Energy Derivatives: Oil, Gas, and Why They Are Different

Wilmott opens Chapter 72 with a confession: “I don’t believe much of what I’m writing, and by the end of the chapter I hope you’ll see why.” That is an unusual way to start a textbook chapter, but it captures something real about energy derivatives. The models are borrowed from other markets, the assumptions are shakier than usual, and the underlying commodities behave in ways that break standard financial theory. But you have to start somewhere.

What Makes Energy Markets Special

Energy markets trade natural gas, various types of oil, and electricity. They share some features with other commodity markets (there is a spot price, a futures curve, and storage costs), but the differences are extreme.

Spot prices are wildly volatile. Not “20% per year” volatile like equities. We are talking about prices that can multiply by 60 in a single day and drop back the next day. Wilmott shows a table of hourly electricity prices where the cost goes from $14.60 at midnight to $900 by 1 PM and back to $19.70 by 11 PM. In one day.

The reason is simple: electricity cannot be easily stored. When demand spikes (hot afternoon, everyone turns on air conditioning), there is no warehouse of electrons to draw from. Generators have to fire up, and if all the cheap ones are already running, you pay whatever the marginal producer demands. When demand drops, the price collapses back.

Natural gas shows similar spikes, though less extreme. Oil is better behaved because it can be stored in tanks and pipelines. The key factors across all energy commodities are:

• Basis risk from location. Electricity in Texas is a different product from electricity in New York. You cannot perfectly hedge one with the other because transmission has limits and costs.
• Basis risk from time. Tomorrow’s electricity is a completely different product from today’s electricity. You cannot store today’s delivery for use tomorrow (at least not cheaply).
• Large jumps and mean reversion. Prices spike and then fall back. The long-term forward curve is much more stable than the spot.
• Seasonality. Heating oil is expensive in winter. Electricity is expensive in summer (in the US). Natural gas futures show a clear seasonal pattern that repeats every year.

Why Black-Scholes Fails Here

The basic Black-Scholes model assumes a lognormal process with constant volatility. For energy, this fails in multiple ways.

First, the actual volatility of the spot price is enormous, especially for electricity. Plugging that volatility into Black-Scholes would give option prices that are absurdly high. But the forward curve for delivery in six months is much less volatile than the spot. The volatility decays rapidly with maturity. Wilmott shows implied volatilities for Brent crude oil options that drop sharply as you move out along the forward curve.

Second, the no-arbitrage framework assumes you can store the underlying and trade it freely. For electricity, storage is either impossible or extremely expensive (pump water uphill, convert back to electricity later). Without storage, the link between spot and forward prices breaks down. You cannot exploit “mispricings” because you cannot hold the commodity.

Third, hedging assumes you can trade the underlying continuously. For electricity delivered to a specific hub on a specific day, there may be no liquid market to hedge with. Different delivery points are correlated but not perfectly so.

All of this means we need something beyond the standard lognormal model.

The Convenience Yield

The convenience yield is a concept borrowed from general commodity theory. It is to energy what dividend yield is to stocks. Specifically, it measures the net benefit of physically holding the commodity minus the cost of storage.

Why would holding a barrel of oil have a benefit beyond its market value? Because if you are a refinery and you need oil next week, having it in your tank right now means you can operate without worrying about supply disruptions. That certainty has value. You would pay a premium for it. That premium, net of storage costs, is the convenience yield.

If the convenience yield were constant, the relationship between forward and spot prices would be simple: the forward price equals the spot price times e raised to the power of (r minus q) times the time to delivery, where r is the interest rate and q is the convenience yield. But the convenience yield is not constant. It varies with the spot price (high when supply is tight and prices are high), with the season, and with general market conditions.

The Pilopovic Two-Factor Model

The chapter presents a model that tries to capture the key features of energy prices. Instead of modeling just the spot price, it uses two factors: the spot price S and a long-term equilibrium price L.

