Interest-Rate Sensitivity of Futures Options Under Black 76
Summary
The document examines interest-rate sensitivity for an option on a futures contract when the option and futures use futures-style settlement. It uses the Black 76 call-pricing expression, in which the option value is discounted by the risk-free rate while the futures price enters the payoff terms. Differentiating that expression with respect to the rate gives rho as minus time to expiry multiplied by the call value, according to the answer.
The explanation also offers an intuition: the interest-rate exposure is already reflected in the futures price, unlike an option priced from spot, where the financing relationship matters directly. The answer cautions that the result depends on the pricing formula used, so another model should be differentiated separately. Its claim of insensitivity needs care: the displayed formula still has a rate-dependent discount factor and a nonzero rho, so it demonstrates a specific sensitivity relationship rather than zero sensitivity.
Key ideas
- The answer analyzes futures options using the Black 76 pricing model.
- It derives call rho as minus time to expiry multiplied by the call price.
- The futures price incorporates interest-rate effects that appear differently in spot-option pricing.
- The stated sensitivity result depends on the pricing model and its assumptions.
Tags
Full text
# Why are options on futures that are subject to futures settlement insensitive to changes in interest rates?
# 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?
## Answer by Kulendra 'KJ' Janaka (score 1)
https://quant.stackexchange.com/a/58222
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).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.