Instructor: Jianfeng Zhang, KAP 248E, (213)740-9805, jianfenz@usc.edu dornsife.usc.edu/jianfeng-zhang/
Grader: Yuxin Xie, yuxinx@usc.edu
Time and location: WF 2:00 – 3:15pm, GFS116
Office hours: W: 12:30-2pm, F: 9:00-10:30am at KAP 248E, or by appointment
Textbooks: (Either Bjork’s book or Shreve’s two books)
- Arbitrage Theory in Continuous Time, 3rd edition by Tomas Bjork, Oxford University Press, 2009
- 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.
Exam Dates:
Midterm Exam: October 4, Friday, in class
Final Exam: December 13, Friday, 2-4pm, in DMC 152
Course Contents:
This course is the first part of a two-semester sequence, 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. The textbooks and the reference books serve as main reference books. Here is a tentative schedule (subject to change):
Chapter 1. Discrete Time Option Pricing Theory
1.1 The one period model
1.2 The multiperiod binomial model
1.3 American options
1.4 Fundamental theorems of mathematical finance
Chapter 2. Stochastic Calculus (in Continuous Time Models)
2.0 From discrete to continuous time models
2.1 Probability space and random variables
2.2 Stochastic processes and filtrations
2.3. Brownian motion and stochastic integration
2.4 Some important theorems in stochastic calculus
Chapter 3. The Black-Scholes-Merton Theory
3.1. The Black-Scholes model
3.2 The Black-Scholes formula for European call option
3.3. Some exotic options
3.4 Fundamental theorems of mathematical finance
3.5 Nonlinear models
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 formula sheet.
Homework problems will be assigned 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 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.
Statement for Students with Disabilities
USC welcomes students with disabilities into all of the University’s educational programs. The Office of Student Accessibility Services (OSAS) is responsible for the determination of appropriate accommodations for students who encounter disability-related barriers. Once a student has completed the OSAS process (registration, initial appointment, and submitted documentation) and accommodations are determined to be reasonable and appropriate, a Letter of Accommodation (LOA) will be available to generate for each course. The LOA must be given to each course instructor by the student and followed up with a discussion. This should be done as early in the semester as possible as accommodations are not retroactive. More information can be found at osas.usc.edu. You may contact OSAS at (213) 740-0776 or via email at osasfrontdesk@usc.edu.
Statement on Academic Integrity
The University of Southern California is foremost a learning community committed to fostering successful scholars and researchers dedicated to the pursuit of knowledge and the transmission of ideas. Academic misconduct is in contrast to the university’s mission to educate students through a broad array of first-rank academic, professional, and extracurricular programs and includes any act of dishonesty in the submission of academic work (either in draft or final form).
This course will follow the expectations for academic integrity as stated in the USC Student Handbook. All students are expected to submit assignments that are original work and prepared specifically for the course/section in this academic term. You may not submit work written by others or “recycle” work prepared for other courses without obtaining written permission from the instructor(s). Students suspected of engaging in academic misconduct will be reported to the Office of Academic Integrity.
Other violations of academic misconduct include, but are not limited to, cheating, plagiarism, fabrication (e.g., falsifying data), knowingly assisting others in acts of academic dishonesty, and any act that gains or is intended to gain an unfair academic advantage.
Academic dishonesty has a far-reaching impact and is considered a serious offense against the university. Violations will result in a grade penalty, such as a failing grade on the assignment or in the course, and disciplinary action from the university itself, such as suspension or even expulsion.
For more information about academic integrity see the student handbook or the Office of Academic Integrity’s website, and university policies on Research and Scholarship Misconduct.
Please ask your instructor if you are unsure what constitutes unauthorized assistance on an exam or assignment or what information requires citation and/or attribution.