# Black (1976) model growth rate input for futures price

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 $$r_f$$)?

• In Black‘s model, $e^{r_s (T-t)}S_t$ should equal the futures price $F_{t,T}$ of the underlying. Thus you could back out $r_s$ given $t,T,F_{t,T}, S_t$. Or you simply plug in $F_{t,T}$ in the first place ... Feb 4, 2021 at 3:31

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.