Course number: CA-MATH-811
Jacobs University, Fall 2022
First class session: Sep. 7, 2022; last class session: Dec. 7, 2022.
All the most recent information about class can be found on this website.
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.
The class material is similar to the following book:
Also, some material is similar to
Some other good books about financial mathematics are
The assessment for this class is a project portfolio. The final grade is weighted as follows:
There will be a final take-home exam (final project). More details will be announced in class.
Each 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: 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 the two worst homework sheets are disregarded in the computation of 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.
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: Immunization
1.5: 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
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. 07, 2022 | Organization, Introduction to git See the information on this website and Introduction to git for academics |
| Sep. 08, 2022 | Introduction to git, Basics of Financial Math (Time Value of Money, General Cash Flows, Annuities) Lyuu Chapters 3.1, 3.2 |
| Sep. 14, 2022 | Introduction to Scientific Python (basics), Basics of Financial Math (Amortization, IRR) Lyuu Chapters 3.3, 3.4; see also the python code examples in the git repository and the Introduction to SciPy |
| Sep. 15, 2022 | 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. 21, 2022 | Bonds; Introduction to Scientific Python (plotting) Lyuu Chapter 3.5. |
| Sep. 22, 2022 | Immunization; Introduction to Scientific Python (plotting and vectorizing functions) Lyuu parts of Chapter 4. See also the python code examples in the git repository and the Introduction to SciPy |
| Sep. 28, 2022 | Spot Rates Selected parts from Lyuu Chapter 5. |
| Sep. 29, 2022 | Options (basics and a binary model) Lyuu Chapter 7; Etheridge Chapters 1.1, 1.3 |
| Oct. 05, 2022 | Research Day (no afternoon classes) |
| Oct. 06, 2022 | Option Pricing with a Binary Model Lyuu Chapters 9.1, 9.2.1; Etheridge Chapter 1.3 |
| Oct. 12, 2022 | Binomial Tree Model Lyuu Chapters 9.2.2, 9.2.3; Etheridge Chapter 2.1 |
| Oct. 13, 2022 | Binomial Tree Method and Calibration Lyuu Chapter 9.3.1 |
| Oct. 19, 2022 | Central Limit Theorem Lyuu Chapter 9.3.1; Parts of Etheridge Chapter 2.6 |
| Oct. 20, 2022 | Black-Scholes Formula, Convergence Rates Lyuu Chapter 9.3; Etheridge Chapter 2.6 |
| Oct. 26, 2022 | Monte-Carlo Method, 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. 27, 2022 | Brownian Motion; Stochastic Integrals Brownian Motion: Lyuu Chapter 13.1 and 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). |
| Nov. 02, 2022 | Stochastic Integrals 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). |
| Nov. 03, 2022 | Stochastic Differential Equations, Euler-Maruyama Method, Weak and Strong Convergence Lyuu Chapters 14.2, 14.2.1 |
| Nov. 09, 2022 | Towards Ito's Lemma Lyuu Chapter 14.2.3 and parts of 14.3; Etheridge Chapter 4.3 (more advanced than the treatment in class). |
| Nov. 10, 2022 | 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). |
| Nov. 16, 2022 | Black-Scholes PDE Lyuu Chapters 15.1, 15.2; a mathematically rigorous derivation can be found in Etheridge Chapters 5.1, 5.2 |
| Nov. 17, 2022 | Black-Scholes PDE and Relation to Black-Scholes Formula Lyuu Chapter 15.2.2 |
| Nov. 23, 2022 | Finite Difference Approximation Lyuu Chapter 18.1 |
| Nov. 24, 2022 | Finite Difference Approximation and Stability; Explicit and Implicit Euler Methods; Application to the Heat Equation 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). |
| Nov. 30, 2022 | 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. |
| Dec. 01, 2022 | Discussion of HW 11 |
| Dec. 07, 2022 | No class |