# Why are options on futures that are subject to futures settlement insensitive to changes in interest rates?

Why is it that, if options are subject to futures-settlement and the underlying futures themselves are also subject to futures settlement, then these options are insensitive to interest-rate changes?

This is closely related to What is the Rho of an option on a futures contract priced using the Black 76 model?.

If you take the Black 76 pricing model, you can see the following as the call price:

$$C = e^{-rT}[FN(d1) - KN(d2)]$$

here r refers to the risk free interest rate, T is the time to expiry, F is the futures price, K is the strik price and, N(x) is the standard normal CDF. the values d1 and d2 (which you can google up), do not have a term that has r, they do have the term F.

Sensitivity to interest rate of the above can by found by differentiating it with respect to r, which gives us:

$$\frac {dC}{dr} = -T.e^{-rT}[FN(d1) - KN(d2)]$$

But if you inspect the part right hand side to the dot (.) you will see that it is the same as C, there fore

$$\frac {dC}{dr} = -TC$$

As the result does not have a term r in it, whatever r happens to be, the result is always dependent only on T and C.

This is of course the mathematical explanation and if you have a different formula than Black76 to price the formula, then you can differentiate it with respect to r and see if the result is the same.

But if you are looking for a more intuitive answer, the reason is because the interest rate risk is already priced into the futures price, but it is not so if you use the spot price (which is relevant for options on spot/cash).

• If I understood correctly, what you have shown is that the "sensitivity to interest rates" (rho) is not a function of interest rates. But this sensitivity exists and is negative. Is that a fair summary? Sep 26 '20 at 6:54
• But aren't futures themselves sensitive to interest rates? Sep 26 '20 at 8:31
• @noob2 it is insensitive to changes in the interest rate. It is a constant. Think of it as something like $y = mx + c$, the rate of change of y is not dependent on x, it is always m. But if you take $y = mx^2 + c$, then the rate that y changes for a small amount of x, changes based on the value of x. In the case of Futures pricing we have this: $F = Se^{rT}$ $\frac {dF}{dr} = STe^{rT} = ST.F$ Therefore Futures pricing also depends on the Future price, similar to that of Options. The key is change of interest rate. Not the interest rate itself. Sep 27 '20 at 10:43