M.A. Program Syllabus: Stochastic Optimal Control and Investment

GRS MF594 - Spring 2008

Prof. Andrew Lyasoff Boston University, Graduate Program in Mathematical Finance

Lecture:

Course Prerequisites:

Linear Algebra (MA242) and all levels of Calculus (MA123, MA124 and MA225), Differential Equations (MA 226), Probability (MA 581), Stochastic Processes (MF 795, or consent of instructor).

Course Description:

Text and such: The course is based on the book INVESTMENT UNDER UNCERTAINTY by AVINASH K. DIXIT and ROBERT S. PINDYCK (Princeton University Press, Princeton, N.J., 1994, ISBN: 0-691-03410-9) plus PROF. LYASOFF's own lecture notes.

All lecture notes, homework and reading assignments, topics for the forthcoming exams, solutions to homework problems, the course calendar, various announcements and other relevant information are available at the course website It can be accessed on the web at  http://andrew.lyasoff.com/mf594/ only by students who are officially registered for this course. Instructions on how to access the website will be sent to the e-mail address that appears on the class-list distributed by the Registrar's Office (presumably, this is the e-mail address students have specified on the BU Student Link).  It is each student's own responsibility to follow the announcements and the course calendar posted to the course website.

What is Stochastic Optimal Control and how does it relate to investment? The following example illustrates the nature of the problems Stochastic Optimal Control is dealing with. Suppose that you need to park your car on a very long (one-way) street with designated parking spots lined up one after another. For each parking spot you know exactly the probability of finding that spot free. You also know exactly how far each spot is from the building you want to go to. As you pass by a spot which happens to be free you need to decide whether to leave your car at that spot (and walk the distance to your destination, which is the price incurred by your decision) or keep driving further down expecting to find a free spot which is closer to your destination (and, consequently, face the possibility of not finding any free spots). There are situations in the world of investment that are completely analogous to (in fact, as mathematical problems, indistinguishable from) the parking problem just described. Indeed, on any given day an investor holding an American option on a particular common stock has to decide whether to exercise his option (take that parking spot and incur the loss/profit) or keep waiting for a better opportunity. Similarly, if you are considering buying a property of any kind or investing in a research and development enterprise you need to choose the right moment to take the money out of your bank account and make the purchase. As a mathematical discipline, Stochastic Optimal Control is a tool which offers a general method for dealing with this type of situation. It also allows you to deal with problems in which a decision is being made continuously (you decide whether to exercise your option – whatever "option" might refer to – at any moment, not just once a day, say). Many examples from engineering and navigation fit this pattern too: a space-probe sent to the surface of Mars needs to choose the right moment to open its parachute (or turn on the rocket engine) so that the impact with the surface is minimal.

Course function The course is intended primarily for students who intend to graduate in Mathematics, Finance, Economics, Engineering or Operations Research. It is a part of the M.A. program in Mathematical Finance, but can also be taken by students not enrolled in that program. MA594 will cover the key topics in Stochastic Optimal Control and will focus mostly on the practical and computational aspects of the theory (down to the actual implementation of the methods on a computer). Students will be introduced to the general principles and available computational techniques almost exclusively by way of exploring particular examples from Finance and Investment.

Reading/Homework Assignments:

Reading and/or homework assignments will be posted to the course website weekly. All homework assignments must be typeset - I will not accept hand-written homework assignments. I will collect the homework assignments at the beginning of the regular-class meeting on the day when the assignment is due. The course website will provide information about the topic to be covered on a given day, homework due dates, exam dates and other relevant information. In general, students will be expected to have read the notes for each lecture before coming to class. It is each student's own responsibility to follow the assignments and the due dates posted to the course website.

Seeking Help:

Prof. Lyasoff may be contacted by phone at 617.353.5785 or by e-mail @ the address «bu-dot-edu» preceded by «alyasoff». His office hours are Monday and Wednesday 2:30PM-4:00PM, or by appointment.

Exams and grading:

There will be three 50 minute exams during the regular class hours on Wednesday, February 27, Wednesday, March 26 and Wednesday, April 30. Please note that these dates may change. There will be no final exam for this course. All homework assignments will be graded. The final grade will be 40% (average) from homework + 60% (average) from the exams.