The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 182 Wednesday, February 4, 2004 3:00 PM

182 The Mathematics of Financial Modeling and Investment Management

omy can belong is progressively reduced. Intuitively, revelation of infor-

mation means the progressive reduction of the number of possible states;

at the end of the period, the realized state is fully revealed. In continuous

time and continuous states, the number of events is infinite at each

instant. Thus its cardinality remains the same. We cannot properly say

that the number of events shrinks. A more formal definition is required.

The progressive reduction of the set of possible states is formally

expressed in the concepts of information structure and filtration. Let’s

start with information structures. Information structures apply only to

discrete probabilities defined over a discrete set of states. At the initial

instant T 0 , there is complete uncertainty on the state of the economy;

the actual state is known only to belong to the largest possible event

(that is, the entire space Ω). At the following instant T 1 , assuming that

instants are discrete, the states are separated into a partition, a partition

being a denumerable class of disjoint sets whose union is the space

itself. The actual state belongs to one of the sets of the partitions. The

revelation of information consists in ruling out all sets but one. For all

the states of each partition, and only for these, random variables assume

the same values.

Suppose, to exemplify, that only two assets exist in the economy

and that each can assume only two possible prices and pay only two

possible cash flows. At every moment there are 16 possible price-cash

flow combinations. We can thus see that at the moment T 1 all the states

are partitioned into 16 sets, each containing only one state. Each parti-

tion includes all the states that have a given set of prices and cash distri-

butions at the moment T 1. The same reasoning can be applied to each

instant. The evolution of information can thus be represented by a tree

structure in which every path represents a state and every point a parti-

tion. Obviously the tree structure does not have to develop as symmetri-

cally as in the above example; the tree might have a very generic

structure of branches.

FILTRATION

The concept of information structure based on partitions provides a

rather intuitive representation of the propagation of information through

a tree of progressively finer partitions. However, this structure is not suffi-

cient to describe the propagation of information in a general probabilistic

context. In fact, the set of possible events is much richer than the set of

partitions. It is therefore necessary to identify not only partitions but also

a structure of events. The structure of events used to define the propaga-