Black 76 Rho for Options on Futures
Summary
The discussion examines rho for a call option priced with the Black 76 model. Under the stated setup, the futures price is modeled without risk-free-rate growth, and the quoted call price discounts the payoff at the risk-free rate. If the futures price and the Black formula’s d-values do not depend on that rate, differentiating the price with respect to the rate gives rho as negative time to expiry multiplied by the call price. This supports the questioner’s proposed expression and rejects the alternative that differentiates only the strike term.
The answer adds an important qualification: cost-of-carry relationships can make the futures price depend on the interest rate. For equity-index forwards with the stated carry relation, the formulation can reduce to the familiar Black–Scholes setting; for VIX futures, that relation need not apply. The result therefore depends on what is held fixed when taking the rate derivative, and the discussion notes that modeling VIX futures as geometric Brownian motion is itself debatable.
Key ideas
- In the stated Black 76 setup, the futures price and d-values are independent of the risk-free rate.
- Differentiating the discounted call price then gives rho equal to negative expiry time times the option price.
- A cost-of-carry relationship can make the futures price rate-dependent and change the derivative setup.
- Whether Black 76 is appropriate depends on the contract and modeling assumptions, including for VIX futures.
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Full text
# What is the Rho of an option on a futures contract priced using the Black 76 model?
# What is the Rho of an option on a futures contract priced using the Black 76 model?
I wanted to quickly confirm some simple calculations for the Black 76 greeks and was making use of the formulas on this website:
http://riskencyclopedia.com/articles/black_1976/
I have an issue with their statement of Rho and I wanted to check with you guys. Their version for a call has:
$$\rho = \tau e^{-r\tau}K\Phi(d_2)$$
Whereas I feel it should be:
$$\rho = \tau e^{-r\tau}(K\Phi(d_2) - F \Phi(d_1))$$
i.e.
$$\rho = - \tau C$$
Where $C$ is the Black76 call option price:
$$C = e^{-r\tau}(F \Phi(d_1) - K\Phi(d_2))$$
Would really appreciate your input as the one person to post on my comment disagrees with me!
## Answer by Richi Wa (score 4, accepted)
https://quant.stackexchange.com/a/11495
I agree with you. In the setting of Black 76 we consider an option on a forward/futures price $F$. $F$ is modelled as geometric Brownian motion and in contrast to the usual BM model the futures price does not grow with the risk-less rate.
Then you formula for the call (C) is correct. Note that in general neither $F$, $d_1$ nor $d_2$ contain the term $r$. Then taking the derivative w.r.t. $r$ we arrive at your formula.
In the case where the forward price can be found by cost-of-carry pricing (e.g. options on equity indices things simplify because then $$F_t = S_0 \exp((r-q)t),$$ which simplifies further if $q=0$ Then the $r$ terms cancles out but then in fact we are back in the BS model. Thus I would say: if cost-of-carry in the sense of the above formula holds, then Black-Scholes and Black 76 are just the same thing.
In the example pf VIX-futures cost-of-carry does not hold and one needs B76 (if one wants to model VIX-futures as GBM which is debateable).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.