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:
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 ):
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:
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:
The Green’s function for where is
constant is:
If the boundary condition is expressed as at , where is continuous and
bounded for all , then the solution is:
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:
where:
Substituting and into the formula for recovers the
Black-Scholes formulae, e.g. for a call option:
Where
and is the cumulative
Normal distribution function, i.e.