How do investors' expectations drive asset prices?

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, G_t■ 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 F_t is a martingale⁴, thus

F = E^p (F_τ ∖Q_t) , O ≤ t ≤ T.

Defining the density process η_t := E^p (∖Q∖G_t), O ≤ t ≤ T, then F follows a
P-martingale if and only if ηF follows a P-martingale, in other words

η_tF_t = E^p (η_τF_τ ∖Q_t) , O ≤ t ≤ T.

Now, for an adapted real-valued process
consider

dΦj^λ> = -Φj^λ∖dW_t ,
= 1,

equation (4) is Φ/ = exp (— ' λ_udW⁷_u — f f∖ ∖λ_u∖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

F = E^p (φ(*F F_τ ∖Q_t^ , O ≤ t ≤ T,                (5)

⁴We consider only forward contracts with maturity time T, thus for notational conve-
nience we write F_t instead of F_tt?.

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