Math 86: Mathemtaical Finance I

Fall 2010

 

Instructor Craig J. Sutton
Lectures MWF 11:15-12:20 (108 Kemeny)
X-Hour Tu 12-12:50 (108 Kemeny)
Office Hours TBD
Office 321 Kemeny Hall
E-mail Craig.J.Sutton AT You Know Where
Phone 603-646-1059
HW, Handouts, Announcements, etc. Blackboard

 

Course Description:

In their simplest form, derivatives can be thought of as insurance policies that protect their holders from financial uncertainties. For instance, an airline company would like to protect itself against large surges in the price of oil, or an investor who holds many shares of XYZ corporation might like to lessen his/her exposure to severe downturns in the stock price. To insure themselves against these events, they purchase derivatives. But, how should the derivative be structured? And what is the ``fair'' price for this insurance policy?

In this course we will consider the discrete-time analogs of these and other questions arising in finance from the mathematician's viewpoint. That is, we will thoroughly and rigorously develop some important mathematical ideas found in discrete probability, and show how these concepts can be used to construct a discrete-time model in which we can explore questions appearing in finance. In short, one can view this as an advanced course in discrete probability that explores applications to finance.

Topics may include some of the following

  • (as always) Reading & Writing Proofs
  • Finite Probability Spaces: sample space, sigma-algebras, random variables, expectation, (discrete-time) stochastic processes, filtrations, conditional expectation, martingales & Markov processes
  • Change of Measure and the (Discrete) Radon-Nikodym Derivative
  • The Binomial Asset Pricing Model
  • No-Arbitrage Pricing and the Risk-Neutral/Equivalent-Martingale Measure
  • Stopping Times and American Derivatives
  • Random Walks: the Discrete-Time Version of Brownian Motion.
  • Stochastic Interest Rates & Fixed Income Derivatives

Prerequisites: a keen interest in mathematics and writing proofs, Math 20/60 & Math 23. Some programming experience will also be useful.

 

Textbook: Stochastic Calculus for Finance I: the Binomial Asset Pricing Model, Steven E. Shreve (Carnegie Mellon Univeristy), Springer 2004. (available at Wheelock Books).

Tentative Syllabus: This syllabus is subject to change, but it should give you a rough idea of the topics we will cover this term.


 

Chapters

Brief Description

Week 1 & 2

1

No Arbitrage Pricing and the bi-nomial asset pricing model; risk-neutral measure

 

Week 2 & 3

2

Discrete probability: finite probability spaces, random variables, conditional expectation, filtrations, martingales & Markov processes

 

Week 3 & 4

2

Discrete probability: finite probability spaces, random variables, conditional expectation, filtrations, martingales & Markov processes

 

Week 4 & 5

3

Change of Measure & the Radon-Nikodym Derivative

 

Week 5 & 6

4

American Derivatives: stopping times, path independent & dependent options



Week 6 & 7

5

Random Walks

 

Week 7 & 8

6
 

Fixed Income Securites

Week 8 & 9

1-6

Review (of Math 86) & Preview (of Math 96)
 
 

Deliverables & (tentative) Grading Guide: The following will comprise the written assignemtns for this term.

Your course grade will probably be computed as follows.

Group Homework
20 %
Midterm Exam
35 %
Cumulative Final Exam
45 %

 

Students with disabilities: If you have a disability and require disability related accomodations please speak to me and Ward Newmeyer, Director of Student Accessibility Services, as soon as possible so we can find a remedy.

 

Last Updated 20 September, 2010