Course Catalog Description
This course provides the mathematical foundation for understanding modern nancial theory. It includes topics such as basic probability, random variables, discrete continous distributions, random processes, Brownian motion, and an introduction to Ito’s calculus. Applications to nancial instruments are discussed throughout the course.
This course is designed for first year graduate students in Financial Engineering. The goal is to learn the foundation on which Financial engineering is built upon. It is highly recommended that students have a strong background in applied mathematics (analysis) and probability. This is a core course for all programs in Financial Engineering.
Prerequisites: Background in probability and applied mathematics
At the end of this course, students will be able to:
1. classify stochastic processes as martingales, Markov, or both/neither
2. simplify stochastic (Ito) integrals
3. determine the differentials of functions of stochastic processes
4. change probability measures to facilitate pricing of derivatives
5. solve stochastic differential equations through transformations to partial differential equations.
Stochastic Calculus for Finance vol II, by Steven E. Shreve, Springer Finance, 2004, ISBN-13: 978-0387401010 (vol II).
Stochastic Calculus for Finance vol I, by Steven E. Shreve, Springer Finance, 2004, ISBN-13: 978-0387249681 (vol I).
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 Integration and Differential Equations, by Philip E. Protter, Springer 2005. ISBN-13: 9783642055607
Stochastic Differential Equation, by Bernt Oksendal, 6th edition, 2010, ISBN-10: 3540047581, ISBN-13: 978-3540047582
Introduction to the Mathematics of Financial Derivatives, by by Salih N Neftci, 2nd ed, Associated Press, 2000, ISBN 0125153929.
The final grade in the class will be determined in the following manner:
- 20% Homeworks
- 30% Midterm
- 50% Final Exam
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 need for a better grade will be considered.
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 and the end of the semester, as I will only direct your attention to this part of the syllabus.
||Probability review: Random variables and vectors. Stochastic processes.
||Ch. 1 and 2
| Week 2
||Random walk. Brownian motion.
| Week 3
||Markov Property, Reflection Principle and Passage Times
| Week 4
| Week 5
||Ito lemma and applications
| Week 6
| Week 7
||Multivariable Stochastic Calculus
| Week 8
| Week 9
||Risk-Neutral Measure and Girsanov
| Week 10
||Multidimensional Stock Model
| Week 11
||PDE's and SDE's
| Week 12
||Poisson Processes and Jump Diffusion
| Week 13
| Week 14
||Review & Catch-up