Black (1976) model growth rate input for futures price

Question

When using the Black 76 model for pricing European index options I've often seen people use 2 different rates: the typical risk free rate used to get the discount factor, and a growth rate used to get the forward price. The adjusted equation for a call option (assuming no dividends) using $r_g$ as the growth rate and $r_f$ as the risk free rate would be

$$C = e^{-r_fT}[e^{r_gT}SN(d_1) - KN(d_2)]$$

I'm not totally sure what rates to use for each of these and I am having trouble finding information about it online. What is the reasoning behind having two separate rates? What would someone typically use for each rate (for example LIBOR forward for $r_g$ and OIS discount for $r_f$)?

Answer 1

As @Kermittfrog said in the comment, in Black formula for options on futures price you need to insert the futures price $F$:

$$C = e^{-rT}[FN(d_1) - KN(d_2)]$$

where $r$ is the discounting rate. Here, $d_1$ depends only on $F$ (no rate involved).

For Black-Scholes formula for options on spot price (assume asset pays no dividend to keep it clean), we have:

$$C = e^{-rT}[e^{r_RT} S N(d_1) - KN(d_2)]$$

where $r$ is the discounting rate and $r_R$ is the financing rate of the underlying asset (if it can be repoed, it will be lower, if not, it will be higher). Here, $d_1$ depends on $S$ and $r_R$.

The discount rate $r$ depends on whether the option itself is collateralized (there is CSA) or not (there is no CSA), so it will be a collateral rate or a funding rate, respectively. See Piterbarg's Funding beyond discounting article.