Spread Options And Kirk's Approximation

A refinery doesn't really care whether crude oil goes up or down in isolation — it cares about the "crack spread," the gap between what it pays for crude and what it sells refined gasoline for, since that gap is its actual profit margin. A spread option is built for exactly that kind of bet: its payoff depends on the difference between two prices, not on either price alone. Two assets can both rise 20% and a spread option on their difference pays nothing, because the gap between them didn't move.

The payoff and why it's hard to price

A spread call on two assets $S_1$ and $S_2$, struck at $K$, pays:

$$\text{Payoff} = \max(S_1 - S_2 - K,\ 0).$$

In words: $S_1$ and $S_2$ are the two underlying prices at expiry (say, gasoline and crude oil), and $K$ is a strike applied to their difference, often zero. The payoff is positive only if $S_1$ ends up more than $K$ above $S_2$. This looks like an ordinary call, but it's much harder to price in closed form: even if $S_1$ and $S_2$ are each individually lognormal (the standard Black-Scholes assumption), their difference is not — the difference of two lognormal variables has no clean, known distribution. That's the problem Kirk's approximation exists to solve.

Kirk's approximation

Kirk's trick is to treat the ratio $S_1 / (S_2 + K)$ as approximately lognormal — a ratio behaves much better than a difference — and plug an adjusted volatility into an otherwise ordinary Black-Scholes-style formula. The adjusted volatility is:

$$\sigma_{\text{Kirk}}^2 = \sigma_1^2 + \left(\frac{S_2}{S_2+K}\sigma_2\right)^2 - 2\rho\, \sigma_1 \left(\frac{S_2}{S_2+K}\sigma_2\right).$$

In words: $\sigma_1$ and $\sigma_2$ are the individual volatilities of the two assets, $\rho$ is the correlation between their returns, and $S_2/(S_2+K)$ is a weighting factor that shrinks $\sigma_2$'s contribution when strike $K$ is large relative to $S_2$. This has the shape of a portfolio-variance formula — variance of a difference is the sum of variances minus twice the covariance — which makes sense: a spread option is a bet on the volatility of a difference, and that volatility shrinks the more correlated the two legs are.

Worked example 1 — how correlation kills the spread's volatility

Gasoline and crude oil both have 30% annualized volatility, correlation $\rho=0.9$ (they usually move together), strike $K$ near zero so the weighting factor is close to 1. $\sigma_{\text{Kirk}}^2 \approx 0.30^2+0.30^2-2(0.9)(0.30)(0.30)=0.018$, so $\sigma_{\text{Kirk}}\approx13.4\%$ — far below either asset's own 30% vol, because high correlation means the two prices rise and fall together, leaving the gap comparatively stable. Now drop correlation to $\rho=0.2$: $\sigma_{\text{Kirk}}^2\approx0.144$, so $\sigma_{\text{Kirk}}\approx38.0\%$ — the spread is now more volatile than either leg alone, because low correlation lets the two prices drift apart largely independently.

Worked example 2 — plugging into the price

Suppose crude is $70/barrel, gasoline (equivalent units) is $78, strike $K$ = $5, one-year maturity, near-zero rates, and $\sigma_{\text{Kirk}}=13.4\%$ from above. Feeding that adjusted vol into a Black-Scholes-style $d_1$, $d_2$ calculation on the effective forward levels gives a spread call value of only a few dollars per barrel — small relative to either leg's own option value, because the low $\sigma_{\text{Kirk}}$ makes the spread hard to move far from its current $8 level within a year.

Drag the correlation slider from near 1 toward 0 and watch the scatter loosen — the same mechanism that inflates $\sigma_{\text{Kirk}}$: tightly correlated points keep the gap narrow and predictable, while a loose scatter lets it wander.

What this means in practice

Spread options are the core instrument in energy trading (crack spreads, spark spreads between power and gas) and show up in equity pairs trading and yield-curve products too. Kirk's approximation is the desk-standard formula because it's fast and closed-form, but it's an approximation — for wide strikes, low correlation, or very different individual volatilities, the lognormal-ratio assumption breaks down and desks fall back to Monte Carlo or numerical integration for anything that actually matters for risk.