Commodity Spread Option with Cointegration

Abstract

We derive the valuation formula of a European call option on the spread of two cointegrated commodity futures prices, based on the Gibson–Schwartz with cointegration (GSC) model. We also analyze the American commodity spread option including the early exercise premium representation and an analytical approximation valuation formulae with cointegration. In the numerical analysis, we compare the spread option values calculated by the GSC model and the Gibson–Schwartz (GS) model that ignores cointegration. Consistent with the intuition that the cointegration prevents the prices from diverging, the GSC model prices the commodity spread option lower than the GS model which have longer maturity of more than 6 years. In other words, the GS model may overprice the commodity spread options for those with longer maturity without taking account of cointegration. Thus, incorporating cointegration is important for valuation and hedging of long-term commodity spread options such as large scale oil refining plant developments.

Appendix

1.1 Expectation and Covariances of Log Futures Return

In this subsection, we derive the futures price and expectation value and covariance of log futures return. We use the future price equation written in terms of spot price and derive the future price process using Ito’s lemma written in terms of futures price. This price process can be explicitly written in terms of futures price levels. Finally, we calculate the expectation and covariance of stochastic terms of futures price using properties of stochastic calculus.

Note that futures price in terms of spot price is¹⁸

$$\begin{aligned} G_i(t, T) = e^{\mu _{X_i}(t, T) + \frac{\sigma _{X_i}^2(t, T)}{2}},\quad i = 1, 2 \end{aligned}$$

where

$$\begin{aligned} \mu _{X_i}(t, T) = \Bigg [ e^{T {\varvec{\beta }}} \Bigg \{ e^{-t {\varvec{\beta }}} {\varvec{X}}(t) + \int _t^T e^{- s {\varvec{\beta }}} {\varvec{\beta }}_0 (s) ds \Bigg \} \Bigg ]_i, \end{aligned}$$

and

$$\begin{aligned} \sigma _{X_i X_j} (t, T) = \Bigg [\int _t^T (e^{(T-s) {\varvec{\beta }}}) {\varvec{\Sigma }} (e^{(T-s) {\varvec{\beta }}})^\top ds \Bigg ]_{i j},\quad i, j = 1, 2. \end{aligned}$$

The partial derivatives are

$$\begin{aligned} \frac{\partial G_i(t, T)}{\partial S_j(t)}= & {} \frac{\left[ e^{(T-t) {\varvec{\beta }}}\right] _{i,j}}{S_j(t)} G_i(t, T),\quad i, j = 1, 2, \\ \frac{\partial G_i(t, T)}{\partial \delta _j(t)}= & {} \left[ e^{(T-t) {\varvec{\beta }}}\right] _{i,2+j} G_i(t, T), \quad i, j = 1, 2, \end{aligned}$$

where we denote $[A]_{i,j}$ as [i, j]th entry of matrix A.

Since the futures price $G_i(t, T)$ is a function of $S_i(t), \delta _i(t)$ and twice differentiable, we can use the Ito’s lemma and the dynamics of future price is

$$\begin{aligned} dG_i(t, T)= & {} \sum _{k=1}^2 \sigma _{S_k} S_k(t) \frac{\partial G_i}{\partial S_k} dW_{S_k}(t) + \sum _{k=1}^2 \sigma _{\delta _k} \frac{\partial G_i}{\partial \delta _k} dW_{\delta _k}(t) \\= & {} G_i(t, T) \left[ \sum _{k=1}^2 \sigma _{S_k} \left[ e^{(T-t) {\varvec{\beta }}}\right] _{i,k} dW_{S_k}(t) \right. \\&\left. + \sum _{k=1}^2 \sigma _{\delta _k} \left[ e^{(T-t) {\varvec{\beta }}}\right] _{i,2+k} dW_{\delta _k}(t) \right] , \end{aligned}$$

where the drift term is 0 since $G_i(t, T)$ is martingale under the risk-neutral probability.

Again, using Ito’s lemma we have,

$$\begin{aligned} d \log G_i(t, T)= & {} - \frac{1}{2} \Bigg \{ \sum _{k,l=1}^2 \sigma _{S_k S_l} \left[ e^{(T-t) {\varvec{\beta }}}\right] _{i,k} \left[ e^{(T-t) {\varvec{\beta }}}\right] _{i,l} \\&+\,2 \sum _{k,l=1}^2 \sigma _{S_k \delta _l} \left[ e^{(T-t) {\varvec{\beta }}}\right] _{i,k} \left[ e^{(T-t) {\varvec{\beta }}}\right] _{i,2+l}\\&+ \sum _{k,l=1}^2 \sigma _{\delta _k \delta _l} \left[ e^{(T-t) {\varvec{\beta }}}\right] _{i,2+k} \left[ e^{(T-t) {\varvec{\beta }}}\right] _{i,2+l} \Bigg \} dt \\&+ \sum _{k=1}^2 \sigma _{S_k} \left[ e^{(T-t) {\varvec{\beta }}}\right] _{i,k} dW_{S_k}(t) + \sum _{k=1}^2 \sigma _{\delta _k} \left[ e^{(T-t) {\varvec{\beta }}}\right] _{i,2+k} dW_{\delta _k}(t). \end{aligned}$$

The futures price can be expressed as follows.

$$\begin{aligned} G_i(T_0, T_i) = G_i(t, T_i) e^{X_{G_i}(t, T_0, T_i)}, \quad t \le T_0 \le T_i, \end{aligned}$$

where

$$\begin{aligned} X_{G_i}(t, T_0, T_i)\equiv & {} \mu _{X_{G_i}}(t, T_0, T_i) \\&+ \int _t^{T_0} \sum _{k=1}^2 \sigma _{S_k} \left[ e^{(T_i-t) {\varvec{\beta }}}\right] _{i,k} dW_{S_k}(u) \\&+ \int _t^{T_0} \sum _{k=1}^2 \sigma _{\delta _k} \left[ e^{(T_i-t) {\varvec{\beta }}}\right] _{i,2+k} dW_{\delta _k}(u). \\ \end{aligned}$$

The expectation value is

$$\begin{aligned} \mu _{X_{G_i}}(t, T_0, T_i)\equiv & {} E_t[X_{G_i}(t, T_0, T_i)] \nonumber \\= & {} - \frac{1}{2} \Bigg \{ \int _t^{T_0} \sum _{k,l=1}^2 \sigma _{S_k S_l} \left[ e^{(T_i-u) {\varvec{\beta }}}\right] _{i,k} \left[ e^{(T_i-u) {\varvec{\beta }}}\right] _{i,l} du \nonumber \\&+\,2\int _t^{T_0} \sum _{k,l=1}^2 \sigma _{S_k \delta _l} \left[ e^{(T_i-u) {\varvec{\beta }}}\right] _{i,k} \left[ e^{(T_i-u) {\varvec{\beta }}}\right] _{i,2+l} du \nonumber \\&+ \int _t^{T_0} \sum _{k,l=1}^2 \sigma _{\delta _k \delta _l} \left[ e^{(T_i-u) {\varvec{\beta }}}\right] _{i,2+k} \left[ e^{(T_i-u) {\varvec{\beta }}}\right] _{i,2+l} du \Bigg \}. \end{aligned}$$

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The covariance of $X_{G_i}(t, T_0, T_i)$ and $X_{G_j}(t, T_0, T_j)$ is

$$\begin{aligned} \sigma _{X_{G_i} X_{G_j}}(t, T_0, T_i, T_j)\equiv & {} {\text{ cov }}_t [X_{G_i}(t, T_0, T_i), X_{G_i}(t, T_0, T_j)] \nonumber \\= & {} \int _t^{T_0} \sum _{k,l=1}^2 \sigma _{S_k S_l} \left[ e^{(T_i-u) {\varvec{\beta }}}\right] _{i,k} \left[ e^{(T_j-u) {\varvec{\beta }}}\right] _{j,l} du \nonumber \\&+ \int _t^{T_0} \sum _{k,l=1}^2 \sigma _{S_k \delta _l} \left[ e^{(T_i-u) {\varvec{\beta }}}\right] _{i,k} \left[ e^{(T_j-u) {\varvec{\beta }}}\right] _{j,2+l} du \nonumber \\&+ \int _t^{T_0} \sum _{k,l=1}^2 \sigma _{S_l \delta _k} \left[ e^{(T_j-u) {\varvec{\beta }}}\right] _{j,l} \left[ e^{(T_i-u) {\varvec{\beta }}}\right] _{i,2+k} du \nonumber \\&+ \int _t^{T_0} \sum _{k,l=1}^2 \sigma _{\delta _k \delta _l} \left[ e^{(T_i-u) {\varvec{\beta }}}\right] _{i,2+k} \left[ e^{(T_j-u) {\varvec{\beta }}}\right] _{j,2+l} du.\nonumber \\ \end{aligned}$$

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1.2 Proof of Proposition 2.4

In this subsection, we prove Proposition 2.4. This is done by the following scheme. What we need to calculate is the expectation which can be expressed in terms of double integrals, since we are only dealing with the bivariate Gaussian processes. We know the expectation and covariance of the stochastic parts as we mentioned previously. The integrals can be calculated using multivariate version of completing squares, decomposing bivariate normal joint distribution in to conditional distribution and marginal distribution, and changing of variables. Finally, collecting all the terms, we have the pricing equation.

