Forget the Spot Price: Black's Formula for Options on Futures

Three years after the paper that made him famous, Fischer Black wrote a shorter, less celebrated piece about commodity futures. It contains a formula that is, today, probably used more often than the original Black-Scholes. Every interest rate cap, every floor, every swaption, every option on a crude oil future, every option on a bond future: they are priced with what the market simply calls Black-76.

The idea behind it is one of those reframings that makes an entire class of problems disappear.

The problem: what is the underlying, really?

Try pricing an option on gold using Black-Scholes and you immediately hit trouble. The model wants to know the cost of holding the underlying. For a share of stock, that is just the financing rate. For gold, you have to pay to store it, insure it, and guard it. For oil, storage is worse and sometimes there is nowhere to put it at all. For an agricultural commodity, there is spoilage. And for all of them there is something economists call the convenience yield: the intangible benefit of physically having the stuff on hand when you need it, which is precisely why a refiner will pay a premium for oil today rather than oil in three months.

Now try an option on a bond. What is the cost of carrying a bond? It pays you coupons, so it is negative, but it varies. And what is the "volatility" of a bond, when the bond is contractually guaranteed to be worth exactly its face value at maturity, so its price volatility must shrink to zero as it ages? Black-Scholes assumes constant volatility forever. A bond flatly refuses to cooperate.

Every one of these asset classes has its own carrying-cost mess. Modelling each mess separately would be a nightmare.

The key idea via analogy: let the futures market do the accounting

Black's insight: do not price off the spot price. Price off the forward or futures price.

Why? Because the futures price has already done all the messy accounting for you. The price of December oil today is not a mystery to be modelled. It is a number you can see on a screen. And that number already contains, baked in, the market's collective view of storage costs, financing costs, convenience yields, seasonality and everything else. The futures market has aggregated all of it into one observable quantity.

So if the option pays off based on where the futures price ends up, and the futures price is directly observable and tradable, why go anywhere near the spot?

Now comes the elegant part. What is the expected drift of a futures price?

Zero. A futures contract costs nothing to enter into. You post margin, but you do not pay a price. If entering a position costs nothing, then in a risk-neutral world it must be expected to make nothing, otherwise you would have a free lunch. The futures price is therefore a martingale: today's futures price is the risk-neutral expectation of tomorrow's.

That is a wonderfully clean starting point. All the drift terms that make commodity and bond modelling horrible have not been solved, they have been absorbed into the observable forward price and thus removed from the problem entirely. What remains is pure volatility.

The formula that follows looks almost exactly like Black-Scholes, but with the spot price replaced by the forward price, no carrying-cost term anywhere, and the whole thing discounted back at the risk-free rate. It is Black-Scholes with the awkward parts amputated.

Why it mattered

• It is the pricing language of the interest rate derivatives market, which is enormous. An interest rate cap is a portfolio of options on future interest rates. A swaption is an option on a swap rate. In both cases the underlying is a forward rate, which is observable and, under the right measure, driftless. Black-76 prices them. For decades, the multi-trillion-dollar rates options market has quoted, traded and risk-managed in Black volatilities. This is not a footnote in Black's career, it may be its most-used product.
• It unified the treatment of everything with a forward market. Commodities, bonds, rates, dividend-paying indices: instead of a separate carrying-cost model for each, you use one formula and let the forward curve carry the information. This is an enormous simplification and it is why the formula spread so fast.
• It handles the bond volatility problem gracefully. You are not modelling the bond's price as if it wanders forever. You are modelling the forward price of the bond for delivery on the option's expiry date, which is a perfectly well-behaved quantity over the option's life.
• It shifted the industry's mental model. Traders in these markets do not think in spot terms. They think in forwards and forward curves. Black-76 is both a cause and a symptom of that.

The honest limitations

• The forward is assumed lognormal, and sometimes it plainly is not. The most spectacular failure came when interest rates went negative. A lognormal model literally cannot represent a negative number, so Black-76 broke, catastrophically and publicly, when European and Japanese rates went below zero after 2014. Entire trading systems had to be rebuilt, either shifting rates upward before applying the formula (the "shifted lognormal" hack) or moving to normal, Bachelier-style models where negative values are permitted. Watching an industry discover that its default model could not represent an observed market price was instructive.
• Volatility is still assumed constant. The forward smile is real, and it is often violent in commodities, where the risk of a supply shock creates enormous fat tails. Anyone who priced oil options with a flat Black volatility in early 2020, when front-month crude futures famously traded below zero, would have been in serious trouble.
• Futures and forwards are not quite the same thing. Futures are marked to market daily, which creates a subtle interaction between the daily cash flows and the interest rate you earn on them. When rates are correlated with the underlying, as they very much are for bond and rate derivatives, futures prices and forward prices differ slightly. Black-76 papers over this, and the correction (the convexity adjustment) is a real, if usually small, item on a rates desk's list of concerns.
• It assumes constant, deterministic interest rates for discounting. For an option on an interest rate, that is an obviously uncomfortable assumption, and it is one of the main reasons more sophisticated rate models exist.
• Commodity forward curves move in complicated ways. They are not a single number but a whole curve, and it twists, steepens and inverts. A model built around one forward price at a time cannot capture the structure of how the curve as a whole moves, which is exactly what many commodity options depend on.

The one-line takeaway

Black's 1976 paper showed that if you price options off the forward price rather than the spot price, all the horrible carrying-cost details of commodities and bonds are already baked into an observable number and simply drop out of the problem, leaving a driftless underlying and a formula so useful that it, rather than Black-Scholes itself, became the standard language of the interest-rate and commodity options markets.