:: Collaborations ::

:: Solving the Black-Scholes Equation ::

We have the Black-Scholes equation for European non-dividend-paying call option

with boundary conditions

and

These conditions  for a European call option with expiry T and exercise price E arise from: an option being worthless if the price of the underlying security is zero itself; the price of an option C(t) cannot be greater than the price of the underlying security S(t) otherwise an aribtrage can achieved by buying the security and selling the call option.

In fact, C(t) <= S(t), i.e. the maximum value of the call option is the difference between the spot price and the exercise price.

The Black-Scholes equation resembles a linear parabolic equation similar to the diffusion equation.  A general linear parabolic equation is of the form:

where a and b are constants.  This can always be reduced to a diffusion equation

by choosing the  substitution

We find that this condition can be satisfied by setting

and

If we assume r and to be constant, the Black-Scholes equation can be reduced to the diffusion equation.  To make this possible we set

The result is

where

Having used this substitution, our initial condition becomes

Now our equation looks more like the diffusion equation and we can obtain the similarity solution using

By substituting this expression for v into the parabolic equation above and choosing values of and such that all terms involving u(x, ) are eliminated, we obtain

with

For now it is just a matter of solving the diffusion equation by methods of Fourier Transform.

The final solution is therefore

where

and

Here N(.) is the Cumulative Normal Distribution Function, the exact form is given by:

A simpler polynomial approximation up to 6 decimal places accurate can be used to represent this function instead:

and

Here

This is the exact solution for calculating the value of an European non-dividend paying call option. For put option of the same expiry T and exercise price E, the put-call parity formula can be used if the value of the call option is first calculated

This gives value of P as

Written by Alessio Farhadi.

Back for more