From Harrison and Kreps (1979) and Harrison and Pliska (1981), the price of commodity spread option at t, which option maturity is $T_0$, futures maturity for $G_1$ and $G_2$ are $T_1$ and $T_2$, respectively, is

$$\begin{aligned} C^E (G_1(t, T_1), G_2(t, T_2), t) = e^{-r(T_0 - t)} E_t[ (h_1 G_1(T_0, T_1) - h_2 G_2(T_0, T_2) - K)^+ ]. \end{aligned}$$

The expectation value can be calculated as follows.

$$\begin{aligned}&E_t [ (h_1 G_1(T_0, T_1) - h_2 G_2(T_0, T_2) - K)^+ ] \\&\quad = \int _D (h_1 G_1(t, T_1) e^{x_1} - h_2 G_(t, T_2) e^{x_2} - K) n({\varvec{x}}| {\varvec{\mu }}_{{\varvec{x}}}, {\varvec{\Sigma }}_{{\varvec{x}}}) d{\varvec{x}}, \end{aligned}$$

where

$$\begin{aligned} {\varvec{x}}= & {} \left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right] , \\ {\varvec{\mu }}_{{\varvec{x}}}= & {} \left[ \begin{array}{cc} \mu _{X_{G_1}}(t, T_0, T_1) \\ \mu _{X_{G_2}}(t, T_0, T_2) \end{array} \right] , \\ {\varvec{\Sigma }}_{{\varvec{x}}}= & {} \left[ \begin{array}{cc} \sigma _{X_{G_1}}^2(t, T_0, T_1) &{} \sigma _{X_{G_1} X_{G_2}}(t, T_0, T_1, T_2) \\ \sigma _{X_{G_1} X_{G_2}}(t, T_0, T_1, T_2) &{} \sigma _{X_{G_2}}^2(t, T_0, T_2) \end{array} \right] , \end{aligned}$$

and

$$\begin{aligned} D= & {} \{ {\varvec{x}} | h_1 G_1(t, T_1) e^{x_1} - h_2 G_2(t, T_2) e^{x_2} - K \ge 0 \} \\= & {} \{ {\varvec{x}} | d(x_2) \le x_1 \}, \\ d(x_2, K)\equiv & {} \ln (h_2 G_2(t, T_2) e^{x_2} + K) - \ln (h_1 G_1(t, T_1)). \end{aligned}$$

We now calculate the integrals. Suppose that ${\varvec{e}}_1$ is unit vector which the 1st entry is 1.

$$\begin{aligned}&\int _D e^{x_1} n({\varvec{x}} | {\varvec{\mu }}_{{\varvec{x}}}, {\varvec{\Sigma }}_{{\varvec{x}}}) d{\varvec{x}} \\&\quad = \int _D (2 \pi )^{-1} |{\varvec{\Sigma }}_{{\varvec{x}}}|^{-\frac{1}{2}} \exp \left\{ {\varvec{e}}_1^\top {\varvec{x}} - \frac{1}{2} ({\varvec{x}} - {\varvec{\mu }}_{{\varvec{x}}})^\top {\varvec{\Sigma }}_{{\varvec{x}}}^{-1} ({\varvec{x}} - {\varvec{\mu }}_{{\varvec{x}}}) \right\} d{\varvec{x}} \\&\quad = \int _D (2 \pi )^{-1} |{\varvec{\Sigma }}_{{\varvec{x}}}|^{-\frac{1}{2}} \exp \left\{ {\varvec{e}}_1^\top {\varvec{\mu }}_{{\varvec{x}}} + {\varvec{e}}_1^\top ({\varvec{x}} - {\varvec{\mu }}_{{\varvec{x}}}) - \frac{1}{2} ({\varvec{x}} - {\varvec{\mu }}_{{\varvec{x}}})^\top {\varvec{\Sigma }}_{{\varvec{x}}}^{-1} ({\varvec{x}} - {\varvec{\mu }}_{{\varvec{x}}}) \right\} d{\varvec{x}} \\&\quad = \int _D (2 \pi )^{-1} |{\varvec{\Sigma }}_{{\varvec{x}}}|^{-\frac{1}{2}} \exp \Biggl \{ {\varvec{e}}_1^\top {\varvec{\mu }}_{{\varvec{x}}} + \frac{1}{2} {\varvec{e}}_1^\top {\varvec{\Sigma }}_{{\varvec{x}}} {\varvec{e}}_1 \\&\qquad - \frac{1}{2} ({\varvec{x}} - {\varvec{\mu }}_{{\varvec{x}}} - {\varvec{\Sigma }}_{{\varvec{x}}} {\varvec{e}}_1)^\top {\varvec{\Sigma }}_{{\varvec{x}}}^{-1} ({\varvec{x}} - {\varvec{\mu }}_{{\varvec{x}}} - {\varvec{\Sigma }}_{{\varvec{x}}} {\varvec{e}}_1) \Biggr \} d{\varvec{x}} \\&\quad = \exp \Biggl \{ \mu _{X_{G_1}}(t, T_0, T_1) + \frac{1}{2} \sigma _{X_{G_1}}^2(t, T_0, T_1) \Biggr \} \\&\qquad \times \int _D (2 \pi )^{-1} |{\varvec{\Sigma }}_{{\varvec{x}}}|^{-\frac{1}{2}} \exp \{ - \frac{1}{2} ({\varvec{x}} - {\varvec{\mu }}_{{\varvec{x}}} - {\varvec{\Sigma }}_{{\varvec{x}}} {\varvec{e}}_1)^\top {\varvec{\Sigma }}_{{\varvec{x}}}^{-1} ({\varvec{x}} - {\varvec{\mu }}_{{\varvec{x}}} - {\varvec{\Sigma }}_{{\varvec{x}}} {\varvec{e}}_1) \} d{\varvec{x}}. \end{aligned}$$

The integral can be expanded as follows. We omit the time prameters for simplicity.

$$\begin{aligned}&\int _D (2 \pi )^{-1} |{\varvec{\Sigma }}_{{\varvec{x}}}|^{-\frac{1}{2}} \exp \Biggl \{ - \frac{1}{2} ({\varvec{x}} - {\varvec{\mu }}_{{\varvec{x}}} - {\varvec{\Sigma }}_{{\varvec{x}}} {\varvec{e}}_1)^\top {\varvec{\Sigma }}_{{\varvec{x}}}^{-1} ({\varvec{x}} - {\varvec{\mu }}_{{\varvec{x}}} - {\varvec{\Sigma }}_{{\varvec{x}}} {\varvec{e}}_1) \Biggr \} d{\varvec{x}} \\&\quad = \int _{-\infty }^{\infty } \int _{d(x_2, K)}^{\infty } (2 \pi )^{-1} (\sigma _{X_{G_1}} \sigma _{X_{G_2}} \sqrt{1 - \rho _{X_{G_1} X_{G_2}}^2})^{-1} \exp \Biggl \{ - \frac{1}{2 (1 - \rho _{X_{G_1} X_{G_2}})} \\&\qquad \times \,\Biggl ( \Biggl ( \frac{x_1 - \mu _{X_{G_1}} - \sigma _{X_{G_1}}^2}{\sigma _{X_{G_1}}} \Biggr )^2 - 2 \rho _{X_{G_1} X_{G_2}} \Biggl (\frac{x_1 - \mu _{X_{G_1}} - \sigma _{X_{G_1}}^2}{\sigma _{X_{G_1}}} \Biggr ) \\&\qquad \times \,\Biggl (\frac{x_2 - \mu _{X_{G_2}} - \sigma _{X_{G_1} X_{G_2}}}{\sigma _{X_{G_2}}} \Biggr ) + \Biggl ( \frac{x_2 - \mu _{X_{G_2}} - \sigma _{X_{G_1} X_{G_2}}}{\sigma _{X_{G_2}}} \Biggr )^2 \Biggr )^2 \Biggr \} dx_1 dx_2\\&\quad = \int _{-\infty }^{\infty } \int _{d(x_2)}^{\infty } (2 \pi (1 - \rho _{X_{G_1} X_{G_2}}^2))^{-\frac{1}{2}} \sigma _{X_{G_1}}^{-1} \\&\qquad \times \exp \left\{ - \frac{\Biggl (x_1 - \mu _{X_{G_1}} - \sigma _{X_{G_1}}^2 - \rho _{X_{G_1} X_{G_2}} \sigma _{X_{G_1}} \frac{x_2 - \mu _{X_{G_2}} - \sigma _{X_{G_1} X_{G_2}}}{\sigma _{X_{G_2}}} \Biggr )^2}{2 (1 - \rho _{X_{G_1} X_{G_2}}^2) \sigma _{X_{G_1}}^2} \right\} dx_1 \\&\qquad \times (2\pi )^{-\frac{1}{2}} \sigma _{X_{G_2}}^{-1} \exp \Biggl \{ - \frac{1}{2} \Biggl ( \frac{x_2 - \mu _{X_{G_2}} - \sigma _{X_{G_1} X_{G_2}}}{\sigma _{X_{G_2}}} \Biggr )^2 \Biggr \} dx_2 \\&\quad = \int _{-\infty }^{\infty } \int _{-d_1(x_2)}^{\infty } (2 \pi (1 - \rho _{X_{G_1} X_{G_2}}^2))^{-\frac{1}{2}} \sigma _{X_{G_1}}^{-1} \\&\qquad \times \exp \Biggl \{ - \frac{y^2}{2} \Biggr \} (1 - \rho _{X_{G_1} X_{G_2}}^2 )^{\frac{1}{2}} \sigma _{X_{G_1}} dy \\&\qquad \times (2\pi )^{-\frac{1}{2}} \sigma _{X_{G_2}}^{-1} \exp \Biggl \{ - \frac{1}{2} \Biggl ( \frac{x_2 - \mu _{X_{G_2}} - \sigma _{X_{G_1} X_{G_2}}}{\sigma _{X_{G_2}}} \Biggr )^2 \Biggr \} dx_2 \\&\quad = \int _{-\infty }^{\infty } \Phi (d_1(x_2)) n(x_1 | \mu _{X_{G_2}} + \sigma _{X_{G_1} X_{G_2}}, \sigma _{X_{G_2}}^2) dx_2, \end{aligned}$$

