The Mathematics of Financial Modelingand Investment Management

15-ArbPric-ContState/Time Page 456 Wednesday, February 4, 2004 1:08 PM

456 The Mathematics of Financial Modeling and Investment Management

Y is a regular deflator, a trading strategy θ is self-financing with respect

to the price process X = (X^1 ,...,XN) if and only if it is self-financing with

respect to the deflated price process

XY = (Y^1 N

tXt , ..., YtXt )

In addition, it can be demonstrated that the price process X =

(X^1 ,...,XN) admits no arbitrage if and only if the deflated price process

XY = (Y^1 N

tXt , ..., YtXt )

admits no arbitrage.

A state-price deflator is a deflator π with the property that the

deflated price process Xπ is a martingale. As explained in Chapter 6, a

martingale is a stochastic process Mt such that its current value equals

the conditional expectation of the process at any future time: Mt =

Et[Ms], s > t. For each price process Xti , the following relationship

therefore holds:

i

πtXt = Et[πX

i

s s ] , s > t

This definition is the equivalent in continuous time of the definition of a

state-price deflator that was given in discrete time in the previous chap-

ter. In fact, recall that we defined a state-price deflator as a process π

such that

T

S^1

t

i

= -----Et ∑ πjdj

i

πt j = t + 1

If there is no intermediate payoff, as in our present case, the previous

relationship can be written as

i i i

πtSt = Et[πTST

i

] = Et[Et + 1 [πTST]] = Et[πt + 1 St + 1 ]

The next proposition states that if there is a regular state-price

deflator then there is no arbitrage. The demonstration of this proposi-

tion hinges on the fact that, as the deflated price process is a martingale,

the following relationship holds: