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