interpretation of the process defining the pricing kernel. Further, we apply
Girsanov’s theorem in a representative investor economy to gain a deeper
understanding of the pricing kernel.
4.1 Brief summary of Girsanov’s theorem
Assuming that the market, in some filtered probability space (Ω, G, Gt■ P) on
which defined a standard Brownian motion W, admits no arbitrage possibil-
ities it is well known that there exists a martingale measure P equivalent to
P under which the forward price Ft is a martingale4, thus
F = Ep (Fτ ∖Qt) , O ≤ t ≤ T.
Defining the density process ηt := Ep (∖Q∖Gt), O ≤ t ≤ T, then F follows a
P-martingale if and only if ηF follows a P-martingale, in other words
ηtFt = Ep (ητFτ ∖Qt) , O ≤ t ≤ T.
A with P ÇJq λjdt < ∞) = 1
(4)
Now, for an adapted real-valued process
consider
dΦjλ> = -Φjλ∖dWt ,
= 1,
then it is clear that Φ^λ i is a P -martingale.
Further, the unique solution to
equation (4) is Φ/ = exp (— ' λudW7u — f f∖ ∖λu∖2 du} for O ≤ t ≤ T. By
construction the process Φ^λh has the properties of a density process.
However, it follows from the no arbitrage assumption that there exists a
process к such that Φ^d defines a probability measure P by
under which F is a martingale. Hence, we have
d^ = φw
dP t
O ≤ t ≤ T, P-a.s.
F = Ep (φ(*F Fτ ∖Qt^ , O ≤ t ≤ T, (5)
4We consider only forward contracts with maturity time T, thus for notational conve-
nience we write Ft instead of Ftt?.