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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.

 

 

 


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