4-PrincipCalculus Page 92 Friday, March 12, 2004 12:39 PM
92 The Mathematics of Financial Modeling and Investment Managementchange of a number of functions—such as polynomials, exponentials,
logarithms, and many more—can be explicitly computed as a closed for-
mula. For example, the rate of change of a polynomial is another poly-
nomial of a lower degree.
The process of computing a derivative, referred to as differentiation,
solves the problem of finding the steepness of the tangent to a curve; the
process of integration solves the problem of finding the area below a
given curve. The reasoning is similar. The area below a curve is approx-
imated as the sum of rectangles and is defined as the limit of these sums
when the rectangles get arbitrarily small.
A key result of calculus is the discovery that integration and deriva-
tion are inverse operations: Integrating the derivative of a function
yields the function itself. What was to prove even more important to the
development of modern science was the possibility of linking together a
quantity and its various instantaneous rates of change, thus forming dif-
ferential equations, the subject of Chapter 9.
A solution to a differential equation is any function that satisfies it.
A differential equation is generally satisfied by an infinite family of func-
tions; however, if a number of initial values of the solutions are
imposed, the solution can be uniquely identified. This means that if
physical laws are expressed as differential equations, it is possible to
exactly forecast the future development of a system. For example,
knowing the differential equations of the motion of bodies in empty
space, it is possible to predict the motion of a projectile knowing its ini-
tial position and speed. It is difficult to overestimate the importance of
this principle. The fact that most laws of physics can be expressed as
relationships between quantities and their instantaneous rates of change
prompted the physicist Eugene Wigner’s remark on the “unreasonable
effectiveness of mathematics in the natural sciences.”^2
Mathematics has, however, been less successful in describing human
artifacts such as the economy or financial markets. The problem is that
no simple mathematical law can faithfully represent the evolution of
observed quantities. A description of economic behavior requires the
introduction of a certain amount of uncertainty in economic laws.
Uncertainty can be represented in various ways. It can, for example,
be represented with concepts such as fuzziness and imprecision or more
quantitatively as probability. In economics, uncertainty is usually repre-
sented within the framework of probability. Probabilistic laws can be
cast in two mathematically equivalent ways:(^2) Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural
Sciences,” Communications in Pure and Applied Mathematics 13, no. 1 (February
1960).