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Financial Derivatives and Partial Differential Equations

Summary: Financial Derivatives and
Partial Differential Equations
Robert Almgren
July, 2001
1. ASSETS AND DERIVATIVES. Assets of all sorts are traded in
financial markets: stocks and stock indices, foreign currencies, loan contracts
with various interest rates, energy in many forms, agricultural products,
precious metals, etc. The prices of these assets fluctuate, sometimes wildly.
As an example, Figure 1 shows the price of IBM stock within a single day.
The picture would look more or less the same across a month, a year, or a
decade, though the axis scales would be different.
If you could anticipate the price fluctutations to any significant extent,
then you could clearly make a great amount of money very quickly. The
fact that many people are trying to do exactly that makes the fluctuations
essentially unpredictable for practical purposes. A fundamental principle
of finance, the efficient market hypothesis [9] asserts that all information
available to anyone anywhere is instantly expressed in the current price,
as market participants race to be the first to profit from new information.
Thus successive price changes may be considered to be uncorrelated random
variables, since they depend on as-yet unrevealed information. This principle


Source: Almgren, Robert F. - Courant Institute of Mathematical Sciences, New York University


Collections: Mathematics