For a heuristic derivation of the equation for
we assume that
is differentiable as many times as
necessary. From a Taylor expansion it follows:
In doing this, we use the fact that
and
is of the size of its standard deviation,
We neglect terms which are of smaller size than
Thus, we can express
by a simpler term:
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Thanks to this identity, calculus rules for stochastic integrals
can be derived from the rules for deterministic functions (as
Taylor expansions for example). Neglecting terms which are of
smaller size than we obtain from (5.9) the
following version of Itôs lemma:
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The logarithm of the stock price is a generalized Wiener process with drift
rate
and variance rate
Since
is
N
-distributed
is itself
lognormally distributed with parameters
and
A generalized version of Itôs lemma for functions
which are allowed to depend on time
is: