The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 399 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Finite-State Models 399

ψj 

ψψψψ= {} = ------

ψ 0 

ˆ ψˆ j

is a set of positive numbers whose sum is one. These numbers can be

interpreted as probabilities. They are not, in general, the real probabili-

ties associated with states. They are called risk-neutral probabilities. We

can then write

S------^1 = Dψψψψ

ˆ

ψ 0

We can interpret the above relationship as follows: The normalized

security prices are their expected payoffs under these special probabili-

ties. In fact, we can rewrite the above equation as

Si

Si = ------ = Ed[]i

ψ 0

where expectation is taken with respect to risk-neutral probabilities. In

this case, security prices are the discounted expected payoffs under these

special risk-neutral probabilities.

Suppose that there is a portfolio θθθθ such that dθ = D′θθθθ = {1,1,...,1}.

This portfolio can be one individual risk-free security. As we have seen

above Sθθθθ= dθθθθψψψψ, which implies that ψ 0 = θθθθS is the discount on riskless

borrowing.

Complete Markets

Let’s now define the concept of complete markets, a concept that plays a

fundamental role in finance theory. In the simple setting of the one-

period finite-state market, a complete market is one in which the set of

possible portfolios is able to replicate an arbitrary payoff. Call span(D)

the set of possible portfolio payoffs which is given by the following

expression:

span ()D ≡ {D′θθθθ: θθθθ∈ R

M

}

A market is complete if span(D) = RM.

A one-period finite-state complete market is one where the equation