e first
assume that the price of some asset is Markovian and has continuous path. We
further assume that we can divide the whole time interval into sufficiently
small subintervals
such that the properties of the price process are uniform within each
.
We further divide such subintervals into smaller intervals of length
.
Let us denote
the price changes over
and denote
the price change over
.
According to the central limit theorem (see the section
(
Central limit theorem
)) applied to the
sum
we
have
or
where the
and
are the volatility and mean of the smaller price changes
.
The above expression must be invariant with respect to any sufficiently large
number of subdivisions
.
Therefore, we must have that
and
for large enough
Consider now two time intervals, put together:
.
We presume that
is still sufficiently small to maintain the identical distribution assumption
for all
. For
the price change
over
the union
we
have
where the variables
and
are independent (Markovian property). The
and
have the same dependence on
as noted above. Therefore we find that the price change at time t over a small
time interval has two components with the structure originated from the CLT.
The first component is deterministic conditionally on
and depends linearly on the length of the time interval. The second component
is a normal random variable (also conditionally on
)
with standard deviation proportional to the square root of the time interval.
Hence we use the Ito process to model the Markovian price process with
continuous path.
If the price does not have continuous path then we use a sum of an Ito process
and a Poisson process with stochastic intensity (see the section on Credit
risk). If the price is not Markovian and the total state variable is
observable then we increase the dimensionality of the model to include the
full information and consider a multidimensional Ito process. If some part of
the filtration is not observable then we base our modelling on the Kalman
filter technique, see section (
Kalman
filter II
). If we are not sure about precise manner of interaction between
the missing information and the price process then we may perform Bayesian
modification of the model possibly combined with the hierarchical techniques,
see the section (
Baysian
statistics
).
It is common in financial analysis to use some limited model because hedging
strategy restricts most sources of uncertainty or because the contract under
consideration is not sensitive to some degrees of freedom. The section on
incomplete markets contains a technique for combining several models into a
consistent pricing policy.
More remarks on the subject of modelling are placed at the beginning of the
part (
Data Analysis part
) and the part
(
Implementation tools II
).
Principal insight into composition of stationary Markov process is given by
the proposition (
Construction
of generic Levy process
).
