The Mathematics of Financial Modelingand Investment Management

15-ArbPric-ContState/Time Page 466 Wednesday, February 4, 2004 1:08 PM

466 The Mathematics of Financial Modeling and Investment Management

which shows that πt is a state-price deflator. The same reasoning in

reverse order demonstrates that if πt is a state-price deflator then:

t

∫ruud

ξ^0 πt

t = e ------

π 0

is a density process for Q.

ARBITRAGE PRICING WITH A PAYOFF RATE

In the analysis thus far, we assumed that there is no intermediate payoff.

The owner of an asset makes a profit or a loss due only to the changes in

value of the asset. Let’s now introduce a payoff-rate process δt

i

for each

asset i. The payoff-rate process must be interpreted in the sense that the

cumulative payoff of each individual asset is

t

Dti = ∫δsi sd

0

We define a gain process

i i

Gt

i

= St + Dt

By the linearity of the Itô integrals, we can write any trading strategy as

t t t

∫θtdGt = ∫ θtdXt + ∫θtdDt

0 0 0

If there is a payoff-rate process, a self-financing trading strategy is a

trading strategy such that the following relationships hold:

i i i

θθθθtSt = ∑θtSt = ∑



θtiSti + ∫

t

θitdGt , t ∈ [0,T]

i i  0 

An arbitrage is, as before, a self-financing trading strategy such that