The concept of options stems from the inherent human desire and need to reduce risks. This course starts with a rigorous mathematical treatment of options pricing, credit default swaps, and related areas by developing a powerful mathematical tool known as Ito calculus. We introduce and use the well-known field of stochastic differential equations to develop various techniques as needed, as well as discuss the theory of martingales. The mathematics will be applied to the arbitrage pricing of financial derivatives, which is the main topic of the course. We treat the Black-Scholes theory in detail and use it to understand how to price various options and other quantitative financial instruments. We also discuss interest rate theory. We further apply these techniques to investigate stochastic differential games, which can be used to model various financial and economic situations including the stock market. Time permitting, we discuss related topics in mechanism designs, a subfield of game theory that is concerned about designing economic games with desired outcome.

Course prerequisites:

Multivariate calculus, linear algebra and matrix theory (e.g., 625.409 Matrix Theory), and a graduate-level course in probability and statistics (such as 625.403 Statistical Methods and Data Analysis).

Course notes:

This class is distinguished from 625.441 Mathematics of Finance: Investment Science (formerly 625.439) and 625.714 Introductory Stochastic Differential Equations with Applications, as follows: 625.441 Mathematics of Finance: Investment Science gives a broader and more general treatment of financial mathematics, and 625.714 Introductory Stochastic Differential Equations with Applications provides a deeper (more advanced) mathematical understanding of stochastic differential equations, with applications in both finance and non-finance areas. None of the classes 625.441 Mathematics of Finance: Investment Science, 625.442 Mathematics of Risk, Options, and Financial Derivatives, and 625.714 Introductory Stochastic Differential Equations with Applications is a prerequisite or co-requisite for the other classes; the classes are intended to be complementary. Feel free to contact the instructor(s) should you have any questions about these courses.

Course instructor:
Pemy