The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 398 Wednesday, February 4, 2004 1:08 PM

398 The Mathematics of Financial Modeling and Investment Management

S 1 d 11 d 12

ψ 1

d 11 ψ 1 + d 12 ψ 2

S 2 = d 21 d 22

ψ 2

= d 21 ψ 1 + d 22 ψ 2

S 3 d 31 d 32 d 31 ψ 1 + d 32 ψ 2

Given security prices and payoffs, state prices can be determined

solving the system:

d 11 ψ 1 + d 12 ψ 2 = S 1

d 21 ψ 1 + d 22 ψ 2 = S 2

d 31 ψ 1 + d 32 ψ 2 = S 3

This system admits solutions if and only if there are two linearly inde-

pendent equations and the third equation is a linear combination of the

other two. Note that this condition is necessary but not sufficient to ensure

that there are state prices as state prices must be strictly positive numbers.

A portfolio θθθθ is characterized by payoffs dθθθθ= D′θθθθ. Its price is given,

in terms of state prices, by: Sθθθθ = Sθθθθ = Dψψψψθθθθ = dθθθθψψψψ.

It can be demonstrated that there is no arbitrage if and only if there is

a state-price vector. The formal demonstration is quite complicated given

the inequalities that define an arbitrage portfolio. It hinges on the Separat-

ing Hyperplane Theorem, which says that, given any two convex disjoint

sets in RM, it is possible to find a hyperplane separating them. A hyper-

plane is the locus of points xi that satisfy a linear equation of the type:

M

a 0 + ∑ aixi = 0

i = 1

Intuitively, however, it is clear that the existence of state prices ensures

that the law of one price introduced in the previous section is automatically

satisfied. In fact, if there are state prices, two identical payoffs have the

same price, regardless of how they are constructed. This is because the price

of a security or of any portfolio is univocally determined as a weighted

average of the payoffs, with the state prices as weights.

Risk-Neutral Probabilities

Let’s now introduce the concept of risk-neutral probabilities. Given a

state-price vector, consider the sum of its components ψ 0 = ψ 1 + ψ 2 + ...

+ ψM. Normalize the state-price vector by dividing each component by

the sum ψ 0. The normalized state-price vector