Black-76 Model – Futures Options

---

Black’s brilliant model was originally developed for pricing European futures options (“The Pricing of Commodity Contracts,” Journal of Financial Economics, 3 (March 1976)). Later, it turned out to be very convenient in practice for spot options as well.

The main assumption of the model is that, in a risk-neutral world, futures prices evolve in the same way as dividend-paying stock prices, if the dividend payouts are equal to the risk-free rate (this can be mathematically proven using the risk-neutral probability formulas).

Difference between Black-Scholes and Black-76:

The main conclusion derived from the key assumption is that, in a risk-neutral world, the drift of the futures price is zero; only volatility exists, and the expected price equals the current price. Intuitively, since entering a futures contract is costless, the futures price cannot systematically drift in a risk-neutral environment.

Accordingly, the model formulas are arranged as follows:

Note: It can be proven that if the storage costs and convenience yield of the asset are only functions of time, the volatility of the futures price is identical to the volatility of the spot price.

Using Black-76 instead of Black-Scholes-Merton:

If the futures and European option contracts expire at the same time, then a spot option and a futures option yield mathematically identical payoffs, because at that point in time the futures and spot prices are equal.

This equality allows us to price spot options using the Black-76 model, which does not require knowledge of the asset’s income expectations (Dividend Yield, Convenience Yield, Cost of Carry). Black-76 leaves the task of determining these complex and sensitive components to the market, because the futures price already incorporates all of them. This is why Black-76 is widely used in practice by traders.

Examples: Excel – Valuing European Futures and Spot Options

P.S.

A few notes regarding American futures options:

• American futures options are generally more valuable than European futures options, because there is a chance that early exercise may be optimal before expiration.
• Even if the futures and option deadlines coincide, for American options, the futures and spot payoffs are not equal, because the spot and futures prices differ before expiration; accordingly, option prices must also differ.
• The logic in the second point also applies when the option and futures deadlines do not coincide. The difference between spot and futures option prices grows as the gap between their expiration dates increases.

Source: Options, Futures & Other Derivatives, John C. Hull

---