American Mathematical Monthly - MAY 1998

MAY 1998

The Modeling and Analysis of Financial Time Series
by James Case
jcase66777@aol.com

This paper begins with a brief account of the mathematical tools with which Wall Street's famed "rocket scientists"-or "financial engineers" as they call themselves-earn their enviable livings. Foremost among those tools are the familiar Bachelier-Wiener model of Brownian motion, the associated Black-Scholes PDE, the discrete time Bernoulli process, the binomial approximation to the normal distribution, and Monte Carlo simulation. The second part of the paper describes certain shortcomings of the Bachelier-Wiener model, some of which have only recently been detected, and draws attention to the extensive class of stochastic processes known as "random evolutions", which may just contain the next generation of financial models.

Merton's Partial Differential Equation and Fixed Point Theory
by Frank Lowenthal, Arnold Langsen, and Clark T. Benson
flowenth@csuhayward.edu, ctbenso@afterlife.ncsc.mil

This paper examines the solution of Merton's boundary value problem for the market nature of risky corporate debt. The solution assumes values that must always be less than par value. Through a method of successive iterations a risky interest rate is determined for which the market value of the debt equals par. Convergence is established by the standard method of fixed point theory and contraction maps. Applications to small, closely held firms and junk bond yields are included.

"Rational Diameters" and the Discovery of Incommensurability
by Wilbur R. Knorr

Graphs and Marriages
by Michel Balinski and Guillaume Ratier
balinski@poly.polytechnique.fr

Consider a two-sided market with two distinct finite sets of players, M and W, such as workers and firms, interns and hospitals, buyers and sellers, students and universities, co-eds and sororities, ... or just plain men and women. Each player has a strict order of preference over those members of the opposite set that she/he/it considers to be "acceptable" (and for the case of firms, hospitals, universities or sororities each has an "acceptance quota").

How should the men and women be matched? How are interns assigned to hospitals?

A reasonable and realistic answer to both questions is that the matchings be stable: no two players not paired can both improve their individual preferences by being paired (in total disregard of all the other players).

This idea, first analyzed by David Gale and Lloyd Shapley years ago, has sired a burgeoning literature in mathematics, economics, and computer science, concerning the existence and the structure of solutions, the behavior of real two-sided markets, and the development of algorithms for the problem and its generalizations.

This paper presents a new approach to the basic problem-the stable marriage problem-in the language of directed graphs. The presentation of the problem and the proofs of the various results are at once unified and simplified. A new algorithm for finding stable matchings, and so a new proof of existence, naturally emerges together with new results.

By and large, pictures do the job. And in the process one touches upon several themes of current mathematics - graphs and doubly-linked lists, algorithms and complexity, algebraic structures, games and convex polyhedra.

On the Second Number of Plutarch
by Laurent Habsieger, Maxim Kazarian, and Sergei Lando
habsiege@math.u-bordeaux.fr, kazarian@ium.kps.ras.ru, lando@lando.mccme.ru

NOTES

Descartes' Rule of Signs Revisited
by Bruce Anderson, Jeffrey Jackson, and Meera Sitharam
banderson@kenteliv.kent.edu

A Conjecture of Erdös
by Ralph Faudree
rfaudree@cc.memphis.edu

Toricelli's Law - An Ideal Example of an Elementary ODE
by R. D. Driver
driver@math.uri.edu

THE EVOLUTION OF...

Does Mathematics Distinguish Certain Dimensions of Spaces? Part II
by Zdislaw Pogoda and Leszek M. Sokolowski
PROBLEMS AND SOLUTIONS

REVIEWS

Computational Economics and Finance: Modeling and Analysis with Mathematica. Edited by Hal R. Varian.
Reviewed by Daniel Schwalbe
schwalbe@macalester.edu

The Pleasures of Counting. By T. W. Kšrner.
Reviewed by Peter Hilton
marge@math.binghamton.edu

What is Mathematics? By Richard Courant and Herbert Robbins, revised by Ian Stewart.
Reviewed by Leonard Gillman
gillman@mail.utexas.edu

TELEGRAPHIC REVIEWS

EDITOR'S ENDNOTES