How do investors' expectations drive asset prices?

Further simplifying the model by assuming that the volatility of the in-
formation process is constant, i.e. the information process is governed
by a geometric Brownian motion and assuming ʌɑC constant, we get

d.F_t = F_tλ^m σ⁷ dt + F_tσ^, dW_t^F

(18)

From equation (18) it is obvious that the asset price follows a geomet-
ric Brownian motion process if the information process has constant
volatility and the risk premium is constant.

In this case the relative risk aversion is constant, too. From

σ¹

∂²

,ʃ (F_τ)

Ox-

∂

--U (F_τ)
ox

F_τ = constant,

we derive the following form of the utility function of the representative
investor for the case ʌɑ = = σ¹

ɪ √1>

U (X) = ^C'^X ʌ,ɪ+ + C₂, X ∈ R⁺,
^{1 -} —

(19)

where C₁, C2 are constants. This utility function belongs to the HARA-
class. In contrast, if ʌɑ = = σ¹, we get

U (X) = Ci ln(X) + C₂ , X ∈ R⁺,           (20)

with Ci, C₂ being two constant parameters. Thus our example illus-
trates that a forward price process governed by a geometric Brownian
motion is consistent with an information process with constant volatil-
ity and a representative investor with a utility function given by equa-
tion (19) or equation (20) where the relative risk aversion is given by
the sharpe ratio divided by the volatility.¹¹

¹¹See also Bick [1], [2] and Franke, Stapleton, Subrahmanyam [12].

20