where

$$\begin{aligned} d_1(x_2) = - \frac{d(x_1) - \mu _{X_{G_1}} - \sigma _{X_{G_1}}^2 - \rho _{X_{G_1} X_{G_2}} \sigma _{X_{G_1}} \frac{x_2 - \mu _{X_{G_2}} - \sigma _{X_{G_1} X_{G_2}}}{\sigma _{X_{G_2}}}}{\sigma _{X_{G_1}} \sqrt{(1 - \rho _{X_{G_1} X_{G_2}}^2)}}, \end{aligned}$$

and we used change of variables in the third equation. Other integrals can be derived in the same manner. For the second integral,

$$\begin{aligned}&\int _D (2 \pi )^{-1} |{\varvec{\Sigma }}_{{\varvec{x}}}|^{-\frac{1}{2}} \exp \Biggl \{ - \frac{1}{2} ({\varvec{x}} - {\varvec{\mu }}_{{\varvec{x}}} - {\varvec{\Sigma }}_{{\varvec{x}}} {\varvec{e}}_2)^\top {\varvec{\Sigma }}_{{\varvec{x}}}^{-1} ({\varvec{x}} - {\varvec{\mu }}_{{\varvec{x}}} - {\varvec{\Sigma }}_{{\varvec{x}}} {\varvec{e}}_2) \Biggr \} d{\varvec{x}} \\&\quad = \int _{-\infty }^{\infty } \Phi (d_2(x_2)) n\left( x_2 | \mu _{X_{G_2}} + \sigma _{X_{G_2}}^2, \sigma _{X_{G_2}}^2\right) dx_2, \end{aligned}$$

where

$$\begin{aligned} d_2(x_2) = - \frac{d(x_1) - \mu _{X_{G_1}} - \sigma _{X_{G_1} X_{G_2}} - \rho _{X_{G_1} X_{G_2}} \sigma _{X_{G_1}} \frac{x_2 - \mu _{X_{G_2}} - \sigma _{X_{G_2}}^2}{\sigma _{X_{G_2}}}}{\sigma _{X_{G_1}} \sqrt{\left( 1 - \rho _{X_{G_1} X_{G_2}}^2\right) }}. \end{aligned}$$

And the last integral is

$$\begin{aligned}&\int _D (2 \pi )^{-1} |{\varvec{\Sigma }}_{{\varvec{x}}}|^{-\frac{1}{2}} \exp \Biggl \{ - \frac{1}{2} ({\varvec{x}} - {\varvec{\mu }}_{{\varvec{x}}})^\top {\varvec{\Sigma }}_{{\varvec{x}}}^{-1} ({\varvec{x}} - {\varvec{\mu }}_{{\varvec{x}}}) \Biggr \} d{\varvec{x}} \\&\quad = \int _{-\infty }^{\infty } \Phi (d_2(x_2)) n\left( x_2 | \mu _{X_{G_2}}, \sigma _{X_{G_2}}^2\right) dx_2. \end{aligned}$$

Collecting all terms, we have

$$\begin{aligned}&C^E (G_1(t, T_1), G_2(t, T_2), t) \\&\quad = h_1 G_1(t, T_1) \exp \{ -r(T_0 - t) \} \\&\qquad \times \int _{-\infty }^{\infty } \Phi (d_1(x_2)) n(x_2 | \mu _{X_{G_2}}(t, T_0, T_2) \\&\qquad +\,\sigma _{X_{G_1} X_{G_2}}(t, T_0, T_1, T_2), \sigma _{X_{G_2}}^2(t, T_0, T_2)) dx_2 - h_2 G_2(t, T_2) \exp \{ -r(T_0 - t) \} \\&\qquad \times \int _{-\infty }^{\infty } \Phi (d(x_2)) n(x_2 | \mu _{X_{G_2}}(t, T_0, T_2) \\&\qquad +\,\sigma _{X_{G_2}}^2(t, T_0, T_2), \sigma _{X_{G_2}}^2(t, T_0, T_2)) dx_2 \\&\qquad - K e^{-r(T_0 - t)} \int _{-\infty }^{\infty } \Phi (d(x_2)) n\left( x_2 | \mu _{X_{G_2}}(t, T_0, T_2), \sigma _{X_{G_2}}^2(t, T_0, T_2)\right) dx_2, \end{aligned}$$

where

$$\begin{aligned}&d(x_2) = - \frac{\ln (h_2 G_2(t, T_2) e^{x_2} + K) - \ln (h_1 G_1(t, T_1)) - \mu _{X_{G_1}}(t, t_0, T_1)}{\sigma _{X_{G_1}}(t, T_0, T_1) \sqrt{1 - \rho _{X_{G_1} X_{G_2}}^2 (t, T_0, T_1, T_2)}} \\&\qquad +\,\frac{\rho _{X_{G_1} X_{G_2}}(t, T_0, T_1, T_2) \sigma _{X_{G_1}}(t, T_0, T_1) \frac{x_2 - \mu _{X_{G_2}}(t, T_0, T_2)}{\sigma _{X_{G_2}}(t, T_0, T_2)}}{\sigma _{X_{G_1}}(t, T_0, T_1) \sqrt{1 - \rho _{X_{G_1} X_{G_2}}^2 (t, T_0, T_1, T_2)}}, \\&d_1(x_2) = d(x_2) + \sigma _{X_{G_1}}(t, T_0, T_1) \sqrt{1 - \rho _{X_{G_1} X_{G_2}}^2 (t, T_0, T_1, T_2)}, \\&\rho _{X_{G_1} X_{G_2}}(t, T_0, T_1, T_2) = \frac{\sigma _{X_{G_1} X_{G_2}}(t, T_0, T_1, T_2)}{\sigma _{X_{G_1}}(t, T_0, T_1) \sigma _{X_{G_2}}(t, T_0, T_2)}. \end{aligned}$$

