Chapter 2: Term Structure of Commodity Futures Prices#

When you look at a barrel of oil today, you see one price. But a contract to deliver that barrel in three months, six months, or two years carries a different price. This chapter explains why those future prices differ, how we model those patterns, and how that knowledge helps you trade, hedge, and value energy contracts.

The Big Picture#

The term structure of futures prices — often called a futures curve — is simply the set of prices for the same commodity across different delivery dates. It tells you whether the market expects scarcity or plenty, how storage costs and yields shape value, and how seasonal forces in power markets twist the curve. Models of this curve help us price derivatives, manage risk, and match real market prices. This chapter builds from simple single‑factor descriptions all the way to rich multi‑factor models and shows you how the same ideas are adapted for electricity, where the forward curve looks very different from oil or gas.

The Shape of the Futures Curve and the Idea of Convenience Yield#

A commodity futures curve can slope upward or downward. When later‑delivery prices are higher than the nearby price, the curve is said to be in contango; when they are lower, it is in backwardation. Neither shape is “normal” — they simply reflect the balance between storage costs, interest rates, and the desire to hold physical inventory.

Imagine you own a tank of crude oil. Storing it costs money — rent, insurance, maybe spoilage. If you sell the oil today and invest the cash, you earn interest. So, in a perfect market, the price for future delivery should be today’s spot price plus storage costs and interest, minus any benefit you get from actually possessing the oil. That benefit is called the convenience yield.

Convenience yield: The hidden benefit you get from holding a physical commodity, not a futures contract. Think of it as a “dividend” from owning inventory — the ability to meet unexpected demand, keep a refinery running, or profit from temporary shortages.

When the convenience yield is high, owning the physical barrel right now is so valuable that the futures price can be lower than the spot price (strong backwardation). When it is low, storage and interest costs dominate and the curve is in contango. Models aim to capture this interplay mathematically, starting with a single driver.

📝 Section Recap: The futures curve shows prices across delivery dates; contango and backwardation reflect the balance of storage costs, interest, and convenience yield.

Black’s Single‑Factor Model#

Fischer Black’s classic 1976 single‑factor model starts with the simplest idea: the spot price of the commodity moves randomly, with a steady growth rate and volatility, just like a stock. That means the spot price changes as

where is the expected drift, is the volatility, and is a random Brownian increment. There is no separate convenience yield process — everything about storage and yield is summarized by a constant net convenience yield . In the risk‑neutral world (the one we use for pricing), the growth rate changes, and the futures price for delivery at time when today is becomes

Here is the risk‑free rate. The futures price simply grows at the difference between the interest rate and the net convenience yield. If , we have contango; if , backwardation. The curve is a smooth exponential function of time to maturity.

This model is simple and gives exact formulas for European options. But real commodity curves are not smooth exponentials. They can hump, dip, and have different volatility at different maturities. One factor cannot create the complex shapes we see.

📝 Section Recap: Black’s one‑factor model treats the spot price as a random walk with a constant net convenience yield, giving a smooth exponential curve — a good start but too simple for real markets.

The Gibson–Schwartz Two‑Factor Model#

To get more realistic curves, Gibson and Schwartz added a second random factor: the convenience yield itself. In their Gibson–Schwartz two‑factor model, the spot price and the instantaneous convenience yield move as

with correlation between the two Brownian motions and . Here is how fast the convenience yield snaps back to its long‑run average , and is how much it jumps around.

The intuition is simple: the spot price grows with an expected drift that now depends on the convenience yield — a high yield pushes the drift down, because owning the physical asset is already providing value. Meanwhile, the convenience yield oscillates around a normal level; when times are tight it spikes, and when storage is plentiful it falls.

In the risk‑neutral world, we adjust for the risk of the convenience yield, and the futures price becomes a formula that depends on both and :

where and are fixed functions of the model’s parameters. Now the futures curve can tilt and change shape as the convenience yield moves. A jump up in pushes the whole curve down (stronger backwardation); a drop flattens it or turns it into contango.

This model is popular because it fits many markets and gives exact formulas for simple options on futures.

📝 Section Recap: The Gibson–Schwartz model adds a mean‑reverting convenience yield as a second factor, allowing the futures curve to change shape over time and matching observed humps and tilts.

The Schwartz–Smith Model: Decomposing Short‑Term and Long‑Term Drivers#

Schwartz and Smith took a different two‑factor approach. Instead of spot price and convenience yield, their Schwartz–Smith model splits the log spot price into two hidden parts:

• A short‑term deviation that mean‑reverts rapidly to zero.
• A long‑term equilibrium level that follows a random walk with drift.

In mathematical form,

with

Here captures short‑lived shocks like a supply glut or a cold snap that quickly fade. is the long‑run price level, driven by things like technology, global demand, and resource depletion — a kind of permanent “fair” price. The two random parts can be correlated.

This split is very useful for hedging and investing. A short‑term trader cares about ; a long‑term project developer looks at . The futures price becomes

where is a fixed function of the risk‑neutral parameters. The model can show contango or backwardation, and by separating mean‑reverting and trending parts, it often fits data with fewer parameters than a three‑factor model. It also connects naturally to Gibson–Schwartz: the short‑term factor is basically a mix of spot and convenience yield, and the long‑term factor is another mix.

