How do investors' expectations drive asset prices?



where the filtration is the one generated by the information process I. Be-
cause of the equality of
Fτ and It the following relationship holds.

Ft = Ep[Fτ ∣σ {IuI 0 иt}] = Ep[Iτ ∣σ {Iu0 иt}] ,   0 tT.

We assume that the transformation from P to P is given by a Girsanov-
functional. More precisely we assume that there is an adapted R2-valued
process
λs = (λ[,1∖λ[,2))z which defines the martingale (∣∣ ∙ ∣∣ is the euclidean
R2-norm)

Φo,t


exp (- i ‘ λ<1W∕ - ∕' ‘ λ<2>dWj’ - Ц‘ ∣∣λ.B2 ds ,   (9)

0 t T,

and the transformed probability measure

4--'

P(A) = E[Φ0,rU] , AFt.

With this definition P and P are mutually absolutely continuous on Fτ and
the process

W wt' \._( Wl + J0' λ<*>ds ∖
(, )■=! w'+ʃo'
f>λA ,

is a 2-dimensional Brownian motion under P. Hence, we have the represen-
tation for F under the probability measure P

Ft = it -


[ λ<1>ZVds - [ λ^Z^ds - [ ZV>dW18 - [ Z<2>dWj

Jt                 Jt                 Jt                Jt

for 0 t T, where Z = (Z^1Z2ψ is the process given by application of
the martingale representation theorem on F. We assume that λ is a smooth
deterministic function that may depend on t,
It and σ{: λ (t, It, σ1t ).

In the following theorem we give a formula for the forward price Ft in
terms of the information process and the market price of risk λ (t,
It, σ1t ).
For the derivation we have to solve a forward-backward stochastic differential
equation. This is done by application of mathematical theorems given in Ma,
Protter, Yong [23] and Ma, Yong [24].

13



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