1.3 Proofs for Characterization and Valuation Formulae of American Commodity Spread Option

In this section, we provide proofs for propositions in Sect. 2.3. As we have seen before,

$$\begin{aligned} d \ln G_i(t, T)= & {} - \frac{1}{2} \Bigg \{ \int _t^{T_0} \sum _{k,l=1}^2 \sigma _{S_k S_l} \left[ e^{(T_i-u) {\varvec{\beta }}}\right] _{i,k} \left[ e^{(T_i-u) {\varvec{\beta }}}\right] _{i,l} du \nonumber \\&\quad +\,2\int _t^{T_0} \sum _{k,l=1}^2 \sigma _{S_k \delta _l} \left[ e^{(T_i-u) {\varvec{\beta }}}\right] _{i,k} \left[ e^{(T_i-u) {\varvec{\beta }}}\right] _{i,2+l} du \nonumber \\&\quad + \int _t^{T_0} \sum _{k,l=1}^2 \sigma _{\delta _k \delta _l} \left[ e^{(T_i-u) {\varvec{\beta }}}\right] _{i,2+k} \left[ e^{(T_i-u) {\varvec{\beta }}}\right] _{i,2+l} du \Bigg \} \\&\quad + \Bigg \{ \sum _{k=1}^2 \sigma _{S_k} \left[ e^{(T-t) {\varvec{\beta }}}\right] _{i,k} dW_{S_k}(t) {+} \sum _{k=1}^2 \sigma _{\delta _k} \left[ e^{(T-t) {\varvec{\beta }}}\right] _{i,2+k} dW_{\delta _k}(t) \Bigg \}. \end{aligned}$$

Since the volatility term is only a linear combination of Brownian motion with time varying parameter, we can redefine this sde as¹⁹

$$\begin{aligned} d\ln G_i(t, T)= & {} \mu _{G_i}(t) dt + \sigma _{G_i}(t) dW_{G_i}(t) \end{aligned}$$

with a new $\sigma _{G_i}(t)$ and 1-dimensional Brownian motion $dW_{G_i}(t)$. The covariance of $\ln G_i(t,T)$ and $\ln G_j(t,T)$ is

$$\begin{aligned} \sigma _{G_i,G_j}(t)= & {} {\varvec{z}}_i^{\top } {\varvec{\rho }} {\varvec{z}}_j\\ {\varvec{z}}_i= & {} \left[ \sigma _{S_1} \left[ e^{(T_i-t){\varvec{\beta }}} \right] _{i,1}, \sigma _{S_2} \left[ e^{(T_i-t){\varvec{\beta }}} \right] _{i,2}, \sigma _{\delta _1} \left[ e^{(T_i-t){\varvec{\beta }}} \right] _{i,3}, \sigma _{\delta _2} \left[ e^{(T_i-t){\varvec{\beta }}} \right] _{i,4} \right] ^{\top }\\ {\varvec{\rho }}= & {} \left[ \begin{array}{cccccc} 1 &{} \rho _{S_1 S_2} &{} \rho _{S_1} &{} \rho _{S_1 \delta _2}\\ \rho _{S_2 S_1} &{} 1 &{} \rho _{S_2 \delta _1} &{} \rho _{S_2 \delta _2}\\ \rho _{\delta _1 S_1} &{} \rho _{\delta _1 S_2} &{} 1 &{} \rho _{\delta _1 \delta _2}\\ \rho _{\delta _2 S_1} &{} \rho _{\delta _2 S_2} &{} \rho _{\delta _2 \delta _1} &{} 1 \end{array} \right] \end{aligned}$$

Suppose $T_0 \le T_1, T_2$. From Karatzas (1988), we have $C^A (G_1, G_2, t)$ which is the value of American call spread option. For $t \le T_0$,

$$\begin{aligned} C^A (G_1, G_2, t) = \sup _{\tau \in \mathcal {S}_{t, T_0}} E_t [ e^{-r(\tau -t)} (h_1 G_1(\tau , T_1) - h_2 G_2(\tau , T_2) - K)^+ ], \end{aligned}$$

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where $\mathcal {S}_{t, T_0}$ is a set of stopping time taking values in $[t, T_0]$ almost surely.²⁰

Before showing some fundamental properties of the commodity spread options, we need the following lemma which is basically same as Lemma 3.9 in Jaillet et al. (1990) but is generalized for a function $\psi$ for two variables.

Lemma 4.1

(Jaillet et al. 1990) Let X(t) be the 2-dimesional vector which satisfies the following stochastic differential equation.

$$\begin{aligned} X(T) = x + \int _t^T \mu (s) ds + \int _t^T \sigma (s) dW(s) \end{aligned}$$

Suppose $\psi (x)$ is a function on ${\varvec{R}}^2$. Suppose there exists $\lambda > 0$ such that $\psi (x) exp(-\lambda ||x||)$ is a bounded, continuous function and admits a bounded weak derivative on ${\varvec{R}}^2$.²¹ Define

$$\begin{aligned} C(x, a)\equiv & {} \sup _{\tau \in \mathcal {S}_{t,T}} E \left[ e^{-r(\tau -t)} \psi \left( x + \int _t^\tau \mu (s)ds + \int _t^\tau \sigma (s)dW(s) \right) \right] . \end{aligned}$$

Then

$$\begin{aligned} C(x, a)= & {} \sup _{\tau _0 \in \mathcal {S}_{0,1}} E \left[ e^{-r(T-t)\tau _0} \psi \left( x + \int _0^{\tau _0} \mu _0(t,a) da + \int _0^{\tau _0} \sigma _0(t,a) dW(a) \right) \right] \end{aligned}$$

with

$$\begin{aligned} \mu _0(t,a)= & {} (T-t)\mu (t+a(T-t)) \\ \sigma _0(t,a)= & {} \sqrt{T-t}\sigma (t+a(T-t)) \end{aligned}$$

Proof

The proof is similar to that of Jaillet et al. (1990), Lemma 3.9. $\square$

Notice that $\psi (x_1, x_2) = (h_1 e^{x_1} - h_2 e^{x_2} - K)^+$ satisfies the condition of the lemma.

1.3.1 Proof of Proposition 2.5

Here we prove Proposition 2.5.²²

Proof

If $h_1 G_1(t, T_1) \le h_2 G_2(t, T_2) + K$ then immediate exercise is suboptimal and thus we have the first assertion.

We prove the second assertion. Using Lemma 4.1, we have

$$\begin{aligned} C^A(G_1, G_2, t)= & {} \sup _{\tau _t \in \mathcal {S}_{0,1}} E\left[ e^{-r \tau _t (T-t)} (h_1 G_1 e^{a_1(t, \tau _t)} - h_2 G_2 e^{a_2(t, \tau _t)} - K)^+\right] \end{aligned}$$

where

$$\begin{aligned} a_i(t, \tau _t)= & {} \int _0^{\tau _t} \mu _{i}(t,a) da + \int _0^{\tau _t} \sigma _{i}(t,a) dW_{G_i}(a)\\ \mu _i(t,a)= & {} (T_0-t)\mu _{G_i}(t+a(T_0-t))\\ \sigma _i(t,a)= & {} \sqrt{T_0-t}\sigma _{G_i}(t+a(T_0-t)), i=1,2. \end{aligned}$$

Consider a new stopping time $\tau _s = \tau _t (T-t)/(T-s)$ for $s>t$. Observe that $\tau _t \in \mathcal {S}_{0,1}$ if and only if $\tau _s \in \mathcal {S}_{0,(T-t)/(T-s)}$ and $\tau _s(T-s)=\tau _t(T-t)$. Now we have

$$\begin{aligned}&C^A(G_1, G_2, t)\\&\quad = \sup _{\tau _t \in \mathcal {S}_{0,1}} E\left[ e^{-r \tau _t (T-t)} (h_1 G_1 e^{a_1(t, \tau _t)} - h_2 G_2 e^{a_2(t, \tau _t)} - K)^+\right] \\&\quad = \sup _{\tau _s \in \mathcal {S}_{0,(T-t)/(T-s)}} E\left[ e^{-r \tau _s (T-s)} (h_1 G_1 e^{a_1(s, \tau _s)} - h_2 G_2 e^{a_2(s, \tau _s)} - K)^+\right] \\&\quad \ge \sup _{\tau _s \in \mathcal {S}_{0,1}} E\left[ e^{-r \tau _s (T-s)} (h_1 G_1 e^{a_1(s,\tau _s)} - h_2 G_2 e^{a_2(s,\tau _s)} - K)^+\right] \\&\quad = C^A(G_1, G_2, s) \end{aligned}$$

where the inequality follows from $\mathcal {S}_{0,1} \subset \mathcal {S}_{0,(T-t)/(T-s)}$. Therefore, if $(G_1, G_2, t) \in \mathcal {E}$ which is equivalent to $C^A(G_1, G_2, t) \le (h_1 G_1 - h_2 G_2 - K)^+$, then $(G_1, G_2, s) \in \mathcal {E}$.

