Math 490 - Special Topics in Mathematics: Introduction to Mathematics of Financial Markets

Class: MW 5:30PM - 6:45PM in Math & Psychology 008 (1/27-5/13/2025), Instructor: Bedrich Sousedik

Course description: This course aims to provide a holistic foundation in mathematical finance, where concepts and intuition are equally important as rigorous mathematical tools. Beginning with the basics of gambles through realistic scenarios, we will lay the basis for understanding the core financial techniques of Options (Puts and Calls), hedging and arbitrage. Then we will recall a few elements of modeling and probability to introduce the centerpiece of state-of-the-art mathematical finance: the Black-Scholes equation. We will discuss its connections to the heat equation, solution and sensitivity analysis that is used in the form of the Greeks. Optional topics include exotic options (American, Asian), bonds and numerical solution of the stochastic differential equations and the Black-Scholes equation.

Recommended Textbook: Donald G. Saari, Mathematics of Finance: An Intuitive Introduction, Springer, series Undergraduate Texts in Mathematics, 2019

Prerequisite: You must complete Math225 and Math251 with a grade of C or better before you can enroll in this class.

The detailed class syllabus (pdf) is here. However, look below to see the actual progress of the class.

The main goals of the course are: 1. Introduce students to the fundamentals of mathematical finance. 2. Develop mathematical intuition as to what are the limitations of the various concepts and conclusions presented throughout the semester and what topics are open for research. 3. Provide a Capstone: the finance topics incorporate many mathematical concepts from earlier classes. So the course will use the opportunity to recall and review relevant concepts with emphasis on their power and utility. 4. Foster creative and innovative thinking for solving problems. 5. Develop skills for communicating ideas in a concise and logical way.

Topics: 1. Gambles: recalling basics of probability, arbitrage, hedging. 2. Options: calls, puts, put-call parity. 3. Modeling in finance and the Efficient Market Hypothesis. 4. More probability and Ito's lemma. 5. Black-Scholes equation - conversion to heat equation - solutions - the Greeks 6. Optional topics: exotic options, bonds, numerics for stochastic differential equations.

1/27 Introduction. 1.1 A football game, hedging, arbitrage.

1/31 1.1 Hedging without arbitrage. 1.2 Expected value and variance, standard form. 1.3 Fair bets and ensuring profit, horse racing. Hw 1 (due 2/5).

2/3 2.1 Calls: long, short (covered).

2/5 2.1 Calls: short (uncovered). 2.2 Puts. 2.3 Hedging: straddle and strangle. Hw 2 (due 2/12).

2/10 2.3 Designing portfolios. 2.4 Present value of money.

2/12 No class.

2/17 2.4 Put-call parity. 2.5 Information gained. Hw 3 (due 2/24).

2/19 2.4 Put-call parity. 2.5 Information gained. Exercises.

2/24 3.1 Assumptions and modeling. 3.2 Efficient market hypothesis. Hw 4 (due 3/3).

2/26 3.2 Efficient market hypothesis, 3.3 Interpretation. Exercises. Standard Normal Distribution Table. Also check this link.

3/3 4.1 Review of some probability.

3/5 No class.

3/10 Exam 1 (Ch 1-2).

3/12 4.2 Ito's lemma. 4.3 Application. Hw 5 (due 3/26).

3/15-23 Spring break.

3/24 5.1 Black-Scholes equation. 5.2 Boundary conditions.

3/26 5.3 Conversion to the heat equation. 5.4 Intuition. Hw 6 (due 4/2).

3/31 Ch 6: Solutions of Black-Scholes.

4/2 Ch 6: Solutions of Black-Scholes (exercises). Ch 7: Partial Information. The Greeks: Hedge ratio.

4/7 Ch 7: Partial Information. The Greeks.

4/9

4/14 No class.

4/16

4/21

4/23

4/28

4/30

5/5

5/7

5/12