SIAM 50th Anniversary and 2002 Annual Meeting

Short Course on Mathematical Models for Finance

Sunday, July 7, 2002
Philadelphia Marriott Hotel, Philadelphia, PA

Rene Carmona, Chair
Department of Operations Research and Financial Engineering
Princeton University

Steven E. Shreve
Department of Mathematical Sciences
Carnegie Mellon University

The Nobel prize-winning work of Markowitz, Black, Scholes and Merton has fundamentally changed the practice of finance. The development of mathematical models to understand the relationship among complicated financial instruments has enabled the proliferation of these instruments which enhance the efficiency of world-wide capital markets. Investment banks, energy firms, and asset management companies employ thousands of people who understand and use numerical methods, partial differential equations, stochastic calculus and statistics, and on the basis of these models billions of dollars are traded daily.

This course is an introduction to the stochastic calculus on which modern financial models are built, and an introduction to the models themselves. The course will also touch on the role of partial differential equations in finance and will indicate some of the areas in which further research is needed.

Level of Material

Course Objectives
The course is intended to provide an overview of the applications of mathematics to finance, provide an introduction to the literature, and suggest areas for research.

Who Should Attend
Students and scientists with little or no previous knowledge of the field.

Recommended Background
Measure theoretic probability is desirable. Knowledge of undergraduate probability and a good background in measure and integration should suffice.

Professor Steven Shreve is the Director of the Ph.D. program in Mathematical Finance and a co-founder of the professional Master's degree in Computational Finance at Carnegie Mellon. He serves as an Advisory Editor for Finance and Stochastics, is a member of the governing Council of the Bachelier Finance Society and a Fellow of the Institute for Mathematical Statistics. He is co-author with I. Karatzas of Brownian Motion and Stochastic Calculus and Methods of Mathematical Finance. He regularly teaches executive education courses to investment bank employees.



8:30 - 10:00

I. Binomial model
a. One-period pricing and hedging
b. Multi-period pricing and hedging
c. Risk-neutral pricing
d. Optimal investment

10:30 - 12:00

II. Stochastic calculus
a. Brownian motion
b. Itô integrals
c. Itô-Doeblin formula
d. Black-Scholes-Merton formula


2:00 - 3:00

III. Change of measure
a. Girsanov theorem
b. Risk-neutral pricing
c. Connections with partial differential equations

3:30 - 4:00

IV. Optimal investment
a. Complete markets
b. Incomplete markets

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