The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 180 Wednesday, February 4, 2004 3:00 PM

180 The Mathematics of Financial Modeling and Investment Management

P(Xt = Yt) = 1

■ Property 3. The process X = X(t,ω) is said to be strongly equivalent to

or indistinguishable from the process Y = Y(t,ω) if

P(Xt = Yt, for all t) = 1

Property 3 implies Property 2, which in turn implies Property 1.

Implications do not hold in the opposite direction. Two processes hav-

ing the same finite distributions might have completely different paths.

However it is possible to demonstrate that if one assumes that paths are

continuous functions of time, Properties 2 and 3 become equivalent.

PROBABILISTIC REPRESENTATION OF FINANCIAL MARKETS

We are now in the position to summarize the probabilistic representation

of financial markets. From a financial point of view, an asset is a contract

which gives the right to receive a distribution of future cash flows. In the

case of a common stock, the stream of cash flows will be uncertain. It

includes the common stock dividends and the proceeds of the eventual

liquidation of the firm. A debt instrument is a contract that gives its

owner the right to receive periodic interest payments and the repayment

of the principal by the maturity date. Except in the case of debt instru-

ments of governments whose risk of default is perceived as extremely

low, payments are uncertain as the issuing entity might default.

Suppose that all payments are made at the trading dates and that no

transactions take place between trading dates. Let’s assume that all

assets are traded (i.e., exchanged on the market) at either discrete fixed

dates, variable dates or continuously. At each trading date there is a

market price for each asset. Each asset is therefore modeled with two

time series, a series of market prices and a series of cash flows. As both

series are subject to uncertainty, cash flows and prices are time-depen-

dent random variables (i.e., they are stochastic processes). The time

dependence of random variables in this probabilistic setting is a delicate

question and will be examined shortly.

Following Kenneth Arrow^9 and using a framework now standard,

the economy and the financial markets in a situation of uncertainty are

described with the following basic concepts:

(^9) Kenneth Arrow, “The Role of Securities in the Optimal Allocation of Risk Bear-
ing,” Review of Economic Studies (April 1964), pp. 91–96.