Use to denote the value of a portfolio of one long option position and a short position in some quantity of the underlying:

As the time to the value of the portfolio will change. This change in value is partly due to the change in the option value and partly to the change in the underlying

From Ito's Lemma we have

Substituting, the value of in the previous equation and grouping for and terms.

, terms are deterministic terms and terms are random.

We can eliminate the random terms by choosing -

,

Thus by choosing a suitable value of , our portfolio change is deterministic and completely *riskless.* This riskfree change must be equal to the growth we would get by putting the equivalent amout in a risk-free asset.

Putting it all together and elimiating and , we get

The Black-Scholes PDE.