We prove the third assertion. It is easy to see that

$$\begin{aligned} (h_1 \lambda G_1 - h_2 G_2 - K)^+= & {} (h_1 G_1 - h_2 G_2 - K + (\lambda - 1) h_1 G_1)^+\\\le & {} (h_1 G_1 - h_2 G_2 - K)^+ + (\lambda - 1) h_1 G_1. \end{aligned}$$

If $(G_1, G_2, t) \in \mathcal {E}$ which implies $h_1 G_1 - h_2 G_2 - K > 0$ and $\lambda > 1$, then

$$\begin{aligned} (h_1 \lambda G_1 - h_2 G_2 - K)^+= & {} h_1 \lambda G_1 - h_2 G_2 - K\\= & {} h_1 G_1 - h_2 G_2 - K + (\lambda - 1) h_1 G_1\\= & {} (h_1 G_1 - h_2 G_2 - K)^+ + (\lambda - 1) h_1 G_1. \end{aligned}$$

Using the above facts, if $(G_1, G_2, t) \in \mathcal {E}$ then

$$\begin{aligned}&C^A(\lambda G_1, G_2, t)\\&\quad = \sup _{\tau \in \mathcal {S}_{0,1}} E\left[ e^{-r \tau (T-t)} (h_1 \lambda G_1 e^{a_1(t, \tau )} - h_2 G_2 e^{a_2(t, \tau )} - K)^+\right] \\&\quad \le \sup _{\tau \in \mathcal {S}_{0,1}} E\left[ e^{-r \tau (T-t)} (h_1 G_1 e^{a_1(t, \tau )} - h_2 G_2 e^{a_2(t, \tau )} - K)^+ + (\lambda - 1) h_1 G_1 e^{a_1(t, \tau )}\right] \\&\quad \le C^A(G_1, G_2, t) + (\lambda - 1) h_1 G_1\\&\quad = (h_1 G_1 - h_2 G_2 - K)^+ + (\lambda - 1) h_1 G_1\\&\quad = (h_1 \lambda G_1 - h_2 G_2 - K)^+ \end{aligned}$$

Therefore $(\lambda G_1, G_2, t) \in \mathcal {E}$.

The proof of the fourth assertion proceeds as before. Since $0 \le \lambda \le 1$,

$$\begin{aligned} (h_1 G_1 - h_2 \lambda G_2 - K)^+= & {} (h_1 G_1 - h_2 G_2 - K + (1 - \lambda ) h_1 G_1)^+\\\le & {} (h_1 G_1 - h_2 G_2 - K)^+ + (1 - \lambda ) h_2 G_2. \end{aligned}$$

If $(G_1, G_2, t) \in \mathcal {E}$ and $0 \le \lambda \le 1$, then we have

$$\begin{aligned} (h_1 G_1 - h_2 \lambda G_2 - K)^+= & {} h_1 G_1 - h_2 \lambda G_2 - K\\= & {} h_1 G_1 - h_2 G_2 - K + (1 - \lambda ) h_2 G_2\\= & {} (h_1 G_1 - h_2 G_2 - K)^+ + (1 - \lambda ) h_2 G_2. \end{aligned}$$

With this inequality, if $(G_1, G_2, t) \in \mathcal {E}$ then

$$\begin{aligned}&C^A(G_1, \lambda G_2, t)\\&\quad = \sup _{\tau \in \mathcal {S}_{0,1}} E\left[ e^{-r \tau (T-t)} (h_1 G_1 e^{a_1(t, \tau )} - h_2 \lambda G_2 e^{a_2(t, \tau )} - K)^+\right] \\&\quad \le \sup _{\tau \in \mathcal {S}_{0,1}} E\left[ e^{-r \tau (T-t)} (h_1 G_1 e^{a_1(t, \tau )} - h_2 G_2 e^{a_2(t, \tau )} - K)^+ + (1 - \lambda ) h_2 G_2 e^{a_2(t, \tau )}\right] \\&\quad \le C^A(G_1, G_2, t) + (1 - \lambda ) h_2 G_2\\&\quad = (h_1 G_1 - h_2 G_2 - K)^+ + (1 - \lambda ) h_2 G_2\\&\quad = (h_1 G_1 - h_2 \lambda G_2 - K)^+ \end{aligned}$$

Thus, $(\lambda G_1, G_2, t) \in \mathcal {E}$.

If $G_2(t, T_2)=0$ then $G_2(u, T_2) = 0$ for all $u \ge t$. This implies that the commodity spread option reduces to a plain option on a single asset $G_1$, with exercise boundary $B_1$ and we have the fifth assertion.

We prove the last assertion which follows from the following inequality.

$$\begin{aligned}&C^A(G(\lambda ), t)\\&\quad = \sup _{\tau \in \mathcal {S}_{0,1}} E_t\left[ e^{-r \tau (T-t)} h_1 \left( \lambda G_1 e^{a_1(t, \tau )} + (1-\lambda ) \tilde{G}_1 e^{a_1(t, \tau )} \right) \right. \\&\left. \qquad -\,h_2 \left( \lambda G_2 e^{a_2(t, \tau )} + (1-\lambda ) \tilde{G}_2 e^{a_2(t, \tau )} - K\right) ^+ \right] \\&\quad \le \sup _{\tau \in \mathcal {S}_{0,1}} E_t\left[ e^{-r \tau (T-t)} \lambda \left( h_1 G_1 e^{a_1(t, \tau )} - h_2 G_2 e^{a_2(t, \tau )} - K\right) ^+\right. \\&\left. \qquad +\,(1-\lambda ) \left( h_1 \tilde{G}_1 e^{a_1(t, \tau )} - h_2 \tilde{G}_2 e^{a_2(t, \tau )} - K\right) ^+ \right] \\&\quad \le \sup _{\tau \in \mathcal {S}_{0,1}} E_t\left[ e^{-r \tau (T-t)} \lambda \left( h_1 G_1 e^{a_1(t, \tau )} - h_2 G_2 e^{a_2(t, \tau )} - K\right) ^+\right] \\&\qquad + \sup _{\tau \in \mathcal {S}_{0,1}} E_t\left[ e^{-r \tau (T-t)} (1-\lambda ) \left( h_1 \tilde{G}_1 e^{a_1(t, \tau )} - h_2 \tilde{G}_2 e^{a_2(t, \tau )} - K\right) ^+ \right] \\&\quad = \lambda C^A(G, t) + (1-\lambda ) C^A(\tilde{G}, t) \end{aligned}$$

where we used the convexity of the payoff function $(h_1 G_1 - h_2 G_2 - K)^+$ in the first inequality. $\square$

1.3.2 Proof of Proposition 2.6

The proof for Proposition 2.6 is provided below.

Proof

The first assertion follows from the continuity of the payoff function $(h_1 G_1(\tau , T_1) - h_2 G_2(\tau , T_2) - K)^+$ and the continuity of the flow of the sdes. Since $(h_1 G_1 - h_2 G_2 - K)^+$ is nondecreasing in $G_1$ and nonincreasing in $G_2$ and applying Proposition 2.18 of Karatzas and Shreve (1991), the second assertion follows. Properties 3 and 4 are proved using the results of Proposition 2.5, 2 and 6, respectively. $\square$

1.3.3 Proof of Proposition 2.7

We prove the variation inequality characterization for spread option. The proof is based on Jaillet et al. (1990), Theorem 3.6 and Broadie and Detemple (1997), Proposition 2.6. However, we emphasize that since the model includes 2 commodities and the parameters depends on time, the proof is enhanced from those above.

Proof

First, let us prove the uniform boundedness of $\partial C^A / \partial G_i, i = 1, 2$.²³ We focus on the derivative relative to $G_1$. The argument for $G_2$ is similar. Consider two asset values $(G_1, G_2, t)$ and $(\tilde{G}_1, G_2, t)$. Here we slightly abuse the notation and define

$$\begin{aligned} \ln \tilde{G}_1(s, T_1)= & {} \ln \tilde{G}_1(t, T_1) + \int _t^s \mu _{G_1}(u) du + \int _t^s \sigma _{G_1}(t) dW_{G_1}(t)\\\equiv & {} \ln \tilde{G}_1(t, T_1) + X_{G_1}(t, s, T_1) \end{aligned}$$

For any stopping time $\tau \in \mathcal {S}_{t, T_0}$ we have

$$\begin{aligned}&|(h_1 G_1(\tau , T_1) - h_2 G_2(\tau , T_2) - K)^+ - (h_1 \tilde{G}_1(\tau , T_1) - h_2 G_2(\tau , T_2) - K)^+| \nonumber \\&\quad \le |(h_1 G_1(\tau , T_1) - h_2 G_2(\tau , T_2)) - (h_1 \tilde{G}_1(\tau , T_1) - h_2 G_2(\tau , T_2))| \nonumber \\&\quad = |h_1 G_1(\tau , T_1) - h_1 \tilde{G}_1(\tau , T_1)| \nonumber \\&\quad = |h_1 G_1(t, T_1) - h_1 \tilde{G}_1(t, T_1)| e^{X_{G_1}(t, \tau , T_1)} . \end{aligned}$$

(17)

