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3.3.1 Wiener process and martingales
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Wiener 
Markov 
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Although it is not possible to predict with any certainty the spot price
of an asset in an efficient market,
sect.2.1.1 demonstrated in that
possible realizations can be simulated with
their probability of occurrence by summing price increments
over
small steps in time
.
Separating the
deterministic
from the
remaining random component
,
the evolution of the random variable
satisfies a
stochastic differential equation

(3.3.1#eq.1) 
where
=0 (alt. 1) chooses between the normal (alt. lognormal)
distribution of increments discussed previously in (2.1.1#eq.1).
Integrating over time, the first term yields a uniform (alt.
exponential) growth with a rate
that accounts for example for
a continuous payment of a fixed dividend (alt. a compounded interest
rate).
The second term reproduces a random walk proportional to the market
volatility
, using a socalled
Wiener process
that has the following properties
 The Wiener increment
over a time step
is a random variable drawn from a normal distribution with zero
mean and a timestep variance
(1.4#eq.5).
Extensions to other distributions (e.g. 2.1.1#eq.3) are possible.
 The Wiener increment is independent of the past. Using the
definition (1.4#eq.3) for the correlation, this is satisfied
when
.
A convenient way of writing this is

(3.3.1#eq.2) 
where different realizations of the random variable
are
generated using random numbers
that are normally distributed.
This construction provides the mathematical foundation for the
MonteCarlo simulations, where the
mean value of the increment

(3.3.1#eq.3) 
and the variance

(3.3.1#eq.4) 
are matched with historical data to forecast possible evolutions into
the future. Third or even higher order moments of the probability
distribution can in principle also be matched; experience, however,
shows that little is to be gained from such a procedure.
A better description of the market is obtained by performing a
principal components analysis
[19], where several imperfectly correlated random variables
are identified and then superposed to drive increments of the form

(3.3.1#eq.5) 
where
is a correlation factor.
The first component typically accounts for 8090% (and the first three
for up to 9599%) of the variance observed in the market price of forward
rates [19].
Note that the forecast value for the price of a share
or an
interest rate
constructed using Wiener increments depends
only the present value: this independence from the past is known in
mathematics as the Markov property.
Also note that a suitable choice of a
numeraire can always be found to normalize
random variables and make them riskneutral by scaling out the drift observed
in the real world: called martingales,
such variables play an important role in the construction of financial models.
SYLLABUS Previous: 3.3 Improved model using
Up: 3.3 Improved model using
Next: 3.3.2 Itô lemma