# Math 86: Mathemtaical Finance I

Fall 2014

 Instructor Craig J. Sutton Lectures MWF 11:15-12:20 (028 Kemeny) X-Hour Tu 12-12:50 (028 Kemeny) Office Hours Monday3-4:30, Thursday 9:30-10:30 AM Office 321 Kemeny Hall E-mail Craig.J.Sutton AT You Know Where Phone 603-646-1059 HW, Handouts, Announcements, etc. Canvas

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:

1. Math 60 or Math 20 and 40
2. Math 23
3. COSC 1

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.

• Midterm Exam: Date, Time and Location TBD (Closed Book)
• Cumulative Final Exam: Friday, Nov. 21, 3-6 PM, Location TBD by Registrar, (Closed Book)
• Weekly Homework: Your assignments should be written neatly and you should present your proofs and solutions using complete sentences. You are encouraged to collaborate with other memebers of the class, but your final write-up must reflect your understanding of the material. You must acknowledge any people with whom you consulted. No late homework will be accepted.

 Homework 15% Term Project 15% Midterm Exam 30% Cumulative Final Exam 40%

I will announce a definitive grading sheme in the coming weeks.

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 15 September, 2014