Without loss of generality, suppose $G_1(t, T_1) > \tilde{G}_1(t, T_1)$. Let $\tau \in \mathcal {S}_{t, T_0}$ represent the optimal stopping time for $(G_1, G_2, t)$. We have

$$\begin{aligned}&|C^A(G_1, G_2, t) - C^A(\tilde{G}_1, G_2, t)|\\&\quad \le \left| E_t\left[ e^{-r(T_0-t)} \left( h_1 G_1(\tau , T_1) - h_2 G_2(\tau , T_2) - K\right) ^+\right] \right. \\&\qquad \left. - E_t\left[ e^{-r(T_0-t)} \left( h_1 \tilde{G}_1(\tau , T_1) - h_2 G_2(\tau , T_2) - K\right) ^+ \right] \right| \\&\quad \le e^{-r(T_0-t)} |h_1 G_1(t, T_1) - h_1 \tilde{G}_1(t, T_1)| E_t[ e^{X_{G_1}(t, \tau , T_1)} ]\\&\quad = e^{-r(T_0-t)} |h_1 G_1(t, T_1) - h_1 \tilde{G}_1(t, T_1)|\\&\quad \le |h_1 G_1(t, T_1) - h_1 \tilde{G}_1(t, T_1)|. \end{aligned}$$

where the first inequality is obtained by definition of the optimal stopping time, the second inequality is obtained by (17), and the first equality is by the expectation of log normal distribution. Hence

$$\begin{aligned} \frac{|C^A(G_1, G_2, t) - C^A(\tilde{G}_1, G_2, t)|}{|h_1 G_1(t, T_1) - h_1 \tilde{G}_1(t, T_1)|} \le 1. \end{aligned}$$

This implies Lipschitz continuous on ${\varvec{R}}_+$. Therefore there exist partial differentiation almost everywhere which is uniformly bounded.

Second, we prove the local boundedness of the time derivative. We have

$$\begin{aligned}&|C^A(G_1(t, T_1), G_2(t, T_2), t) - C^A(G_1(t, T_2), G_2(t, T_2), s)|\\&\quad = \left| E_t\left[ e^{-r \tau (T_0-t)} \left( h_1 G_1(t, T_1) e^{a_1(t)} - h_2 G_2(t, T_2) e^{a_2(t)} - K\right) ^+\right. \right. \\&\left. \left. \qquad -\,e^{-r \tau (T_0-s)} \left( h_1 G_1(t, T_1) e^{a_2(s)} - h_2 G_2(t, T_2) e^{a_2(s)} - K\right) ^+ \right] \right| \\&\quad \le E_t\left[ \left| e^{-r \tau (T_0-t)} - e^{-r \tau (T_0-s)} \right| \left( h_1 G_1(t, T_1) e^{a_1(t)} - h_2 G_2(t, T_2) e^{a_2(t)} - K\right) ^+ \right. \\&\qquad +\,e^{-r \tau (T_0-s)} \left| \left( h_1 G_1(t, T_1) e^{a_1(t)} - h_2 G_2(t, T_2) e^{a_2(t)} - K\right) ^+\right. \\&\left. \left. \qquad -\,\left( h_1 G_1(t, T_1) e^{a_1(s)} - h_2 G_2(t, T_2) e^{a_2(s)} - K\right) ^+\right| \right] . \end{aligned}$$

where we denoted

$$\begin{aligned} a_i(u)= & {} \int _0^{\tau } \mu _{i}(u,a) da + \int _0^{\tau } \sigma _{i}(u,a) dW_{G_i}(a)\\ \mu _i(t,a)= & {} (T_0-t)\mu _{G_i}(t+a(T_0-t))\\ \sigma _i(t,a)= & {} \sqrt{T_0-t}\sigma _{G_i}(t+a(T_0-t)),\quad i=1,2. \end{aligned}$$

and used Lemma 4.1 in the first equality. Since $e^{x}$ is convex which implies $(e^{x})^\prime$ is monotonically increasing, we have

$$\begin{aligned} |e^{-r \tau (T-t)} - e^{-r \tau (T-s)}| \le \max _{u \in [0,1], v \in [0, T]} (r u e^{-ru(T-v)}) |t-s| \equiv M_1 |t-s| \end{aligned}$$

(18)

Furthermore,

$$\begin{aligned}&E_t\left[ \left( h_1 G_1(t, T_1) e^{a_1(t)} - h_2 G_2(t, T_2) e^{a_2(t)} - K\right) ^+\right] \nonumber \\&\quad \le E_t\left[ h_1 G_1(t, T_1) e^{a_1(t)} + h_2 G_2(t, T_2) e^{a_2(t)}\right] \nonumber \\&\quad = h_1 G_1(t, T_1) + h_2 G_2(t, T_2) \end{aligned}$$

(19)

We also have

$$\begin{aligned}&\left| \left( h_1 G_1(t, T_1) e^{a_1(t)} - h_2 G_2(t, T_2) e^{a_2(t)} - K\right) ^+ \right. \nonumber \\&\left. \quad - \left( h_1 G_1(t, T_1) e^{a_1(s)} - h_2 G_2(t, T_2) e^{a_2(s)} - K\right) ^+ \right| \nonumber \\&\quad \le \left| \left( h_1 G_1(t, T_1) e^{a_1(t)} - h_2 G_2(t, T_2) e^{a_2(t)}\right) \right. \nonumber \\&\left. \qquad - \left( h_1 G_1(t, T_1) e^{a_1(s)} - h_2 G_2(t, T_2) e^{a_2(s)}\right) \right| \nonumber \\&\quad \le \left| h_1 G_1(t, T_1) e^{a_1(t)} - h_1 G_1(t, T_1) e^{a_1(s)} \right| \nonumber \\&\qquad + \left| h_2 G_2(t, T_2) e^{a_2(t)} - h_2 G_2(t, T_2) e^{a_2(s)} \right| \nonumber \\&\quad \le h_1 G_1(t, T_1) e^{|a_1(t)| + |a_1(s)|} |a_1(t) - a_1(s) | \nonumber \\&\qquad + h_2 G_2(t, T_2) e^{|a_2(t)| + |a_2(s)|} |a_2(t) - a_2(s) | \nonumber \\&\quad \le \left( h_1 G_1(t, T_1) + h_2 G_2(t, T_2)\right) e^{|a_1(t)| + |a_1(s)| + |a_2(t)| + |a_2(s)|} \nonumber \\&\qquad \times (|a_1(t) - a_1(s) | + |a_2(t) - a_2(s) |) \end{aligned}$$

(20)

where the third inequality follows from the convexity of the exponential function. Moreover,

$$\begin{aligned}&\exp \left( |a_1(t)| + |a_1(s)| + |a_2(t)| + |a_2(s)|\right) \nonumber \\&\quad \le \exp \Bigg ( \int _0^{\tau _0} |\mu _1(t, a)| da + \left| \int _0^{\tau _0} \sigma _1(t, a) dW_{G_1}(a) \right| \nonumber \\&\quad \qquad + \int _0^{\tau _0} |\mu _1(s, a)| da + \left| \int _0^{\tau _0} \sigma _1(s, a) dW_{G_1}(a) \right| \nonumber \\&\quad \qquad + \int _0^{\tau _0} |\mu _2(t, a)| da + \left| \int _0^{\tau _0} \sigma _2(t, a) dW_{G_2}(a) \right| \nonumber \\&\quad \qquad + \int _0^{\tau _0} |\mu _2(s, a)| da + \left| \int _0^{\tau _0} \sigma _2(s, a) dW_{G_2}(a) \right| \Bigg ) \nonumber \\&\quad \le M_2 \exp \Bigg ( \left| \int _0^{\tau _0} \sigma _1(t, a) dW_{G_1}(a) \right| + \left| \int _0^{\tau _0} \sigma _1(s, a) dW_{G_1}(a) \right| \nonumber \\&\quad \qquad + \left| \int _0^{\tau _0} \sigma _2(t, a) dW_{G_2}(a) \right| + \left| \int _0^{\tau _0} \sigma _2(s, a) dW_{G_2}(a) \right| \Bigg ) . \end{aligned}$$

