How does Black's model for options on futures differ from the standard Black-Scholes model?

Black's model (1976) is essentially Black-Scholes adapted for options where the underlying is a futures contract rather than a spot asset. The key simplification is that the futures price already incorporates the cost of carry, so you don't need to worry about dividends, storage costs, or the risk-free rate in the drift.

The Formula

Call on Futures:

c = e^{-rT} [F_0 N(d1) - K N(d2)]

Put on Futures:

p = e^{-rT} [K N(-d2) - F_0 N(-d1)]

Where:

• d1 = [ln(F_0/K) + (sigma^2 / 2) T] / [sigma sqrt(T)]
• d2 = d1 - sigma sqrt(T)
• F_0 = current futures price
• K = strike price
• sigma = volatility of the futures price
• T = time to option expiry
• r = risk-free rate

Key Differences from Standard BSM

Why zero drift? In a risk-neutral world, the futures price is already an unbiased estimate of the expected future spot price. It doesn't need to grow at the risk-free rate because no upfront investment is required to enter a futures position.

Worked Example

Silverstone Trading is pricing a 3-month call option on crude oil futures.

• Current futures price F_0 = $78.50
• Strike K = $80.00
• Volatility sigma = 28%
• Risk-free rate r = 5.0%
• T = 0.25 years

d1 = [ln(78.50/80) + (0.28^2/2)(0.25)] / [0.28 x sqrt(0.25)]

d1 = [-0.01897 + 0.0098] / [0.14]

d1 = -0.00917 / 0.14 = -0.0655

d2 = -0.0655 - 0.14 = -0.2055

N(-0.0655) = 0.4739, N(-0.2055) = 0.4186

c = e^{-0.05 x 0.25} [78.50 x 0.4739 - 80 x 0.4186]

c = 0.9876 x [37.20 - 33.49]

c = 0.9876 x 3.71 = $3.66

Black's model is widely used in practice for caps, floors, swaptions, and commodity options. For your FRM exam, know when to apply it versus standard BSM.

Dive deeper with our FRM derivatives course.