📝 Section Recap: The Schwartz–Smith model splits the log spot price into a quickly decaying short‑term deviation and a persistent long‑term trend, making it ideal for risk management across different time horizons.

Moving to Three Latent Factors#

Two factors explain a lot, but studies show three factors often work even better. A latent three‑factor model adds a third hidden state variable — often a random long‑run average for the convenience yield, random volatility, or an extra short‑term cycle. The log spot price is a sum of the three hidden states, and each follows a normal (Gaussian) process in the risk‑neutral world.

A generic representation is

where is a vector of three state variables (the hidden factors) following

with controlling how fast each factor reverts to its mean, giving the long‑run levels, and the volatility matrix. The futures price is then an exponential‑affine function of the state variables:

where and solve ordinary differential equations. The three factors are usually interpreted, after rotation, as a level, slope, and curvature of the futures curve — similar to the principal components we will see next.

These models can fit many curve shapes and volatility patterns, but they need careful fitting to liquid futures and options, often using Kalman filters or Bayesian methods because the factors are hidden.

📝 Section Recap: Three‑factor models add an extra hidden variable, giving more flexibility to match the level, slope, and curvature of real curves while still giving exact formulas.

Principal Component Analysis of Convenience Yield and Curve Dynamics#

When we have a history of futures prices for many delivery dates, we can back out the convenience yields from the cost‑of‑carry formula and see how they move together. Principal Component Analysis (PCA) finds the main patterns of movement without assuming any economic model.

A typical PCA on weekly changes in convenience yields across maturities shows that the first three components explain over 95% of total variance. The first component is a parallel shift — all maturities move up or down together, matching a change in the overall level of the curve. The second component is a slope factor, tilting the curve around the middle. The third adds curvature, making the belly move relative to the ends.

These results directly support using two‑ and three‑factor models. If the first three components capture almost all movement, a model with three random factors should be enough. The PCA loadings also suggest how to interpret the hidden variables: the level factor matches a long‑term price, the slope factor matches near‑term supply‑demand tightness, and the curvature matches temporary imbalances at medium maturities. Fitting to liquid futures then makes sure the model’s factors behave like these real‑world components.

📝 Section Recap: PCA of convenience yield changes shows that a few factors — level, slope, curvature — drive most of the variation, providing a data‑driven justification for multi‑factor term‑structure models.

The Electricity Price Forward Curve#

Electricity is a flow commodity; you can’t store it cheaply. That makes its forward curve — often called the Price Forward Curve (PFC) — very different from storable commodities. There is no simple cost‑of‑carry formula. Instead, the curve comes from expected spot prices, risk premiums, and the physical details of generation and demand.

The PFC is built from a set of liquid forward contracts: baseload monthly, quarterly, and annual forwards, plus peak and off‑peak products. Because electricity is delivered continuously over a period, the forward price for a contract covering delivery interval is

where is the instantaneous spot price under the risk‑neutral measure . This integral smooths out hourly spikes, so the PFC is usually much less jumpy than the spot price.

To estimate a smooth PFC from a handful of traded block forwards, we assume a parametric form for the instantaneous forward price . A popular choice models as a sum of a long‑term baseline and seasonal components:

where is the time of year. We choose the coefficients so that the average of over each contract period exactly matches the market forward prices. Since we have only a few traded contracts, we often add a smoothness penalty (like a roughness penalty) to avoid unrealistic wiggles.

This fitting is usually a linear inverse problem: given forward quotes for delivery periods , we solve for the seasonal parameters by minimising

The parameter controls how smooth the curve is. The resulting PFC shows strong seasonal patterns: higher in winter and summer (heating and cooling demand) and lower in spring and fall. It can also show hourly shapes if we add daily profiles.

When we put this into a multi‑factor spot model, the forward curve factors (level, slope, seasonal strength) become state variables. The factor loadings for each contract tell us how sensitive the forward price is to each factor. For example, a level factor has a loading of 1 at all maturities; a seasonal factor has loadings that follow a cosine wave. Fitting the loadings together with the seasonal function makes sure the model matches both the observed forward curve and the expected factor behavior.

📝 Section Recap: The electricity PFC is built by fitting a smooth, seasonal forward function to a few liquid block contracts, using a roughness penalty to keep it realistic. Factor loadings then connect this curve to the state variables of a multi‑factor spot model.

Summary#

We started with the simplest one‑factor model of a commodity futures curve and moved all the way to rich hidden‑factor models and the unique challenges of electricity. The key idea: a small number of random drivers — level, slope, curvature — can describe most of the movement in futures prices across different delivery dates. Black’s model gave us a first formula, but adding a mean‑reverting convenience yield or splitting the price into short‑ and long‑term parts made it much more realistic. When three factors are needed, PCA shows they capture almost all the variation. Electricity makes us think differently: without storage, the forward curve is a fitted mix of expected spot and seasonal patterns, with factor loadings linking it back to a dynamic model. This framework is the foundation for pricing, hedging, and risk management in commodity and energy markets.