Why Black’s Futures Option Equation Omits the Financing Term
Summary
The document explains the difference between the Black-Scholes equation for an option on a security and Black’s equation for an option on a futures contract. The security equation includes a term involving the underlying price, the option’s sensitivity to that price, and an interest rate. Black’s futures equation omits that financing term, prompting the question of why the term vanishes even though the futures price and option delta are not zero.
The answer interprets delta multiplied by the futures price as the value of the hedging position. Financing that position would create an interest cost, but a futures contract itself has no upfront financing cost. Accordingly, the interest rate associated with financing the hedge is set to zero, removing that term. The explanation distinguishes this rate from the separate rate that remains elsewhere in the equation. It is a conceptual account of the equation’s assumptions, not a broader derivation or discussion of situations where funding costs may differ.
Key ideas
- The Black-Scholes equation for a security option contains a financing term absent from Black’s futures option equation.
- The term represents the financing cost of the position used to hedge the option.
- A futures contract has no upfront financing cost, so the relevant rate for this term is set to zero.
- The interest rate that remains elsewhere in the equation has a different interpretation.
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Full text
# In Black-1976, why is the differential equation missing a term relative to B-S?
# 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 $xr = 0$?
## Answer by nbbo2 (score 1, accepted)
https://quant.stackexchange.com/a/78362
$w_1$ 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.
In other words the term disappears because we set the appropriate interest rate to 0.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.