16Commodity and Energy Derivatives

In the second week of February 2021 a cold snap froze gas wells across Texas and the southern plains. Spot gas at the Waha hub in West Texas reached 206.19 dollars per million British thermal units on 16 February; at one Oklahoma hub the average price on 17 February was 1 192 dollars, against under 3 dollars in the first week of the month, according to the US Energy Information Administration. Henry Hub, the benchmark, went from 2.88 dollars on 1 February to 23.86 on the 17th and back under 3 by the 23rd. For a week, the value of gas in the ground depended on how fast it could be taken out: an owner of a salt cavern that can empty in days could sell at the peak, one with a slow reservoir could not. This chapter values that flexibility. One Quant Book 3 described the forward curves, storage economics and structured contracts of commodity markets; here they get models: a forward curve that moves in two factors, options on the spread between two prices, and the valuation of storage and swing rights by optimisation under uncertainty.

16.1 Factor models of the forward curve

A commodity has a forward curve rather than a price: each delivery month trades, with seasonal shape and a convenience yield (One Quant Book 3, chapter 10). A model must move the whole curve, and it must let short-dated contracts move more than long-dated ones.

Both factors are Gaussian and simulated exactly, and each forward is lognormal: an option on the forward $F(\cdot,T)$ expiring at $t$ is priced by Black’s formula with the implied volatility $\sqrt{V(t,T)/t}$.

16.2 The Samuelson effect and forward volatility

In the model the instantaneous volatility of a forward with time $\tau$ to delivery is

68.7% at delivery, 27.2% one year out and 20.0% five years out for the calibrated parameters. This is the Samuelson effect of One Quant Book 3 as a model property: shocks to supply and demand move near contracts, which the short-term factor carries, while far contracts move only with the long-term level (Figure 16.1). The correlation between two contracts falls with the distance between their deliveries, which is what makes spread options worth something.

16.3 Spread options

With $K = 0$ and lognormal prices the option is an exchange option, priced exactly by Margrabe’s formula with the volatility of the ratio, $\sigma^2 = \sigma_1^2+\sigma_2^2-2\rho\sigma_1\sigma_2$. With $K\ne0$ the difference is not lognormal and there is no closed form.

def kirk(F1: float, F2: float, K: float, T: float, s1: float, s2: float, rho: float, df: float = 1.0) -> float:
    """Kirk's approximation to a call on F1 - F2 - K: F2 + K treated as lognormal with vol s2 F2 / (F2 + K)."""
    b = F2 / (F2 + K)
    s = math.sqrt(s1 * s1 - 2 * rho * s1 * s2 * b + (s2 * b) ** 2)
    d1 = (math.log(F1 / (F2 + K)) + 0.5 * s * s * T) / (s * math.sqrt(T))
    return df * (F1 * ND.cdf(d1) - (F2 + K) * ND.cdf(d1 - s * math.sqrt(T)))

16.4 Storage valuation

A storage facility buys gas when it is cheap, holds it and sells it when it is dear, within limits on capacity and on injection and withdrawal rates, and at a cost per unit moved. Its owner holds a portfolio of calendar spread options linked by the inventory.

def lsm(fac: Facility, spot: np.ndarray, dfs: list[float], degree: int = 2) -> float:
    """Least-squares Monte Carlo value of the facility on simulated spot (paths x periods)."""
    npaths, n = spot.shape
    cap = fac.capacity
    V = np.full((npaths, cap + 1), -1e18)
    V[:, fac.end if fac.end >= 0 else slice(None)] = 0.0
    for m in range(n - 1, -1, -1):
        S = spot[:, m]
        X = np.column_stack([S ** d for d in range(degree + 1)])
        cont = np.empty_like(V)
        for j in range(cap + 1):
            if V[0, j] <= -1e17 and (V[:, j] <= -1e17).all():
                cont[:, j] = -1e18
            else:
                beta, *_ = np.linalg.lstsq(X, V[:, j], rcond=None)
                cont[:, j] = X @ beta
        W = np.full_like(V, -1e18)
        for i in range(cap + 1):
            best_est = np.full(npaths, -np.inf)
            chosen = np.zeros(npaths)
            for a in fac.actions(i):
                if (V[:, i + a] <= -1e17).all():
                    continue
                now = dfs[m] * fac.cash(a, S)
                est = now + cont[:, i + a]
                real = now + V[:, i + a]
                better = est > best_est
                best_est = np.where(better, est, best_est)
                chosen = np.where(better, real, chosen)
            W[:, i] = np.where(np.isfinite(best_est), chosen, -1e18)
        V = W
    return float(V[:, fac.start].mean())

16.5 Swing contracts

A swing contract (One Quant Book 3, chapter 12) is storage without injection: the holder has a number of rights to take gas at a fixed price, limited per period and in total. The same dynamic programme values it, with the inventory counting rights left.

16.6 Tutorial: storage under a two-factor curve

16.7 Build: the energy model

16.8 Exercises

16.9 Problem: The Salt Cavern in February 2021

16.10 Interview questions