(1) |
(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