
Additional material from the lectures: Estimate volatility, Monte Carlo for the Greeks
Some old exams can be found here: Old_Exams.zip
Exam March 2022 Download Exam March 2022
Re-exam June 2022 Download Re-exam June 2022
Re-exam Aug 2022 Download Re-exam Aug 2022
Re-exam June 2025 Download Re-exam June 2025
Kurssammanfattning:
| Datum | Information | Sista inlämningsdatum |
|---|---|---|