Spread Option Pricing & Kirk's Approximation Lab

Model two-asset European spread options with non-zero strikes using Kirk's (1995) analytical approximation. Quantify crack spreads, spark spreads, cross-asset Greeks, and correlation sensitivities.

Market Presets:
Contract & Asset Parameters
Kirk Spread Value

$5.3284

Analytical Approximation
Intrinsic Value

$3.6000

Spread Minus Strike
Time (Extrinsic) Value

$1.7284

Volatility Premium
Effective Vol ($\sigma_K$)

16.42%

Kirk Weight-Adjusted
Margrabe Limit ($K=0$)

$28.9142

Pure Exchange Value
Daily Theta ($\Theta$)

-$0.0215

1-Calendar-Day Decay
Cross-Asset Risk Sensitivities (Greeks)
Delta 1 ($\Delta_1$) Delta 2 ($\Delta_2$) Gamma 1 ($\Gamma_1$) Gamma 2 ($\Gamma_2$) Cross-Gamma ($\Gamma_{12}$) Vega 1 ($\mathcal{V}_1$) Vega 2 ($\mathcal{V}_2$) Rho-Corr ($\frac{\partial C}{\partial \rho}$)
+0.6842 -0.5189 0.038241 0.022194 -0.029104 $0.1182 $0.0715 -$0.1245

Plots the option's current fair value curve against intrinsic expiration payoff as Asset 1 ($S_1$) varies while Asset 2 ($S_2$) and Strike ($K$) are held fixed.

Highlights the negative slope of spread option value with respect to correlation $\rho$. As correlation approaches +1.0, the variance of the difference collapses toward zero.

Compares Delta 1 (long sensitivity to $S_1$), Delta 2 (short sensitivity to $S_2$), and Vega 1 across the Asset 1 price spectrum.

5x5 Sensitivity: Asset 1 Price vs. Asset 2 Price on Option Price ($)
5x5 Sensitivity: Correlation ($\rho$) vs. Volatility 1 ($\sigma_1$) on Value ($)

Quantitative Foundations of Two-Asset Spread Options & Kirk's Formula

1. The Two-Asset Spread Pricing Challenge

A European spread option confers the right, but not the obligation, to receive the difference between two asset prices $S_1$ and $S_2$ above a pre-specified strike $K$ at maturity $T$:

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

When both assets follow correlated Geometric Brownian Motion (GBM): $$dS_i(t) = (r - q_i)S_i(t)dt + \sigma_i S_i(t) dW_i(t), \quad dW_1(t)dW_2(t) = \rho dt$$ the terminal spread $S_1(T) - S_2(T)$ is a difference of two lognormal variables. Because the difference of two lognormal variables has no closed-form probability distribution, an exact analytical formula is non-trivial.

2. Kirk's (1995) Lognormal Approximation

E. Kirk (1995) introduced a remarkably accurate approximation by treating the combined denominator $F_2 + K$ as a single pseudo-asset. Setting effective forward $F = \frac{F_1}{F_2 + K}$, the option is priced as:

$C = e^{-rT} [F_1 N(d_1) - (F_2 + K) N(d_2)]$

where $F_1 = S_1 e^{(r-q_1)T}$, $F_2 = S_2 e^{(r-q_2)T}$, and the effective volatility $\sigma_K$ is weighted by $w = \frac{F_2}{F_2 + K}$: $$\sigma_K = \sqrt{\sigma_1^2 + w^2 \sigma_2^2 - 2 \rho w \sigma_1 \sigma_2}$$ $$d_1 = \frac{\ln\left(\frac{F_1}{F_2 + K}\right) + \frac{1}{2}\sigma_K^2 T}{\sigma_K \sqrt{T}}, \quad d_2 = d_1 - \sigma_K \sqrt{T}$$

3. The Margrabe (1978) Boundary Condition

When the strike $K = 0$, the option becomes a pure exchange option to swap $S_2$ for $S_1$. In this case, $w = \frac{F_2}{F_2 + 0} = 1.0$, and the effective volatility becomes:

$\sigma_{\text{Margrabe}} = \sqrt{\sigma_1^2 + \sigma_2^2 - 2\rho\sigma_1\sigma_2}$

Kirk's approximation reduces exactly to William Margrabe's (1978) exact closed-form solution. This mathematical consistency is what makes Kirk's formulation the preferred benchmark on commodity, energy, and FX trading desks.

4. Cross-Asset Risk & Negative Correlation Greek

Unlike standard vanilla options where vega is always positive, a spread option has a strictly negative correlation sensitivity ($\frac{\partial C}{\partial \rho} < 0$):

$\frac{\partial C}{\partial \rho} = -e^{-rT} F_1 N'(d_1) \sqrt{T} \cdot \frac{w \sigma_1 \sigma_2}{\sigma_K} < 0$

Higher correlation causes the assets to move together, shrinking the probability that the spread widens beyond $K$. Furthermore, the negative Cross-Gamma ($\Gamma_{12} = \frac{\partial^2 C}{\partial S_1 \partial S_2} < 0$) indicates that a rally in $S_2$ diminishes the delta of $S_1$.

Spread Option Mastery Quiz

1. When the strike price $K = 0$, what does Kirk's approximation formula reduce to?

2. What happens to the value of a long European spread call option as cross-asset correlation $\rho$ approaches +1.0?

3. In a thermal power generation Spark Spread, what does the spread represent?

4. Why is the Cross-Gamma ($\frac{\partial^2 C}{\partial S_1 \partial S_2}$) of a spread call option negative?

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Frequently Asked Questions

A spread option is a derivative contract whose terminal payoff depends on the difference between two asset prices relative to a fixed strike: $\max(S_1 - S_2 - K, 0)$. Because the difference between two correlated lognormal random variables is not itself lognormally distributed, no exact simple closed-form formula exists when $K$ is non-zero. Kirk's (1995) approximation elegantly solves this by treating the denominator $(F_2 + K)$ as a single lognormal variable, yielding highly accurate prices across commodities, energy, and equity pairs.

Margrabe's formula prices an exchange option where one asset is exchanged for another with zero strike ($K = 0$). When $K = 0$, Kirk's formula collapses exactly to Margrabe's formula: the weighting factor $w = F_2 / (F_2 + K)$ becomes 1.0, and the effective volatility $\sigma_K$ becomes the exact two-asset diffusion volatility $\sqrt{\sigma_1^2 + \sigma_2^2 - 2\rho\sigma_1\sigma_2}$.

Spread options derive their value from the volatility of the spread ($S_1 - S_2$). When correlation $\rho$ is high, both assets move in lockstep, dampening the variance of their difference. When correlation is low or negative, the assets diverge widely, creating large swings in the spread and dramatically boosting the option's value. Consequently, the correlation Greek (Rho-Corr) of a long spread option is strictly negative.

Spread options are fundamental to physical commodity risk management: (1) Crack spreads allow oil refiners to lock in the margin between crude oil inputs and refined gasoline/diesel outputs; (2) Spark spreads allow power plant operators to hedge the margin between fuel (natural gas) and generated electricity; (3) Crush spreads allow agricultural processors to manage the margin between raw soybeans and processed oil/meal.

Cross-Delta measures the rate of change of option value with respect to each underlying asset ($\Delta_1 > 0$ for call on $S_1$, $\Delta_2 < 0$ for short $S_2$). Cross-Gamma is the second cross-partial derivative ($\frac{\partial^2 C}{\partial S_1 \partial S_2}$). Because an increase in $S_2$ reduces the delta with respect to $S_1$, Cross-Gamma is negative, creating an interaction effect that requires dynamic joint rebalancing.