The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 434 Wednesday, February 4, 2004 1:08 PM

434 The Mathematics of Financial Modeling and Investment Management

M

∑qjwi()j =^1

j = 1

N

∑qj =^1

j = 1

There are M equations and N + 1 unknowns with M ≤N + 1. This

system will either determine unique risk-neutral probabilities or will be

unsolvable. Therefore, the market will be either complete and arbitrage-

free or will exhibit arbitrage opportunities. Note that in this case we

cannot use the constructive procedure used in the previous case.

What is the economic meaning of the condition that the number of

states be less than or equal to the number of assets? To illustrate this

point, assume that the number of states is M = 2K ≤N + 1. This means

that we can choose K assets whose independent price processes identify

all the states as in the previous case. Now add one more asset. This asset

will go up or down not in specific states but in events formed by a num-

ber of states. Suppose it goes up in the event A and goes down in the

event B. These events are determined by the value of the first K assets. In

other words, the new asset will be a function of the first K assets. An

interesting case is when the new asset can be expressed as a linear func-

tion of the first K assets. We can then say that the first K assets are fac-

tors and that any other asset is expressed as a linear combination of the

factors.

Consider that, given the first K assets, it is possible to determine

state-price deflators. These state-price deflators will not be uniquely

determined. Any other price process must be expressed as a linear com-

bination of state-price deflators to avoid arbitrage. If all price processes

are arbitrage-free, the market will be complete if it is possible to deter-

mine uniquely the risk-neutral probabilities.

If we let the number of states become very large, the number of

assets must become large as well. Therefore it is not easy to develop

simple heuristic arguments in the limit of a large economy. What we can

say is that in a large discrete economy where the number of states is less

than or equal to the number of assets, if there are no arbitrage opportu-

nities the market might be complete. If the market is complete and arbi-

trage-free, there will be a number of factors while all other processes

will be linear combinations of these factors. These considerations will

be further developed in Chapter 18.