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4.3.1 Transformation to lognormal variables
[ SLIDE
transformation 
VIDEO
modem 
LAN 
DSL]
The lognormal distribution of the price increments (
in
3.3.1#eq.1) chosen to derive the BlackScholes equation (3.4#eq.4)
shows that the asset price
and the time
are in fact not a
natural choice of variables for the price of an option that expires
at a time
.
This motivates a transformation from financial variables
to lognormal variables
defined by

(4.3.1#eq.1) 
Substitute these in the BlackScholes equation (be careful with the
second derivative)

(4.3.1#eq.2) 
showing that only two dimensionless parameters in fact characterize the problem

(4.3.1#eq.3) 
With a little more algebra, you can verify that further scaling by

(4.3.1#eq.4) 
transforms BlackScholes into a normalized diffusion equation

(4.3.1#eq.5) 
which bears a strong resemblance with the heatequation from
engineering sciences.
This equation has to be solved for
,
using boundary
,
and initial
conditions
that have to be derived from noarbitrage arguments
with financial variable
via the transformations (4.3.1#eq.1
and 4.3.1#eq.4).
.
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Next: 4.3.2 Solution of the