How do investors' expectations drive asset prices?

where by tr {■} we denote the trace of a 2 × 2-matrix and by (∙, ■) the inner
product of the Euclidean space R². Hence, we have for 0 ≤ t ≤ T

0 = u_t(t,x) — ʌɑ(t, x) x_tx₂u_{x 1}(t, x) — λ^²(t, x) σ^v(t,x) u_x2(t,x')
+ j (-H-'d "ʃ _xι(t,x) + (σ^v(t,x))² u_x2x2(t,x^ + b(t,x) u_x(tt,x)
u(T,x) = x_t.

We jump directly to Step ⅛, omitting Step 3: We define Y_t = u(t,X_t) and
Z_t = σ^τ (t, X_t,u(t, X_t)) у u(t,X_t), then (X,Y, Z) is an adapted solution of
(11) and (12).                                                                ■

Theorem 1 shows the close relationship between the asset price process
and the information process. Given the information process and the pricing
kernel Theorem 1 establishes a characterization of the asset price process
as a function of the information process and the pricing kernel. The drift
of the asset price process is governed by the market price of risk λ_t and
the diffusion of the asset price process Z_t. The diffusion Z_t depends on the
information process I_t itself, on the volatility process of I_t, i.e. σ¹_t, and on
the first derivatives w.r.t. x_t and X2 of the function characterizing the asset
p^rice: ⅛ & ^i, σ¹_t ),    (t, I_t, σ¹_t ). Thus, with Theorem 1 we have an explicit

representation of the asset price process in terms of the information process
and the pricing kernel for a 2-dimensional market model. The application of
Theorem 1 to п-dimensional market models is straight forward.

It is obvious from Theorem 1 that the drift of the asset price process
depends on the volatility of the information process. Thus, empirical stud-
ies implicitly assuming non stochastic volatility of the information process
may find unexplainable variations in the drift. Further, neglecting stochastic
volatility of the information process leads to only one risk premium in the
asset price process, i.e. σY = 0 provides Z_t² = = 0.

To gain some better understanding of the implications of Theorem 1 in
the remainder of this section we discuss the case when all the coefficients are
only functions in t. Thus, with this assumption equation (10) simplifies to

0 = u_t(t,x₁,x₂) — λ^¹ ( (t) x₁x₂u_{x 1}(t,x₁,x₂)                  (13)

— λ^( (t) σ^y(t) u_x2(t,x₁,x₂)

⁺ 2 ^^xl^x2 ^uχ_tχ_t(t,x₁,x₂ ) + (σ^v (t))^2
ua¾a¾ ^(t, ^xl, ^X-)^

+b(t) u_x2(t,X₁,X₂) ,   0 ≤ t ≤ T,

u(T, x) = x₁

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