Stochastic Methods Lab

Course number: CA-MATH-811

Jacobs University, Fall 2020

News

• The last regular class session is on Fri, Oct. 30. After that, there will be two more sessions: On Fri, Nov. 6 (to discuss the solution to HW sheet 8, and to introduce the last HW sheet 9), and on Fri, Nov. 13 (to discuss the solution to HW sheet 9, and to introduce the final project / take-home exam). Both sessions will be from 9:45-11:00.
• We got bigger rooms. The Tuesday sessions are now in the Res. III lecture hall, the Friday sessions in the Res. I lecture hall. The first TA question session is on Thursday, Sep. 24, 11:15-12:30 in Westhall 4 (+ online via MS Teams).
• Class recordings can be found on MS Teams under the "F20_CA-MATH-811_Stochastic Methods Lab" channel.
• During the week from Sep. 7-11, class times are Mon 9:45-12:30, Tue 14:15-17:00, and Fri 8:15-11:00. After that, starting on Sep. 14, the classes are regularly on Tue 14:15-17:00 and Fri 8:15-11:00.
• First class is on Tuesday, Sep. 1, in East Hall 8. If you plan to take this class, but could not register yet, please come anyway. Please bring your laptop to every class session. This class is offered in person by default, but the format can be adapted in case students are not on campus yet or the rules for in-person classes change.

Contact Information

Time and Place

Note: This class only runs from Sep. 1 till Oct. 31. More details will be announced in class.

First class session: Sep. 1, 2020; last class session: Oct. 30, 2020.

Syllabus

All the most recent information about class can be found on this website.

Official Course Description

This module is a first hands-on introduction to stochastic modeling. Examples will mostly come from the area of Financial Mathematics, so that this module plays a central role in the education of students interested in Quantitative Finance and Mathematical Economics. The module is taught as an integrated lecture-lab, where short theoretical units are interspersed with interactive computation and computer experiments.
Topics include a short introduction to the basic notions of financial mathematics, binomial tree models, discrete Brownian paths, stochastic integrals and ODEs, Ito's Lemma, Monte-Carlo methods, finite differences solutions, the Black-Scholes equation, and an introduction to time series analysis, parameter estimation, and calibration. Students will program and explore all basic techniques in a numerical programming environment and apply these algorithms to real data whenever possible.

Resources

• For an introduction to using and properly setting up git, see Introduction to git for academics.
• For an introduction to Scientific Python, please read this Introduction, written by Marcel Oliver (html link). Here is another more general introduction to Python, Python Numpy Tutorial.
• The public git course repository can be found here.
• Some standard single and multi-variable calculus is assumed as background knowledge for this class, as well as the basics of vector and matrix manipulation. For example, the lecture notes of Advanced Calculus and Calculus and Linear Algebra II (see in particular Sessions 1 to 10) provide a good background.

Textbooks

The class material is similar to the following book:

• Lyuu - Financial Engineering and Computation - Principles, Mathematics, Algorithms (Cambridge University Press).

Also, some material is similar to

• Etheridge - A Course in Financial Mathematics (Cambridge University Press),

which is, however, more mathematically involved than this class.

Some other good books about financial mathematics are

• Ross - An Introduction to Mathematical Finance: Options and Other Topics (Cambridge University Press)
• Hull - Options, Futures and other Derivatives (Pearson)

Grading

The assessment for this class is a project portfolio. The final grade is weighted as follows:

Exams

There will be a final take-home exam (final project). More details will be announced in class.

Homework Sheets

Each week there will be a homework assignment (starting on Sep. 7; last sheet is handed out on Nov. 2). The homework assignments have to be uploaded individually on each student's own branch on the bitbucket server via git (details are announced in class). The due date is usually one week after it has been handed out, and is stated on each homework sheet. Note: It is encouraged to discuss the exercise sheets with your classmates (e.g., discuss how to come up with the solution or what the right way of approaching the problem is). On the other hand, the solutions must be written down and handed in individually! Copying the solutions from somebody else is a violation of Academic Integrity!

Note that only the best 7 out of 9 homework sheets are used to compute the homework grade. This also means that there will be no extensions of homework submission deadlines and no excuses from the homework obligation, of course with the exception of illness that lasts over several days.

