Math 7380, Fall 2001

Ambar Sengupta, Office: Lockett 324

e-mail: sengupta@math.lsu.edu

ASSIGNMENTS:

NOTES:

Class will meet MWF 8:40am-9:30am, Lockett 111.

Office hours: 10am-12 noon, Mondays and Wednesdays, and by appointment.

This course is devoted to mathematical techniques for pricing financial derivative instruments. The plan of the course will remain somewhat flexible to allow for development in either the theoretical side or for a closer study of real-world issues. This may also be viewed as a stochastic processes course developed around a particular application (finance).

We will begin with a careful examination of the risk neutral measure, the fundamental mathematical notion that underlies pricing theory. Stochastic differential equations, treated in a pragmatic way, will be used to describe the evolution of prices. The Feynman-Kac formula helps transform the pricing problem into a partial differential equation. We will develop general model-independent pricing formulas for standard instruments such as bond options, swaps, and swaptions. Though our focus will be interest rate instruments, we will also prove the celebrated Black-Scholes formula for stock option pricing.

The course will not follow any text but a guide to the literature will be provided. Knowledge of basic probability theory is necessary for this course, but we shall try to keep the technical side as self-contained as reasonably possible.

Useful books include:

• Options, Futures, and Other Derivative Securities by J. Hull, Prentice Hall, 2nd Edition.
• Dynamic Asset Pricing Theory by Darrell Duffie, Princeton University Press, 2nd Edition.
• Foundations for Financial Economics by Chi-fu Huang and Robert H. Litzenberger

Problems will be assigned as we progress through the topics and grades will be decided on the basis of performance on these problems. Satisfactory work on eighty percent or more of the problems will correspond to a grade of A. Sixty to eighty percent will correspond to B. Forty to sixty percent will be needed to get a C. Alternatives, such as oral/in-class presentation of work on problems, may, when appropriate, be considered as substitute for written submission of work.

The following tentative plan will be followed in a flexible way.

Week 1. Economics Fundamentals
• Overview of Markets, concepts from Economics, Equilibrium
Week 2. Probability basics
1. Probability basics: sigma-algebras, probability measure
2. Probability basics: random variables
3. Probability basics: Gaussian random variables
Week 3. Risk-Neutral Measure/Market Equilibrium
1. The Risk-Neutral Measure for a Numeraire
2. The Risk-Neutral Measure
3. The Risk-Neutral Measure
Week 4. Options and Pricing Bond instruments
1. Options
2. Zero-coupon Bonds
3. The log-normal case
Week 5. Forwards, Futures
1. Forward Prices
2. Forward Prices
3. Futures
1. Swaps
2. Swaps
3. Swaptions
Week 7. Stochastics
1. Stochastic differentials
2. Stochastic differentials
3. Feynman-Kac formula
Week 8. Some Interest Rate Models
1. One-Factor Hull-White Model
2. Two-factor Hull-White Model
3. Chi-squared model
Week 9.
1. PDES and numerical methods
2. Stochastics and numerical methods
3. Hedging issues