Stochastic Modeling and Financial Mathematics

Constructor University, Fall 2025

Official Class Description from Campusnet

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. Towards the end, the Fokker-Planck equation, Ornstein-Uhlenbeck processes, and nonlinear Stochastic Partial Differential Equations are discussed, and connections to applications in physics and other areas of mathematics are made. Students will program and explore all basic techniques in a numerical programming environment and apply these algorithms to real data whenever possible.

News

First class is on Friday, Sep. 5, in East Hall 1. 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 only.

Contact Information

Instructor: Prof. Sören Petrat

Email: spetrat AT constructor.university

Office: 112, Research I

Teaching Assistant: Irfan Basheer

Time and Place

Lecture/Lab sessions (instructor):

Fri 9:45-11:00 and 11:15-12:30, East Hall 1

Tutorial, homework help (teaching assistants):

Will be announced.

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 Elements of Linear Algebra and Elements of Calculus provide a good background.

Textbooks

Much of 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 portfolio assessment, consisting of in-class quizzes, assignments, presentations, and a final project (with an in-class and a take-home component). The final grade is weighted as follows:

Quizzes: 20%

Assignments: 20%

Presentations: 20%

Final Project part 1 (in-class): 20%

Final Project part 2 (take-home): 20%

More explanations of the portfolio parts:

Quizzes: There will be 6 quizzes as announced below in the schedule. The two worst quizzes are excluded in the computation of the grade. The quizzes are in-class, and are solved fully by yourself, without any help.
Assignments: Usually once per week there will be a homework assignment. 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 that the two worst homework sheets are disregarded in the computation of the homework grade. AI policy: For these assignments, you are free to use the help of any AI such as chatgpt. You need to clearly state which resources or AI helpers you have used to solve the assignment!
Presentations: Each student needs to present their solutions to half of a homework sheet at least once during class. The presentation schedule will be discussed in class. During and after the presentation questions will be asked, and the instructor might ask for small real-time modifications of the code. For the presentations no AI or other helpers are allowed.
Final Project part 1 (in-class): The in-class part of the final project will take place during the last two class sessions on Dec 5. For this part, no AI or other helpers are allowed, you need to solve the problem fully by yourself. Details of the project will be announced in class.
Final Project part 2 (take-home): The problem setting for this project will be announced during the last week of classes. The due date is Dec 22, 2025 (no extensions except for documented illness!), and you are free to use the help of any AI such as chatgpt. You need to clearly state which resources or AI helpers you have used in your project!

Chapter 0: Introduction to git and Scientific Python
0.1: git
0.2: Scientific Python

Chapter 1: Basics of Financial Math
1.1: Time Value of Money
1.2: General Cash Flows
1.3: Bonds
1.4: Spot Rates

Chapter 2: Options and Binomial Tree Models
2.1: Option Basics
2.2: Binary Model
2.3: Binomial Tree Models
2.4: Binomial Tree and Calibration
2.5: Central Limit Theorem
2.6: Black-Scholes Formula
2.7: Convergence Rates
2.8: Monte-Carlo Method

Chapter 3: Continuous Time Models
3.1: Brownian Motion
3.2: Stochastic Integrals
3.3: Stochastic Differential Equations
3.4: Itô's Lemma

Chapter 4: Black-Scholes Equation and Finite Difference Schemes
4.1: Derivation of the Black-Scholes Equation
4.2: Connection between Black-Scholes Equation and Formula
4.3: Finite Difference Method
4.4: Stability of Time-stepping Methods
4.5: Application to the Heat Equation

Chapter 5: Parameter Estimates for Time Series

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. 05, 2025	Organization, Introduction to git, Basics of Financial Math (Time Value of Money, General Cash Flows, Annuities, Amortization) See the information on this website and Introduction to git for academics. Lyuu Chapters 3.1, 3.2.
Sep. 12, 2025	Introduction to Scientific Python (basics), Root Finding Algorithms, Basics of Financial Math (IRR, Bonds) Lyuu Chapters 3.3, 3.4, 3.5; see also the python code examples in the git repository and the Introduction to SciPy.
Sep. 19, 2025	Introduction to Scientific Python (plotting), Basics of Financial Math (Bonds, Spot Rates) Quiz 1 at 9:45 See the python code examples in the git repository and the Introduction to SciPy.
Sep. 26, 2025	Introduction to Scientific Python (vectorizing functions), Options (basics and a binary model) Lyuu Chapter 7; Etheridge Chapters 1.1, 1.3
Oct. 03, 2025	No classes (public holiday: German Unity Day) Reading material: Spot Rates Reading material: Option pricing in a binary model Spot rates: Selected parts from Lyuu Chapter 5. Option pricing with a binary model: Lyuu Chapters 9.1, 9.2.1; Etheridge Chapter 1.3.
Oct. 10, 2025	Binomial Tree Model Quiz 2 at 9:45 Lyuu Chapters 9.2.2, 9.2.3; Etheridge Chapter 2.1
Oct. 17, 2025	Binomial Tree Method and Calibration, Central Limit Theorem, Black-Scholes Formula Extra material: A summary of option pricing with binomial trees Extra material: Notes on convergence rates Quiz 3 at 9:45 Lyuu Chapter 9.3, Etheridge Chapter 2.6
Oct. 24, 2025	Monte-Carlo Method, Brownian Motion, Stochastic Integrals Quiz 4 at 9:45 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). Stochastic Integration: Lyuu Chapter 14.1 and 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. 31, 2025	No classes (public holiday: Reformation Day) Reading material: Stochastic Differential Equations, Euler-Maruyama Method, Weak and Strong Convergence Lyuu Chapters 14.2, 14.2.1
Nov. 07, 2025	Ito's Lemma and its application to Geometric Brownian Motion Quiz 5 at 9:45 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).
Nov. 14, 2025	Topics tba Quiz 6 at 9:45
Nov. 21, 2025	No class (workshop Mathematical Physics in the Heart of Germany)
Nov. 28, 2025	Topics tba
Dec. 05, 2025	Final Project