price process of the asset Ft must be equal at time T. Thus, by definition
Fτ = Iτ. Hence, by postulating such an information process It the value of
the asset in T which is equal to the cash flow at date T is given. Equation
(1) is a generalization of the information process considered in Franke, Sta-
pleton, Subrahmanyam [12]. While their information process is modeled as
a geometric Brownian motion, equation (1) admits constant, time varying,
deterministic or stochastic volatility.
In the remainder of this paper σ {Iu∣ O ≤ и ≤ t} represents the filtration
generated by It which by assumption represents all the information available
to investors in t. We will now turn to the economic meaning of different
volatility models. First take the case of constant volatility, i.e. σ{ = σθ for
all t ∈ [O, T]. Hence, the logarithm of It is normally distributed with ex-
pectation Ep (In Iτ∣σ {Iu∣ O ≤ и ≤ t}) = In (It) —
(T — t) and variance
var (In It∣σ {Iu∣ O ≤ и ≤ t}) = (σθ)2 (T — t) for O ≤ t ≤ T. Since the price
of the asset in T is equal to It this implies that the uncertainty about the
final value of the asset is a linearly decreasing function of time.
This constant rate of uncertainty resolution over time implies some special
information flow. The intensity of information arrival must be constant over
time, i.e. there are no periods where more information is getting into the
market than during other periods. Deterministic but time varying volatility
would allow for periods with a more intense information flow, but uncertainty
resolution is still a deterministic function of time. Such deterministic time
patterns might be explained by some sort of clustering of the information
flow; companies announcing their results in certain periods, e.g. at the end
of a year, many macroeconomic announcements such as monthly economic
information releases occur at certain week-days (see Ederington and Lee [7]
for a related econometric study). These facts may explain to some degree a
deterministic time pattern of the volatility of the information process. Hence,
under the assumption of time varying volatility the conditional variance of
It is no more linear in t. But the volatility and the process of conditional
variances of It can still be perfectly forecasted.
Since the information process is governed by such scheduled information
releases but also by unforeseen information events we consider some ran-
domness in the volatility of the information process to get a more realistic
model for the conditional expectations. First assume that volatility is a borel