6-ConceptsProbability Page 165 Wednesday, February 4, 2004 3:00 PM

CHAPTER

6

Concepts of Probability

P

robability is the standard mathematical representation of uncertainty in

finance. In this chapter we present concepts in probability theory that

are applied in many areas in financial modeling and investment manage-

ment. Here are just a few applications: The set of possible economic states

is represented as a probability space; prices, cash flows, and other eco-

nomic quantities subject to uncertainty are represented as time-dependent

random variables (i.e., stochastic processes); conditional probabilities are

used in representing the dynamics of asset prices; and, probability distribu-

tions are used in finding the optimal risk-return tradeoff.

REPRESENTING UNCERTAINTY WITH MATHEMATICS

Because we cannot build purely deterministic models of the economy, we

need a mathematical representation of uncertainty. Probability theory is the

mathematical description of uncertainty that presently enjoys the broadest

diffusion. It is the paradigm of choice for mainstream finance theory. But it

is by no means the only way to describe uncertainty. Other mathematical

paradigms for uncertainty include, for example, fuzzy measures.^1

Though probability as a mathematical axiomatic theory is well

known, its interpretation is still the subject of debate. There are three

basic interpretations of probability:

■ Probability as “intensity of belief” as suggested by John Maynard

Keynes.^2

(^1) Lotfi A. Zadeh, “Fuzzy Sets,” Information and Control 8 (1965), pp. 338–353.
(^2) John Maynard Keynes, Treatise on Probability (McMillan Publishing, 1921).
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