Program Overview

Program Overview

The Math 86: Mathematical Finance I program is designed to provide students with a thorough understanding of discrete-time analogs of questions arising in finance from a mathematical viewpoint.

Course Description

In their simplest form, derivatives can be thought of as insurance policies that protect their holders from financial uncertainties. The course will consider the discrete-time analogs of these and other questions arising in finance, developing important mathematical ideas found in discrete probability and showing how these concepts can be used to construct a discrete-time model in which to explore questions appearing in finance.

Topics Covered

• 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 University), 2004

Tentative Syllabus

| Chapters |Brief Description
---|---|---
Week 1 & 2 |1 | No Arbitrage Pricing and the binomial 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 Securities Week 8 & 9 |1-6 | Review (of Math 86) & Preview (of Math 96)

Deliverables & Grading Guide

• 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: Assignments should be written neatly and presented using complete sentences. Collaboration is encouraged, but final write-ups must reflect individual understanding. Acknowledgment of consulted individuals is required. No late homework will be accepted.
• Term Project: 15%
• Homework: 15%
• Midterm Exam: 30%
• Cumulative Final Exam: 40%

Students with Disabilities

If you have a disability and require disability-related accommodations, please speak with the instructor or Ward Newmeyer, Director of Student Accessibility Services, as soon as possible to find a remedy.