The spot price mean-reverts toward L. The speed of mean reversion is fast, which explains why spikes die out quickly. Meanwhile, L itself follows a geometric Brownian motion with a slow drift, representing the gradual changes in the fundamental value of the commodity (new discoveries, shifting demand patterns, geopolitics).

The model equations look like this. S reverts to L with speed alpha, and alpha is much larger than the long-term drift mu of L. When S is above L, it tends to fall. When below, it tends to rise. The two noise sources are uncorrelated, giving the model two independent dimensions of randomness.

The convenience yield in this framework depends on the ratio S/L. When S is much higher than L (supply crunch), the convenience yield is high. When S is close to L or below, the convenience yield is low. This is modeled as a function q(S, L, t) that is calibrated to match the observed forward curve.

The forward price equation derived from this model has a nice structure. The forward price F can be written as S raised to a time-dependent power times L raised to another time-dependent power, times an exponential correction. The powers approach zero and one respectively as the delivery date gets farther away, meaning long-dated forwards depend mainly on L (the equilibrium price) and barely on S (the volatile spot).

This captures the key observation: short-dated forwards are volatile because they track the volatile spot. Long-dated forwards are stable because they track the stable equilibrium price.

Fitting the Model

To make the model practical, you need to calibrate it to the observed forward curve and its volatility structure. The forward curve gives one equation relating the convenience yield parameters to observables. The volatility term structure gives another.

Seasonality is layered on top. You might add Fourier series terms (sinusoidal components) to the spot price model to capture the annual cycle in natural gas or heating oil prices.

Types of Energy Derivatives

The energy markets trade many of the exotic structures we have seen in other chapters, adapted to the peculiarities of the market.

One-day options on electricity are popular in the US. Given the price swings we saw in the hourly data, these are almost pure bets on whether a spike happens. Some contracts include timewise averaging to smooth out the craziest moves.

Asian options are very common because averaging reduces the impact of price spikes. There are two flavors: averaging over realized (settled) spot prices and averaging over unsettled forward prices. The first is a standard path-dependent problem. The second requires an accurate forward curve model because the payoff depends on the future shape of the curve, not just the spot price.

Caps and floors limit the price paid for delivery. They often come with averaging features. A cap at $50/MWh means you never pay more than that for your electricity, regardless of spikes.

Cheapest to deliver contracts let the seller choose which delivery point to use. They will obviously pick the cheapest one. This is a multi-asset problem where the correlation between prices at different hubs matters a lot.

Basis spreads are bets on the price difference between two similar but distinct commodities, like two grades of oil. Again, correlation is the key parameter.

Swing options give the buyer flexibility to vary delivery amounts within limits, and perhaps to choose which days to take delivery. This is a stochastic control problem similar to the passport options from Chapter 27.

Spread options are options on the difference between the price of fuel and the price of electricity. If you own a gas-fired power plant, your profit is roughly the electricity price minus the gas price. A spread option hedges this gap. The “spark spread” (gas-to-electricity efficiency) is a closely watched quantity.

The Honest Assessment

Wilmott ends the chapter the way he started it: with skepticism. Energy derivatives are fascinating, the models are interesting, but there is a long way to go before they become truly satisfactory.

The fundamental problems are hard. You cannot store the underlying easily. You often cannot hedge. Prices jump by factors of 10 or more and then snap back. Mean reversion is real but the reversion speed might itself be random. Seasonality adds another layer of complexity.

The Pilopovic model is a good starting point. It captures mean reversion to an equilibrium price, the term structure of volatility, and (with extensions) seasonality. But it assumes you can hedge, which is dubious for electricity. Wilmott suggests that mean-variance models, which acknowledge unhedgeable risk, might be more appropriate.

For anyone working in energy markets, this chapter is less of a recipe and more of an orientation. Here are the key features of the market, here is one reasonable model, and here is why you should be careful about trusting it too much.

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