The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 400 Wednesday, February 4, 2004 1:08 PM

400 The Mathematics of Financial Modeling and Investment Management

D′θθθθ= ξ: ξ ∈ RM

always admits a solution. Recall from Chapter 5 on matrix algebra that

this is the case if and only if the rank of D is M. This means that there

are at least M linearly independent payoffs—that is, there are as many

linearly independent payoffs as there are states. Let’s write down explic-

itly the system in the two-state three-security market.

D′θθθθ= ξ

θ 1

d 11 d 21 d 31

θ 2 =

ξ 1

d 12 d 22 d 32  

θ 3

ξ 2

d 11 θ 1 + d 21 θ 2 + d 31 θ 3 = ξ 1

d 12 θ 1 + d 22 θ 2 + d 32 θ 3 = ξ 2

Recall from Chapter 5 that this system of linear equations admits

solutions if and only if the rank of the coefficient matrix is 2. This con-

dition is not verified, for example, if the securities have the same payoff

in each state. In this case, the relationship ξ 1 = ξ 2 must always be veri-

fied. In other words, the three securities can only replicate portfolios

that have the same payoff in each state.

In this simple setting it is easy to associate risk-neutral probabilities

with real probabilities. In fact, suppose that the vector of real probabili-

ties p is associated to states so that pi is the probability of the i-th state.

For any given M-dimensional vector x, we write its expected value

under the real probabilities as

M

E [] x = px = (^) ∑ pixi
i = 1
It can be demonstrated that there is no arbitrage if and only if there
is a strictly positive M-vector ππππsuch that: S = E[Dππππ]. Any such vector ππππ
is called a state-price deflator. To see this point, define
ψi
πi = -----
pi