(21)

where we used the boundedness of $\mu _i(u,a)$. Since $\mu _i(u,a)$ have a bounded derivative, it is Lipschitz continuous which implies

$$\begin{aligned}&|a_1(t) - a_1(s) | + |a_2(t) - a_2(s)| \nonumber \\&\quad \le \left| \int _0^{\tau _0} \mu _1(t, a) - \mu _1(s, a) da \right| + \left| \int _0^{\tau _0} \sigma _1(t, a) - \sigma _1(s, a) dW_{G_1}(a) \right| \nonumber \\&\qquad + \left| \int _0^{\tau _0} \mu _2(t, a) - \mu _2(s, a) da \right| + \left| \int _0^{\tau _0} \sigma _2(t, a) - \sigma _2(s, a) dW_{G_2}(a) \right| \nonumber \\&\quad \le \int _0^{\tau _0} |\mu _1(t, a) - \mu _1(s, a)| da + \left| \int _0^{\tau _0} \sigma _1(t, a) - \sigma _1(s, a) dW_{G_1}(a) \right| \nonumber \\&\qquad + \int _0^{\tau _0} |\mu _2(t, a) - \mu _2(s, a)| da + \left| \int _0^{\tau _0} \sigma _2(t, a) - \sigma _2(s, a) dW_{G_2}(a) \right| \nonumber \\&\quad \le M_3 |t-s| + \left| \int _0^{\tau _0} \sigma _1(t, a) - \sigma _1(s, a) dW_{G_1}(a) \right| \nonumber \\&\qquad + \left| \int _0^{\tau _0} \sigma _2(t, a) - \sigma _2(s, a) dW_{G_2}(a) \right| . \end{aligned}$$

(22)

We use the following two inequalities.

$$\begin{aligned} E \Bigg [ \exp \Bigg ( A \left| \int _0^{\tau _0} \sigma _i(u, a) dW_{G_i}(a) \right| \Bigg ) \Bigg ]\le & {} E \Bigg [ \exp \Bigg ( A \left| \int _0^1 \sigma _i(u, a) dW_{G_i}(a) \right| \Bigg ) \Bigg ] \nonumber \\\le & {} 2 \exp \Bigg ( \frac{A^2 \int _0^1 \sigma _i^2(u, a) da}{2} \Bigg ) \nonumber \\\le & {} M_4. \end{aligned}$$

(23)

where we used the submartingale property of $\exp \left( A \left| \int _s^t \sigma _i(u, a) dW_{G_i}(a) \right| \right) _{s \ge t}$. For the other inequality,

$$\begin{aligned} |\sigma _1(t, a) - \sigma _1(s, a)|= & {} \left| \frac{\partial \sigma _1(u,a)}{\partial t} (t-s) \right| \nonumber \\\le & {} \sup _{u \in [s,t], a \in [0,1]} \left| \frac{\partial \sigma _1(u,a)}{\partial t} \right| |t - s| = M_5 |t-s| \nonumber \\\Rightarrow & {} \int _0^{1} (\sigma _1(t, a) - \sigma _1(s, a))^2 da \le M_5^2 |t-s|^2. \end{aligned}$$

Therefore, we have

$$\begin{aligned} E \Bigg [ \left| \int _0^{\tau _0} \sigma _i(t, a) - \sigma _i(s, a) dW_{G_i}(a) \right| ^2 \Bigg ]= & {} E \Bigg [ \int _0^{\tau _0} (\sigma _i(t, a) - \sigma _i(s, a))^2 da \Bigg ] \nonumber \\\le & {} \int _0^{1} (\sigma _i(t, a) - \sigma _i(s, a))^2 da \nonumber \\\le & {} M_5^2 |t-s|^2. \end{aligned}$$

(24)

Using inequalities (18)–(24) and the Cauchy–Schwartz inequality, we have

$$\begin{aligned}&|C^A(G_1, G_2, t) - C^A(G_1, G_2, s)|\\&\quad \le (h_1 G_1(t, T_1) + h_2 G_2(t, T_2)) \times ( M_1 |t-s|\\&\qquad +\,E \Bigg [ M_2 \exp \Bigg ( \left| \int _0^{\tau _0} \sigma _1(t, a) dW_{G_1}(a) \right| + \left| \int _0^{\tau _0} \sigma _1(s, a) dW_{G_1}(a) \right| \\&\qquad + \left| \int _0^{\tau _0} \sigma _2(t, a) dW_{G_2}(a) \right| + \left| \int _0^{\tau _0} \sigma _2(s, a) dW_{G_2}(a) \right| \Bigg )\\&\qquad \times \Bigg ( M_3 |t-s| + \left| \int _0^{\tau _0} \sigma _1(t, a) - \sigma _1(s, a) dW_{G_1}(a) \right| \\&\qquad + \left| \int _0^{\tau _0} \sigma _2(t, a) - \sigma _2(s, a) dW_{G_2}(a) \right| \Bigg ) \Bigg ]\le M_6 |t-s|\\&\qquad +\,M_7 |t-s| E \Bigg [ \exp \Bigg ( \left| \int _0^{\tau _0} \sigma _1(t, a) dW_{G_1}(a) \right| + \left| \int _0^{\tau _0} \sigma _1(s, a) dW_{G_1}(a) \right| \\&\qquad + \left| \int _0^{\tau _0} \sigma _2(t, a) dW_{G_2}(a) \right| + \left| \int _0^{\tau _0} \sigma _2(s, a) dW_{G_2}(a) \right| \Bigg )\Bigg ]\\&\qquad +\,M_8 E \Bigg [ \exp \Bigg ( \left| \int _0^{\tau _0} \sigma _1(t, a) dW_{G_1}(a) \right| + \left| \int _0^{\tau _0} \sigma _1(s, a) dW_{G_1}(a) \right| \\&\qquad + \left| \int _0^{\tau _0} \sigma _2(t, a) dW_{G_2}(a) \right| + \left| \int _0^{\tau _0} \sigma _2(s, a) dW_{G_2}(a) \right| \Bigg )\\&\qquad \times \Bigg ( \left| \int _0^{\tau _0} \sigma _1(t, a) - \sigma _1(s, a) dW_{G_1}(a) \right| \\&\qquad + \left| \int _0^{\tau _0} \sigma _2(t, a) - \sigma _2(s, a) dW_{G_2}(a) \right| \Bigg ) \Bigg ] \end{aligned}$$

$$\begin{aligned}&\le M_6 |t-s| + M_7 |t-s| \Bigg [ S(1,t,2) S(1,s,4) S(2,t,8) S(2,s,8)\\&\quad + M_8 \Bigg ( E \Bigg [ \left| \int _0^{\tau _0} \sigma _1(t, a) - \sigma _1(s, a) dW_{G_1}(a) \right| ^2 \Bigg ]^{1/2}\\&\quad \times \,S(1,t,4) S(1,s,8) S(2,t,16) S(2,s,16)\\&\quad + E \Bigg [ \left| \int _0^{\tau _0} \sigma _2(t, a) - \sigma _2(s, a) dW_{G_2}(a) \right| ^2 \Bigg ]^{1/2}\\&\quad \times \,S(1,t,4) S(1,s,8) S(2,t,16) S(2,s,16) \Bigg )\\&\le M_9 |t-s| + M_{10} \Bigg [ \Bigg ( \int _0^{1} (\sigma _1(t, a) - \sigma _1(s, a))^2 da \Bigg )^{1/2}\\&\quad + \Bigg ( \int _0^{1} (\sigma _2(t, a) - \sigma _2(s, a))^2 da \Bigg )^{1/2} \Bigg ]\le M_{11} |t-s| \end{aligned}$$

where we denoted

$$\begin{aligned} S(i,u,n) = E \Bigg [ \exp \Bigg ( n \left| \int _0^{\tau _0} \sigma _i(u, a) dW_{G_i}(a) \right| \Bigg ) \Bigg ]^{1/n}. \end{aligned}$$

Therefore $C^A(G_1, G_2, t)$ is Lipschitz continuous and there exist $\partial C^A/ \partial t$ almost everywhere which is locally bounded.

Applying Theorem 3.2 in Jaillet et al. (1990), we can see that $C^A$ satisfies the variational inequalities (9).

