The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 431 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Finite-State Models 431

T – 1

N × ∑ Mt

t = 0

equation plus the probability normalizing equation. Consider that the

previous system can be broken down, at each date t, into separate

blocks formed by N equations (one for each asset) of the following type:

T d

j

i

*

∑ qω ∑ --------- = SAkt

ω ∈ Akt j = t +^1 Rtj,

q* qω

ω = ----------------------

∑ qω

ω ∈ Akt

Each of these systems can be solved individually for the conditional

probabilities q*

ω. Recall that a system of this type admits a solution if

and only if the coefficient matrix and the augmented coefficient matrix

have the same rank. If the system is solvable, its solution will be unique

if and only if the number of unknowns is equal to the rank of the coeffi-

cient matrix.

If the above system is not solvable then there are arbitrage opportuni-

ties. This occurs if the payoffs of an asset are a linear combination of those

of other assets, but its price is not the same linear combination of the prices

of the other assets. This happens, in particular, if two assets have the same

payoff in each state but different prices. In these cases, in fact, the rank of

the coefficient matrix is inferior to the rank of the augmented matrix.

Under the assumption

T – 1

MN » ×∑ Mt

t = 0

this system, if it is solvable, will be undetermined. Therefore, there will

be infinite equivalent risk-neutral probabilities and the market will not

be complete. Going to the limit of an infinite number of states, the

above reasoning proves, heuristically, that a discrete-time continuous-

state market with a finite number of securities is inherently incomplete.

In addition, there will be arbitrage opportunities only if the random

variable that represents the payoff of an asset is a linear combination of

the random variables that represent the payoffs of other assets, but the

random variables that represent prices are not in the same relationship.