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4.3.1 Transformation to log-normal variables
[ SLIDE
transformation ||
VIDEO
modem -
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The log-normal distribution of the price increments (
in
3.3.1#eq.1) chosen to derive the Black-Scholes 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 log-normal variables
defined by
|
(4.3.1#eq.1) |
Substitute these in the Black-Scholes 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 Black-Scholes into a normalized diffusion equation
|
(4.3.1#eq.5) |
which bears a strong resemblance with the heat-equation from
engineering sciences.
This equation has to be solved for
,
using boundary
,
and initial
conditions
that have to be derived from no-arbitrage arguments
with financial variable
via the transformations (4.3.1#eq.1
and 4.3.1#eq.4).
.
SYLLABUS Previous: 4.3 Methods for European
Up: 4.3 Methods for European
Next: 4.3.2 Solution of the