The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 397 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Finite-State Models 397

The market value Sθθθθof a portfolio θθθθat time 0 is a scalar given by

the scalar product:

N

Sθθθθ= Sθθθθ= ∑ Siθi

i = 1

Its payoff dθθθθat time 1 is the M-vector:

dθθθθ= D′θθθθ

The price of a security and the market value of a portfolio can be negative

numbers. In the previous example of a two-state, three-security market

we obtain

Sθθθθ= Sθθθθ= S 1 θ 1 + S 2 θ 2 + S 3 θ 3

dθθθθ D′θθθθ

d 11 d 21 d 31

d 12 d 22 d 32

θ 1

θ 2

θ 3

d 11 θ 1 + d 21 θ 2 + d 31 θ 3

d 12 θ 1 + d 22 θ 2 + d 32 θ 3

= = =

Let’s introduce the concept of arbitrage in this simple setting. As we

have seen, arbitrage is essentially the possibility of making money by trad-

ing without any risk. Therefore, we define an arbitrage as any portfolio θ

which has a negative market value Sθ = Sθθθθ< 0 and a nonnegative payoff

Dθ = D′θθθθ≥ 0 or, alternatively, a nonpositive market value Sθ = Sθθθθ≤ 0 and

a positive payoff Dθ = D′θθθθ> 0.

State Prices

Next we define state prices. A state-price vector is a strictly positive M-

vector ψψψψsuch that security prices can be written as S = Dψψψψ. In other

words, given a state-price vector, if it exists, security prices can be

recovered as a weighted average of the securities’ payoffs, where the

state-price vector gives the weights. In the previous two-state, three-security

example we can write:

ψ 1

ψψψψ=

ψ 2

S = Dψ