The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 396 Wednesday, February 4, 2004 1:08 PM

396 The Mathematics of Financial Modeling and Investment Management

The finite number of states represents uncertainty. There is uncer-

tainty because the world can be in any of the M states. At time 0 it is not

known in what state the world will be at time 1. Uncertainty is quanti-

fied by probabilities but a lot of arbitrage pricing theory can be devel-

oped without any reference to probabilities. Suppose there are N

securities. Each security i pays dij number of dollars (or of any other

unit of account) in each state of the world j. The payoff of each security

need not be a positive number. For instance, a derivative instrument

might have negative payoffs in some states of the world. Therefore, in a

one-period setting, the securities are formally represented by an N×M

matrix D = {dij} where the dij entry is the payoff of security i in state j.

Recall from Chapter 5 that the matrix D can also be written as a set of

N row vectors:

d 1

D = · , di =

dN

di 1 · diM

where the M-vector di represents the payoffs of security i in each of the

M states.

Each security is characterized by a price S. Therefore, the set of N

securities is characterized by an N-vector S and an N×M matrix D. Sup-

pose, for instance, there are two states and three securities. Then the

three securities are represented by

S 1



S = S 2 , D =

S 3

d 11 d 12

d 21 d 22

d 31 d 32

Every row of the D matrix represents one security, every column one

state. Note that in a one-period setting, prices are defined at time 0

while payoffs are defined at time 1. There is no payoff at time 0 and

there is no price at time 1. A portfolio is represented by a N-vector of

weights θθθθ. In our example of a market with two states and three securi-

ties, a portfolio is a 3-vector:

θθθθ=

θ 1

θ 2

θ 3