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

MPENM   sandy.abonaser96@gmail.com      Sandy Abonaser
MPENM   dumovskizlatko@proton.me        Zlatko Dumovski
MPENM   philipgifting02@gmail.com       Philip Gifting
MPCAS   marcus.olsson.2002@gmail.com    Marcus Olsson
MPCAS   erik.o.j.steen@gmail.com        Erik Steen

 

Literature

Basic financial concepts (PDF ). Read this by yourself during the first week.

Stochastic Calculus, Financial Derivatives and PDE's. (PDF) 

Additional recommended (optional) reading:

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

 

Program

The schedule of the course is in TimeEdit

The week plan is preliminary and might be changed during the course.

Lectures

 

Week	Notes	Sections	Slides
3		Introduction. Probability spaces, random variables, distribution functions (Ch 1, 2)	TMA285_MMA711_230117.pdf
		Lebesgue integral, expectation. Change of measure.	TMA285_MMA711_230118.pdf
		Conditional expectation. Stochastic processes. Brownian motion, quadratic variation (Ch 2, 3)	TMA285_MMA711_230119.pdf
		Brownian motion. Martingales. Markov processes (Ch 3)	TMA285_MMA711_230120.pdf
4		Itô's integral (Ch 4)	TMA285_MMA711_230124.pdf
		Itô's formula. Diffusion processes(Sec 4.6)	TMA285_MMA711_230125.pdf
		Diffusion processes. Girsanov's Theorem (Sec 4.5)	TMA285_MMA711_230126.pdf
		Exercises 2.8, 2.15, 3.3, 3.27, 4.4, 4.5, 4.6
5		Stochastic differential equations Kolmogorov PDE (Sec 5.2)	TMA285_MMA711_230131 .pdf
		Kolmogorov PDE, Markov property, and transition density (Sec 5.2)  Exercise 3.33, 5.7	TMA285_MMA711_230201.pdf
		Arbitrage-free markets (Sec 6.1) Risk-neutral formula in discrete case
		Exercises
6		Risk-neutral pricing formula for European derivatives (Sec 6.2)	TMA285_MMA711_230203.pdf
		Black-Scholes price of standard European derivatives (Sec 6.3)
		Black-Scholes price of standard European derivatives (Sec 6.3)	No notes because I followed Sec 6.3
		The Asian option. (Sec 6.4). Finite difference solutions of PDE's (Sec 5.4)	TMA285_MMA711_230210.pdf
7		Finite difference for heat equation. Crude Monte Carlo method. Control variate Monte Carlo method (Sec 6.4)	Compendium page 134-137
		Local volatility models. CEV model (Sec 6.6)	TMA285_MMA711_230215.pdf
		Local volatility models (Sec 6.6)
		Stochastic volatility models.
8		Work on projects
		Work on projects
		Work on projects
		Variance swaps (Sec 6.6)	See compendium, Sec 6.6
9		Yield curve. Classical approach to ZCB pricing.	TMA285_MMA711_230228.pdf
		Classical approach to ZCB pricing. HJM model (Sec 6.7).	TMA285_MMA711_230301.pdf
		Interest rate swaps (Sec 6.7) (Sec 6.8)	TMA285_MMA711_230302.pdf
		Forwards (Sec 6.8)	TMA285_MMA711_forwards.pdf
10		Futures (Sec 6.8).	TMA285_MMA711_futures.pdf
		Repetition/questions

 

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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. 

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

 

Some old exams can be found here: Old_Exams.zip 

Exam March 2022

Re-exam June 2022

Re-exam Aug 2022

 

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