(1) |

A change in

(2) |

In writing the Black-Scholes equation, we will find the value of the price
of the call option
*w*(*x*,*t*) necessary to allow the hedge equity to grow at the same rate
as investing the equity value in an interest account or instrument
at the fixed interest rate *r* per day so that

The Taylor expansion for the the change is

It is now assumed that the variance
comes from a random
walk in the fractional price, and is therefore proportional to
, giving

Putting Eq. 4 and Eq. 5 into Eq. 3,
cancelling the , dividing by and multiplying by
gives the Black-Scholes equation