Topic 5 – Black–Scholes vs Black-76

1. What are they, in one sentence each?

• Black–Scholes (BS): Model + closed-form formula for pricing European options on a spot asset (equity-style), assuming lognormal spot with constant volatility and continuous trading.

• Black-76 (a.k.a. Black model): Same idea, but for European options on a forward/futures price. It takes the forward $F_{0,T}$ and discount factor $D(0,T)$ as inputs.

Mental shortcut:

Black-76 = Black–Scholes with “spot” replaced by forward and everything discounted by $D(0,T)$.

2. Black–Scholes: what you actually feed it

Inputs for a call on spot:

• $S_0$: current spot price.
• $K$: strike.
• $T$: time to maturity (in years).
• $r$: risk-free rate (continuously compounded).
• $q$: continuous dividend yield (or carry).
• $\sigma$: volatility (assumed constant).

Formula (risk-neutral world):

• Dynamics under $\mathbb{Q}$: $$\frac{dS_t}{S_t} = (r - q)\,dt + \sigma\,dW_t$$

• Forward under this model: $$F_{0,T} = S_0 e^{(r - q)T}$$

• European call price at 0: $$C_0 = S_0 e^{-qT} N(d_1) - K e^{-rT} N(d_2)$$

• Put price: $$P_0 = K e^{-rT} N(-d_2) - S_0 e^{-qT} N(-d_1)$$

where

$$d_1 = \frac{\ln(S_0/K) + (r - q + \tfrac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T}.$$

Practitioner instinct:

• BS is a spot model: you need a clean spot, a dividend / carry rate, and a vol.
• In power/gas, spot is often messy; that’s why we like Black-76 more.

3. Black-76: natural for futures/forwards (commodities)

For an option on a forward/future with:

• $F_{0,T}$: current forward/futures price for expiry $T$,
• $K$: strike,
• $D(0,T)$: discount factor to $T$,
• $\sigma$: volatility of the forward,
• $T$: time to option expiry (same as underlying delivery date if European).

Black-76 call:

$$C_0 = D(0,T)\,\big(F_{0,T} N(d_1) - K N(d_2)\big)$$

Put:

$$P_0 = D(0,T)\,\big(K N(-d_2) - F_{0,T} N(-d_1)\big)$$

with

$$d_1 = \frac{\ln(F_{0,T}/K) + \tfrac{1}{2}\sigma^2 T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T}.$$

Same structure as BS, except:

• No explicit $r, q$: they are hidden inside $F$ and $D$.
• Cleaner when you already have:
  • a discount curve $D(0,T)$,
  • a forward curve $F(0,T)$.

In power/commodities this is the default.

“Option on Cal-27 Baseload? Use Black-76 with the Cal-27 forward price and discount factor to that date.”

4. When do practitioners actually use these?

5. Limitations – especially bad in power & commodities

6. How to talk about this in an interview

Short, calm sentences:

• When used:

  “For plain European options on forwards or futures, I’d typically use Black-76: discount factor, forward price, and a volatility. In equity-style settings with clean spot and dividends, Black–Scholes is the equivalent.”

• Why forwards & Black-76 in commodities:

  “In power and gas, forwards are the primary liquid object; spot is noisy and delivery-based. Black-76 naturally takes the traded forward and the discount curve as inputs, so it aligns with how the market quotes.”

• Limitations:

  “Both models assume lognormal prices and constant volatility. That’s already questionable in equities and particularly wrong in power, where you have spikes, negative prices, and strong seasonality. In practice we use Black-76 vols as a quoting surface, but for risk and complex structures we move to more realistic models or Monte Carlo.”