Course Catalog Description
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.
There will be around 5 homework assignments throughout the semester. Collaboration is encouraged as it can be helpful to understand some of these concepts. Do not confuse collaboration for academic misconduct. Attempt each problem on your own before seeking help from another person. Make sure that you understand the entire assignment that you turn in, and could reproduce the work or solve a similar problem. Do not think that you can simply copy another person's assignment and expect to understand the material. Late homeworks will be accepted under the following policy. If the homework is turned in within one week of the original due date, it will receive of its score, going down by a third each week it is late. The homeworks will have a very firm deadline, of 11:55 PM on the due date. When I say this is a firm date, I mean that if a homework is submitted online at 11:56 PM, its late, no exceptions. Plan ahead and submit your homework early to avoid problems due to internet or computer issues.
Homework assignments can be considered to consist of two separate components. The required component which will be graded, and the recommended component which will just be scanned to confirm it was attempted. Both parts need to be your own work and should not be copied in any way. For the recommended part, if all of the problems are attempted, you can earn up to 5 bonus points per assignment.
There will be one midterm and one final exam given in the class. If you miss an exam, you must provide a written explanation signed by proper authorities in order to be allowed the chance to take a replacement exam. If you just don't show up for an exam you will receive a 0, no exceptions (again unless you have a valid note). Do not schedule your final flight home or to anywhere else until you see the final exam schedule posted on the registrar's website (not the date posted within Canvas, that is just an old date). The midterm and final exam are closed book, but each student can bring one hand written page of notes to the midterm and two hand written pages of notes to the final. Calculators are permitted and encouraged, but cell phones and notebook computers are not allowed.
Extensions and Re-submits
It is my policy to never give a student an extension after a deadline is missed. You know before you actually miss a deadline that you will miss the deadline. If you reach out to me before an assignment is due and let me know why you will miss the deadline, if its a valid reason (and this does not have to include medical issues, I realize that sometimes life just gets in the way) I have no problem providing an extension if necessary.
I will under very rare circumstances allow a student to re-submit an assignment that has already been submitted, but it is my policy to not allow a re-submit of an assignment once it has been graded. Once the assignment has been graded, you now possess much better information than the average student and it is therefore unfair to the others in the class to allow this.
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:
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