function of t, It and σ1t, hence it is a deterministic function of t, It and σ(:
σt = σ10 + [ b(s, Is, σ1s) ds , 0 ≤ t ≤ T.
(3)
Jo
This formulation of the volatility is, for example, consistent with a leverage
effect, i.e. volatility increases with decreasing asset value. The easiest way
to model the leverage effect, is to assume a constant elasticity of variance
model (CEV)
σf = σ It “ , 0 ≤ t ≤ T ,for some a with 0 ≤ a ≤ 1 and σ > 0 constant.
The model (equation (3)) includes all variations of the volatility of the in-
formation process which can be described by deterministic functions of t,
It and σ{. Notice that with our model (equation (3)) σ{ is random since it
is a function of It, but it is σ {Iu ∣ 0 ≤ и ≤ t}-measurable, thus the current
volatility is known. Economically this means that the current (short-term
or myopic) risk is known but the long-term risk evolves stochastically over
time.
Even this more general model neglects some kind of uncertainty. Many
news about the economy or politics as well as about markets and companies
are published completely erratically so that stochastic terms have to be con-
sidered explicitly in the volatility process. Therefore to include these facts
in the model, the volatility is governed by a separate stochastic differential
equation with a stochastic term (equation (2)). It will be obvious from The-
orem 1 that modeling the volatility of the information process by a separate
stochastic differential equation with a stochastic term has an important ef-
fect on the asset price process. With this model, the volatility risk of the
information process is priced, i.e. a risk premium is paid. Hence, all other
variations in the volatility of the information process are not priced.
We can conclude that in general we have to consider stochastic volatility
which exhibits some time pattern. The quantification of these facts is an
empirical task and is closely related to the estimation of the volatility of
asset prices.
4 Characterization of the Pricing Kernel
To derive the asset price process we need the Girsanov-theorem. In this
section we give a brief summary of Girsanov’s theorem and an economic