The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 404 Wednesday, February 4, 2004 1:08 PM

404 The Mathematics of Financial Modeling and Investment Management

dθ

t = θt – 1 (St + dt) – θtSt

An arbitrage is a trading strategy whose payoff process is nonnega-

tive and not always zero. In other words, an arbitrage is a trading strat-

egy which is never negative and which is strictly positive for some

instants and some states. Note that imposing the condition that payoffs

are always nonnegative forbids any initial positive investment that is a

negative payoff.

A consumption process is any nonnegative adapted process. Mar-

kets are said to be complete if any consumption process can be obtained

as the payoff process of a trading strategy with some initial investment.

Market completeness means that any nonnegative payoff process can be

replicated with a trading strategy.

State-Price Deflator

We will now extend the concept of state-price deflator to a multiperiod

setting. A state-price deflator is a strictly positive adapted process πt

such that the following set of M equations hold:

T

S^1

t

i = -----E

t ∑ πjdj

i

πt j = t + 1

In other words, a state-price deflator is a strictly positive process such

that prices Sti are random variables equal to the conditional expectation

of discounted payoffs with respect to the filtration ℑ. As noted above, in

this finite-state setting a filtration is equivalent to an information struc-

ture It. Note that in the above stochastic equation—which is a set of M

equations, one for each state, the term on the left, the prices Sti , is an

adapted process that, as mentioned, assumes constant values on each set

of the partition It. The term on the right is a conditional expectation

multiplied by a factor 1/πt. The process πt is adapted by definition and,

therefore, assumes constant values πAit on each set of the partition It.

In this finite setting, conditional expectations are expectations com-

puted with conditional probabilities. Recall from Chapter 6 that condi-

tional expectations are adapted processes. Therefore they assume one

value at t = 0, Mj values for t = j, and M values at the last date.

To illustrate the above, let’s write down explicitly the above equa-

tion in terms of the notation d i

A

i

jt and SAjt. Note first that