6-ConceptsProbability Page 168 Wednesday, February 4, 2004 3:00 PM
168 The Mathematics of Financial Modeling and Investment Managementsent prices as continuous functions, the probability that a price assumes
any particular real number is strictly zero, though the probability that
prices fall in a given interval might be other than zero.
Probability theory is a set of rules for inferring the probability of an
event from the probability of other events. The basic rules are surprisingly
simple. The entire theory is based on a few simple assumptions. First, the
universe of possible outcomes or measurements must be fixed. This is a
conceptually important point. If we are dealing with the prices of an
asset, the universe is all possible prices; if we are dealing with n assets, the
universe is the set of all possible n-tuples of prices. If we want to link n
asset prices with k economic quantities, the universe is all possible (n +
k)-tuples made up of asset prices and values of economic quantities.
Second, as our objective is to interpret probability as relative frequen-
cies (i.e., percentages), the scale of probability is set to the interval [0,1].
The maximum possible probability is one, which is the probability that
any of the possible outcomes occurs. The probability that none of the out-
comes occurs is 0. In continuous probability models, the converse is not
true as there are nonempty sets of measure zero. In other words, in con-
tinuous probability models, a probability of one is not equal to certainty.
Third, and last, the probability of the union of disjoint events is the
sum of the probabilities of individual events.
All statements of probability theory are logical consequences of these
basic rules. The simplicity of the logical structure of probability theory
might be deceptive. In fact, the practical difficulty of probability theory
consists in the description of events. For instance, derivative contracts
link in possibly complex ways the events of the underlying with the events
of the derivative contract. Though the probabilistic “dynamics” of the
underlying phenomena can be simple, expressing the links between all
possible contingencies renders the subject mathematically complex.
Probability theory is based on the possibility of assigning a precise
uncertainty index to each event. This is a stringent requirement that
might be too strong in many instances. In a number of cases we are sim-
ply uncertain without being able to quantify uncertainty. It might also
happen that we can quantify uncertainty for some but not all events.
There are representations of uncertainty that drop the strict requirement
of a precise uncertainty index assigned to each event. Examples include
fuzzy measures and the Dempster-Schafer theory of uncertainty.^6 The
latter representations of uncertainty have been widely used in Artificial(^6) See G. Schafer, A Mathematical Theory of Evidence (Princeton, NJ: Princeton Uni-
versity Press, 1976); Judea Pearl, Probabilistic Reasoning in Intelligent Systems: Net-
works of Plausible Beliefs (San Mateo, CA: Morgan Kaufmann, 1988); and, Zadeh,
“Fuzzy Sets.”