# In Black-1976, why is the differential equation missing a term relative to B-S?

In the notation of the original Black-Scholes paper, let $$w(x, t)$$ be the price of an option with underlying priced at $$x$$, and let $$w_1$$ denote the derivative of $$w$$ w.r.t. to $$x$$ and $$w_2$$ denote the derivative of $$w$$ w.r.t. to $$t$$. In Black-Scholes, they derive the differential equation for the value of an option as:

$$w_2 = rw - rxw_1 - \frac{1}{2} v^2 x^2 w_{11}$$

But in the Black paper, the differential equation is

$$w_2 = rw - \frac{1}{2} v^2 x^2 w_{11}$$

And Black's justification is:

Note that this is like the differential equation for an option on a security, but with one term missing. The term is missing because the value of a futures contract is zero, while the value of a security position is positive.

Notice that the term that is missing is, slightly more verbose notation:

$$\frac{\partial w}{\partial x} x r \stackrel{???}{=} 0.$$

where now $$x$$ refers to the forward price.

My question is: why is this quantity zero? I understand that one does not pay for a futures contract up front, and that since it settles into the underlying, it is worth the underlying at expiration. Formally, this can't be that $$x = 0$$, though, because we see $$x^2$$ elsewhere! I also don't think delta of the Black model (here $$w_1$$) is zero. Is it that $$xr = 0$$?

$$w_1$$ is also known as $$\Delta$$. And $$x \Delta$$ is the value of the hedging position, so $$r x \Delta$$ is the instantaneous cost of financing the hedging position. Since as you say futures have no financing cost, this particular $$r$$ can be set to zero (but the other $$r$$ which has a different interpretation cannot). Thus the whole term $$-r x w_1$$ can be left out of the equation.