Math 514   Fall 2025

Instructor

Dr. Lili Ju
Office: LeConte College 327
Phone: (803) 576-5797
Email: ju@math.sc.edu
Office hours: Tu,Th 9:30am-11:00am or by appointment.
URL: http://people.math.sc.edu/ju/Homepage_files/teaching/math514_F25.html

Course Description and Meeting Times

Math 514, Financial Mathematics I, Cr. 3.
Prerequisite: A grade of C or better in Math 241. The most important calculus requirements are basic differentiation and integration techniques, the exponential and logarithmic functions, and partial derivatives including the chain rule. No prior knowledge of probability or finance will be assumed. A calculator is required.
Lectures: Tu,Th 2:50pm-4:05pm, LeConte College 348

Textbook

TEXTBOOK: An Elementary Introduction to Mathematical Finance, by Sheldon M. Ross, 3rd Edition, Cambridge University Press, 2011

Learning Outcomes

The primary goal of this course is to teach students some necessary mathematical techniques and how to apply them to the fundamental concepts and problems in financial mathematics and their solution. The main contents include: Introduction to probability theory, random variable, probability density, mean, and variance of a random variable. The applications include interest rate, coupon bonds, arbitrage, Brownian motion, geometric Brownian motion for mathematical models on stock price, etc.

Course Outline

Tentatively Covered Contents
Lecture Times Contents
Lectures 1-4 Probability (Chapter 1): Probability spaces. Outcomes and Events. Conditional probability. Random variables. Bernoulli and binomial random variables. Expected value. Variance and standard deviation. Conditional expectation.
Lectures 5-8 Normal Random Variables (Chapter 2): Probability density functions. Cumulative distribution functions. The normal distribution. Sums of independent normal random variables. Discussion of the Central Limit Theorem.
Lectures 8-12 Geometric Brownian Motion (Chapter 3): Brownian Motion and Geometric Brownian Motion viewed as limits of random walks. The drift and volatility parameters. The maximum variables. The Cameron-Martin Theorem.
Lectures 13-16 Present Value Analysis (Chapter 4): Interest Rates. Present value of an income stream. Abel summation and its application to present value analysis. Coupon and zero-coupon bonds. Continuously varying interest rates and the yield curve.
Lectures 17-19 Pricing Contracts via Arbitrage (Chapter 5): The No Arbitrage Principle. Options pricing. The Law of One Price. Pricing via arbitrage arguments. Forward contracts. Simple bounds for options prices. Payoff diagram. The Put-Call Option Parity Formula.
Lectures 20-21 Arbitrage Theorem (Chapter 6): Risk-neutral valuation. The multiperiod binomial model.
Lectures 22-26 Black-Scholes Formula (Chapter 7): The Black-Scholes Formula and properties. Delta hedging arbitrage strategy. Partial derivatives.

** The instructor reserves the right to give a quiz at any time.
** Exam 1 will cover Chapters 1,2,3 (Tentative date: 9/25/2025, Thursday) and Exam 2 covers Chapters 4,5,6 (Tentative date: 11/13/2025, Thursday).
** Final exam is 4:00pm-6:30pm on December 9, 2025 (Tuesday).

Important Dates

The deadline to drop the course without a grade of "W" being recorded is Monday, August 25. 2025.
The deadline to drop the course without a grade of "WF" being recorded is Wednesday, November 5. 2025.

Reading

Reading the textbook in advance of the lecture is strongly encouraged. Benefits of this preparation include obtaining a familiarity with the terminology and concepts that will be encountered (so you can distinguish major points from side issues), being able to formulate questions about the parts of the presentation that you do not understand, and having a chance to review the skills and techniques that will be needed to apply the new concepts.

Grading Policy

Course grades will be determined from student performance on exams and quizzes. There will be weekly 10-15 minute quizzes and three exams: two midterm exams and a final exam. The two lowest quiz scores will be dropped and no make-up quizzes will be given. The two in-class exams, which are indicated on the syllabus above, will be given during the time normally used for lecture. Each of two midterm exams test only the material covered since the previous exam. In contrast the final exam, which will be given during the week of final exams, is a cumulative exam. No make-up quizzes will be given. Reason for missing an exam must be properly documented and any missed exam must be made up within a week. Homework will be assigned on a daily basis and should be done before the next class. Although not collected, these homework assignments are an essential part of the course for learning and understanding the course material. They should be thought of as required for success in the course. The overall grades for the course are determined as follows:

 Exam 1  25%  Exam 2  25%  Final Exam  30%  Quizzes  20% -----  Total  100%

Undergraduate --
90-100: A 86-89: B+ 80-85: B 76-79: C+ 70-75: C 66-69: D+ 60-65: D <60: F

Graduate --
92-100: A 88-91: B+ 82-87: B 78-81: C+ 72-77: C 68-71: D+ 62-67: D <62: F

Attendance and Academic Honesty

Attendance at every class meeting is important and expected. Students missing more than 10% of the class meetings (3 times) can have their grade lowered. Cheating and plagiarism will not be tolerated. Violations of this policy will be dealt with according to University guidelines.

Generative Artificial Intelligence Policy

In this course, every element of class assignments must be fully prepared by the student. The use of generative AI tools for any part of your work will be treated as plagiarism. If you have questions, please contact me.