Deriving the Black-Scholes Option Pricing
Formulae using Ito (stochastic) calculus and partial differential equations
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The following partial differential equation is satisfied by
the price of any derivative on
, given the assumptions
underlying the Black-Scholes world:
![](I/BlackScholesDerivation2_files/image002.png)
This partial differential equation is a second-order, linear
partial differential equation of the parabolic type. This type of
equation is the same as used by physicists to describe diffusion of heat. For
this reason, Gauss-Weiner or Brownian processes are also often commonly called diffusion
processes.
If
,
and
are
constant then we can solve it by transforming it into a standard form which
others have previously solved, namely (for some constant
):
![](I/BlackScholesDerivation2_files/image007.png)
This can be achieved by replacing
by
where
(as long as
is
constant) and by making the following double transformation (assuming that
,
and
are
constant):
This transformation removes one of the terms in the partial
differential equation:
![](I/BlackScholesDerivation2_files/image011.png)
The partial differential equation then simplifies to
, with
. Prices of different
derivatives all satisfy this equation and are differentiated by the imposition
of different boundary conditions. A common tool for solving partial
differential equations subject to such boundary conditions is the use of Green’s
functions. This expresses the solution to a partial differential equation
given a general boundary condition applicable at some boundary
, formed say by the
curve
, as an expression of
the following form, in which
is called the Green’s
function for that partial differential equation:
![](I/BlackScholesDerivation2_files/image017.png)
The Green’s function for
where
is
constant is:
![](I/BlackScholesDerivation2_files/image018.png)
If the boundary condition is expressed as
at
, where
is continuous and
bounded for all
, then the solution is:
![](I/BlackScholesDerivation2_files/image023.png)
For a European call option, with strike price
, we
have, after making the substitutions described above,
. After some further
substitutions we find that this implies that:
![](I/BlackScholesDerivation2_files/image026.png)
where:
![](I/BlackScholesDerivation2_files/image027.png)
![](I/BlackScholesDerivation2_files/image028.png)
Substituting
and
into the formula for
recovers the
Black-Scholes formulae, e.g. for a call option:
![](I/BlackScholesDerivation2_files/image032.png)
Where
![](I/BlackScholesDerivation2_files/image033.png)
![](I/BlackScholesDerivation2_files/image034.png)
and
is the cumulative
Normal distribution function, i.e.
![](I/BlackScholesDerivation2_files/image036.png)