Class Schedule

Will be updated while class is progressing.

Below, please click on the date to download the lecture notes of this day.

Note that the book references given below offer only a rough orientation. Sometimes, only parts of a particular chapter are covered in class.

Date	Topics
Sep. 01, 2020	Organization, Introduction to git See the information on this website and Introduction to git for academics
Sep. 07, 2020	Introduction to git, Basics of Financial Math (Time Value of Money, General Cash Flows, Annuities) Lyuu Chapters 3.1, 3.2
Sep. 08, 2020	Introduction to Scientific Python (basics), Basics of Financial Math (Annuities, Amortization, IRR) Lyuu Chapters 3.2, 3.3, 3.4; see also the python code examples in the git repository and the Introduction to SciPy
Sep. 11, 2020	Introduction to Scientific Python (basics), Root Finding Algorithms Lyuu Chapter 3.4; see also the python code examples in the git repository and the Introduction to SciPy
Sep. 15, 2020	Introduction to Scientific Python (plotting), Bonds, Spot Rates Lyuu Chapter 3.5 and a few selected parts from Chapter 5; see also the python code examples in the git repository and the Introduction to SciPy
Sep. 18, 2020	Options (basics and a binary model) Lyuu Chapter 7; Etheridge Chapters 1.1, 1.3
Sep. 22, 2020	Option Pricing with a Binary Model Lyuu Chapters 9.1, 9.2.1; Etheridge Chapter 1.3
Sep. 25, 2020	Binomial Tree Method and Calibration Lyuu Chapters 9.2.2, 9.2.3, 9.3.1; Etheridge Chapter 2.1
Sep. 29, 2020	Central Limit Theorem Lyuu Chapter 9.3.1; Etheridge Chapter 2.6
Oct. 02, 2020	Black-Scholes Formula, Convergence Rates Lyuu Chapter 9.3; Etheridge Chapter 2.6
Oct. 06, 2020	Monte-Carlo Method, Brownian Motion, Geometric Brownian Motion Lyuu Chapter 13.3 (see also Chapter 13.1 for more on stochastic processes in general) and Chapter 18.2; Etheridge Chapter 3.1 (this is much more detailed than what we covered in class)
Oct. 09, 2020	Stochastic Integrals Lyuu Chapter 14.1; Etheridge Chapter 4.2 (this is much more detailed than what we covered in class, but very good if you would like to understand the mathematical background more)
Oct. 13, 2020	Stochastic Differential Equations, Euler-Maruyama Method, Weak and Strong Convergence Lyuu Chapters 14.2, 14.2.1
Oct. 16, 2020	Ito's Lemma and its application to Geometric Brownian Motion Lyuu Chapter 14.2.3 and parts of 14.3; Etheridge Chapter 4.3 (more advanced than the treatment in class); a nice introduction to numerical methods for SDEs, covering the class topics from Brownian motion up to Ito's lemma is given in the article by Higham - An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations (alternative link).
Oct. 20, 2020	Black-Scholes PDE Lyuu Chapters 15.1, 15.2; a mathematically rigorous derivation can be found in Etheridge Chapters 5.1, 5.2
Oct. 23, 2020	Black-Scholes PDE and Relation to Black-Scholes Formula; Finite Difference Approximation Lyuu Chapter 15.2.2; and beginning of Lyuu Chapter 18.1
Oct. 27, 2020	Finite Difference Approximation and Stability; Implied Volatilities, Visualizing Binomial Tree Models in Python Lyuu Chapter 18.1; a nice summary of all the methods for valuating options that we discussed in class can be found in the article by Higham - Black-Scholes Option Valuation for Scientific Computing Students (alternative link).
Oct. 30, 2020	Time Series and Parameter Estimation; Autocorrelation More about time series for finance can be found in the article by Aas and Dimakos - Statistical modelling of financial time series: An introduction. Even more background can be found in the book by Tsay - Analysis of Financial Time Series.
Nov. 06, 2020	Discussion of HW 8 solution and introduction to HW 9
Nov. 13, 2020	Discussion of HW 9 solution and discussion of final project