Introduction

This course is designed for undergraduate students in Quantitative Finance. In order for students to be able to price derivatives, they must first understand the behavior of the underlying. To this end, the students will need to understand how many of the most common stock models are created and utilized. At the core of this is understanding Brownian motion and other adapted stochastic processes from a conceptual level, which requires the knowledge of quadratic variation, martingales, and filtrations. Using these building blocks, many different structures of stock and interest rate processes can be developed: geometric Brownian motion, stochastic volatility models, Cox-Ingersoll-Ross, and Vasicek to name a few. After this course, the students will be able to price simple derivatives based on some of these models. In addition, through the use of stochastic integration, students will have a better understanding of portfolio valuation. By the end of this course, the student will be able to derive in its entirety the Black-Scholes-Merton European option pricing formula for Geometric Brownian Motion. The students are supposed to have a strong background in applied mathematics (analysis and calculus) and probability at an undergraduate level. Any student who does not already have this previous knowledge will have much greater difficulty learning the material.

Course Outcomes

By the end of this course, the students will be able to:

• Price derivative securities using the Binomial Asset Pricing Model
• Classify stochastic processes as martingales, Markov, or both/neither
• Simplify stochastic (Ito) integrals
• Determine the differentials of functions of stochastic processes
• Derive Black-Scholes-Merton’s European option pricing model

Textbook

Stochastic Calculus for Finance vol I and II, by Steven E. Shreve, Springer Finance, 2004, ISBN-13: 978-0387249681 (vol I) and 978-0387401010 (vol II).

Additional References

Optional Readings:

Introduction to Probability Models, 10th edition, by Sheldon M. Ross, Academic Press, 2009, ISBN-10: 0123756863, ISBN-13: 978-0123756862.

Probability and Random Processes, by Geoffrey Grimmett and David Stirzaker, Oxford University Press 2001. Stochastic Differential Equations, by Bernt Oksendal, 6th edition, 2010, ISBN-10: 3540047581, ISBN-13: 978-3540047582. Introduction to the Mathematics of Financial Derivatives, by Salih N Neftci, 2nd ed, Associated Press, 2000, ISBN 0125153929.

Grading Policies

Assignments: There will be weekly homework assignments throughout the semester, with problems inspired from your textbook and on necessary skills.

The assignments and their weights are as shown below:

• Homework - 30%
• Midterm - 30%
• Final Exams - 40%
TOTAL - 100%

Please note that your grade will be determined solely on the work you present over the course of the semester. No consideration such as your desire for a better grade for academic standing or job offers will be considered.

Extra Credit: On some of the homework assignments and possibly on the exams, there will be the occasional extra credit problem. This is the only source of extra credit for the course. There are no "extra assignments" that students can do to raise their average outside of the ones assigned. There are no exceptions, don't even bother coming to me and asking about extra work at the end of the semester, as I will only direct your attention to this part of the syllabus.