Math 503 Spring 2006 Syllabus

Math 503, Stochastic Calculus for Finance, Spring 2006

Instructor: Jianfeng Zhang, DRB 228, (213)740-9805
   jianfenz@usc.edu     http://almaak.usc.edu/~jianfenz
Time and location: MWF 10:00pm - 10:50pm, SOS B44.
Office hours: M: 2-3 (Math Center), WF: 1-2 (DRB 228)
Textbook:
  • Arbitrage Theory in Continuous Time, second edition by Tomas Bjork, Oxford University Press, 2004

    Reference Books:
  • Introduction to the Economics and Mathematics of Financial Markets, By Cvitanic and Zapatero, MIT Prss, 2004
  • Martingale Methods in Financial Modeling, second edition, by Musiela and Rutkowski, Springer 2005
  • 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
  • Stochastic Calculus and Financial Applications, by M. Steele, Springer 2001

    Prerequisites: Stochastic Processes (e.g. Math 506) and some knowledge on options and financial markets
    Exam Dates:
  • Midterm Exam: 3/6, Monday (New!!!)
  • Final Exam: 5/8, Monday, 8:00am-10:00pm
    • Course Content:
    The course provides mathematical theory and probabilistic tools for modeling and analyzing security markets. A very brief review of stochastic integrals, Ito's rule, Girsanov theorem, will be given. The main topics include discrete and continuous-time stochastic models for security prices, notions of derivative securities, contingent claims, complete and incomplete markets. The notion of hedging portfolios, the fair price and two of its representations - as an expected value and as a solution to a Partial Differential Equation of Feynman-Kac type. Examples: Black-Scholes formula, binomial models. The Fundamental Theorem: equivalence between the absence of arbitrage opportunities and existence of equivalent martingale measures. American and Exotic options. Term structure of interest rate models, Heath-Jarrow-Morton framework, change of numeraire technique. Time permitting, we will also touch upon utility maximization/portfolio optimization problems, as well as risk minimization problems.
    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.

    There is no fixed quotas of A's, B's etc., and the number of points needed for a particular grade is not fixed in advance. The grade cutoffs will be based on the instructor's overall judgement according to the students' achievements.
    The one hour 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 in lectures, and collected weekly. 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 very useful to get feedback and questions, both inside and outside class. You are very welcome to visit me during my office hours. You can also make appointments to see me at other time.