Math 530A, Stochastic Calculus and Mathematical Finance, Fall 2011
Instructor: Jianfeng Zhang, DRB 228, (213)740-9805 firstname.lastname@example.org http://almaak.usc.edu/~jianfenz
Time and location: WF 2:00 - 3:15pm, GFS118
Office hours: W. 11:00-12:00, 12:50-1:50, F: 12:50-1:50 in KAP 248 E
Textbook: Arbitrage Theory in Continuous Time, second edition by Tomas Bjork, Oxford University Press, 2004
Introduction to the Economics and Mathematics of Financial Markets, By Cvitanic and Zapatero, MIT Prss, 2004
Stochastic calculus for finance. I. The binomial asset pricing model, by Shreve, Springer 2004
Stochastic calculus for finance. II. Continuous-time models, by Shreve, Springer 2004
Prerequisites: High level of undergraduate probability theory (e.g. Math 407) is required. Some knowledge on stochastic processes, partial differential equations and financial derivatives (e.g. options) will be very helpful.
Midterm Exam: 10/7, Friday
Final Exam: 12/9, Friday, 2:00pm-4:00pm
This course is the first part of a two-semester sequence, formerly known as 503, which provides the mathematical theory and probabilistic tools for modeling and analyzing security markets. In this semester, we shall focus on the basic materials, and more advanced topics will be provided next semester in 530B. We will start with the discrete time option pricing and hedging theory, which covers most financial topics we are interested in but requires only elementary probability theory. We next introduce the basic theory of Stochastic Calculus, for which the discrete time model also provides the perfect motivation. Finally we study the continuous time option pricing and hedging theory, in particular the Black Scholes model.
Some important financial concepts include: contingent claims, self-financing portfolios, hedging strategy, risk neutral measure, arbitrage free markets, complete and incomplete markets, American type options. Some topics of Stochastic Calculus are: Brownian Motion, filtration, stochastic integration, Ito's formula, Girsanov transformation, martingales, martingale representation theorem, stochastic differential equations, and possibly some basic materials of backward stochastic differential equations.
I will use my own lecture notes, which more or less follow Bjork's book.
Grading and Examination Policies
30% of the grade will be based on homework assignments, 25% on the midterm exam, and 45% on the final exam. The grade cutoffs will be decided after the final exam, based on the students' overall achievements.
The 75 minutes Midterm Exam will be given in regular class time. The Final Exam will be comprehensive, with an emphasis on the materials covered after the Midterm Exam. All exams are closed book, but students are allowed to bring one sheet of formulas.
Homework problems will be assigned biweekly. No late homework will be accepted, but missed homework with valid reasons can be excused. You are permitted and even encouraged to discuss homework problems with classmates. However, you are not permitted to copy solutions from others.
Feedback and Questions
It is extremely important for me to get feedback and questions, both inside and outside class. You are very welcome to visit me during my office hours, and/or make appointments to see me at other time.