Using the transformation $G_1 = e^{x_1}$ and $G_2 = e^{x_2}$ we can rewrite Eq. (9) as

$$\begin{aligned}&\frac{1}{2} \Bigg ( \frac{\partial ^2 C^A}{\partial x_1^2} \sigma _{G_1}^2(t) + 2 \frac{\partial ^2 C^A}{\partial x_1 \partial x_2} \rho _{G_1, G_2}(t) \sigma _{G_1}(t) \sigma _{G_2}(t) + \frac{\partial ^2 C^A}{\partial x_2^2} \sigma _{G_2}^2(t) \Bigg )\nonumber \\&\quad \le rC^A + \frac{\sigma _{G_1}^2(t)}{2} \frac{\partial C^A}{\partial x_1} + \frac{\sigma _{G_2}^2(t)}{2} \frac{\partial C^A}{\partial x_2} - \frac{\partial C^A}{\partial t}. \end{aligned}$$

(25)

From Proposition 2.6, $C^A$ is convex and therefore we have²⁴

$$\begin{aligned} 0\le & {} (\rho _{G_1, G_2}(t) \sigma _{G_1}(t), \sigma _{G_2}(t)) \left( \begin{array}{cc} \frac{\partial ^2 C^A}{\partial x_1^2} &{} \frac{\partial ^2 C^A}{\partial x_1 \partial x_2} \\ \frac{\partial ^2 C^A}{\partial x_1 \partial x_2} &{} \frac{\partial ^2 C^A}{\partial x_2^2} \end{array}\right) \left( \begin{array}{c} \rho _{G_1, G_2}(t) \sigma _{G_1}(t) \\ \sigma _{G_2}(t) \end{array}\right) \\= & {} \rho _{G_1, G_2}(t) \sigma _{G_1}^2(t) \frac{\partial ^2 C^A}{\partial x_1^2} + 2 \rho _{G_1, G_2}(t) \sigma _{G_1}(t) \sigma _{G_2}(t) \frac{\partial ^2 C^A}{\partial x_1 \partial x_2} + \sigma _{G_2}^2 \frac{\partial ^2 C^A}{\partial x_2^2} \end{aligned}$$

This implies

$$\begin{aligned} 0\le & {} \left( 1 - \rho _{G_1, G_2}^2(t)\right) \sigma _{G_1}^2(t) \frac{\partial ^2 C^A}{\partial x_1^2} \nonumber \\\le & {} \sigma _{G_1}^2(t) \frac{\partial ^2 C^A}{\partial x_1^2} + 2 \rho _{G_1, G_2}(t) \sigma _{G_1}(t) \sigma _{G_2}(t) \frac{\partial ^2 C^A}{\partial x_1 \partial x_2} + \sigma _{G_2}^2 \frac{\partial ^2 C^A}{\partial x_2^2}. \end{aligned}$$

(26)

Inequalities (25) and (26) implies

$$\begin{aligned} 0\le & {} \frac{1}{2} \left( 1 - \rho _{G_1, G_2}^2(t)\right) \sigma _{G_1}^2(t) \frac{\partial ^2 C^A}{\partial x_1^2} \nonumber \\\le & {} r C^A + \frac{\sigma _{G_1}^2(t)}{2} \frac{\partial C^A}{\partial x_1} + \frac{\sigma _{G_2}^2(t)}{2} \frac{\partial C^A}{\partial x_2} - \frac{\partial C^A}{\partial t} \end{aligned}$$

(27)

Since we have proved the local boundedness of $\partial C^A / \partial t$ and the uniform boundedness of $\partial C^A / \partial G_i, i = 1, 2$, the second partial derivative $C_{11}^A$ are locally bounded. Similar argument derives the local boundedness of other second partial derivatives. $\square$

1.3.4 Proof of Proposition 2.8

With the same argument provided by Detemple (2006), we decompose the valuation of American spread option using the early exercise premium.²⁵

Proof

Since

$$\begin{aligned}&h_1 G_1(s, T_1) - h_2 G_2(s, T_2)\\&\quad = h_1 G_1(t, T_1) - h_2 G_2(t, T_2)\\&\qquad +\,h_1 \int _t^s G_1(u, T_1) \sum _{k=1}^2 \sigma _{S_k} \left[ e^{(T-u) {\varvec{\beta }}}\right] _{1,k} dW_{S_k}(u)\\&\qquad +\,h_1 \int _t^s G_1(u, T_1) \sum _{k=1}^2 \sigma _{\delta _k} \left[ e^{(T-u) {\varvec{\beta }}}\right] _{1,2+k} dW_{\delta _k}(u)\\&\qquad -\,h_2 \int _t^s G_2(u, T_2) \sum _{k=1}^2 \sigma _{S_k} \left[ e^{(T-u) {\varvec{\beta }}}\right] _{2,k} dW_{S_k}(u)\\&\qquad -\,h_2 \int _t^s G_2(u, T_2) \sum _{k=1}^2 \sigma _{\delta _k} \left[ e^{(T-u) {\varvec{\beta }}}\right] _{2,2+k} dW_{\delta _k}(u) \end{aligned}$$

holds, $h_1 G_1(t, T_1) - h_2 G_2(t, T_2)$ is a continuous semimartingale process and thus we can apply Tanaka’s formula.²⁶

$$\begin{aligned}&(h_1 G_1(s, T_1) - h_2 G_2(s, T_2) - K)^+\\&\quad = (h_1 G_1(t, T_1) - h_2 G_2(t, T_2) - K)^+\\&\qquad + \int _t^s 1_{h_1 G_1(u, T_1) - h_2 G_2(u, T_2) > K} dM(u) + L_K(t, G) \end{aligned}$$

where

$$\begin{aligned} M(u)= & {} h_1 \int _t^u G_1(v, T_1) \sum _{k=1}^2 \sigma _{S_k} \left[ e^{(T-v) {\varvec{\beta }}}\right] _{1,k} dW_{S_k}(v)\\&+\,h_1 \int _t^u G_1(v, T_1) \sum _{k=1}^2 \sigma _{\delta _k} \left[ e^{(T-v) {\varvec{\beta }}}\right] _{1,2+k} dW_{\delta _k}(v)\\&-\,h_2 \int _t^u G_2(v, T_2) \sum _{k=1}^2 \sigma _{S_k} \left[ e^{(T-v) {\varvec{\beta }}}\right] _{2,k} dW_{S_k}(v)\\&-\,h_2 \int _t^u G_2(v, T_2) \sum _{k=1}^2 \sigma _{\delta _k} \left[ e^{(T-v) {\varvec{\beta }}}\right] _{2,2+k} dW_{\delta _k}(v) \end{aligned}$$

and $L_K(t, G)$ is the local time associated with $G(t) = h_1 G_1(t, T_1) - h_2 G_2(t, T_2)$. Let us define

$$\begin{aligned} D(s) = e^{-r(s-t)} (h_1 G_1(s, T_1) - h_2 G_2(s, T_2) - K)^+. \end{aligned}$$

Therefore, our concern is the following Snell envelope.

$$\begin{aligned} C^A(G_1, G_2, t)= & {} \sup _{\tau \in \mathcal {S}_{t,T_0}} E_t [ D(\tau ) ] \end{aligned}$$

Using Ito’s lemma, we have

$$\begin{aligned} dD(s)= & {} d \left( e^{-r(s-t)} (h_1 G_1(s, T_1) - h_2 G_2(s, T_2) - K)^+ \right) \\= & {} e^{-r(s-t)} \left( - r (h_1 G_1(s, T_1) - h_2 G_2(s, T_2) - K)^+ ds + dL_K(s, G) + dM_s\right) . \end{aligned}$$

Denote

$$\begin{aligned} \tau _t = \inf \left\{ s \in [t,T]: C^A(G_1, G_2, s) = e^{-r(s-t)} (h_1 G_1(s, T_1) - h_2 G_2(s, T_2) - K)^+ \right\} \end{aligned}$$

where the following condition holds.²⁷

$$\begin{aligned} C^A(G_1, G_2, \tau _t) = e^{-r(\tau _t-t)} (h_1 G_1(\tau _t, T_1) - h_2 G_2(\tau _t, T_2) - K)^+ \end{aligned}$$

This implies

$$\begin{aligned} C^A(G_1, G_2, t)= & {} E_t \left[ e^{-r(\tau _t,t)} (h_1 G_1(\tau _t, T_1) - h_2 G_2(\tau _t, T_2) - K)^+\right] \\= & {} E_t [ C^A(G_1, G_2, \tau _t) ]. \end{aligned}$$

Furthermore, we have $\tau _s=\tau _t$ for $t \le s < \tau _t$ or otherwise $\tau _t$ is dominated by some other stopping time $\tau _s$ on $A \in \mathcal {F}_s$, that is,

$$\begin{aligned} E_s[ C^A(G_1, G_2, \tau _t) ] < E_s[ C^A(G_1, G_2, \tau _s) ], A \in \mathcal {F}_s \end{aligned}$$

which yields

$$\begin{aligned}&E_s[ C^A(G_1, G_2, \tau _t) 1_A ] < E_s[ C^A(G_1, G_2, \tau _s) 1_A ]\\&\quad \Rightarrow E_t[ C^A(G_1, G_2, \tau _t) ] < E_t[ C^A(G_1, G_2, \tau _s) 1_A + C^A(G_1, G_2, \tau _t) (1 - 1_A) ] \end{aligned}$$

and this contradicts the optimality of $\tau _t$. Therefore,

$$\begin{aligned} C^A(G_1, G_2, s)= & {} E_s[ e^{-r(\tau _s-t)} (h_1 G_1(\tau _s, T_1) - h_2 G_2(\tau _s, T_2) - K)^+ ]\\= & {} E_s[ C^A(G_1, G_2, \tau _t) ], s \in [t, \tau _t). \end{aligned}$$

Thus, $C^A(G_1, G_2, s)$ is a martingale for $s \in [t, \tau _t)$. Now define

$$\begin{aligned} N(s)= & {} C^A(G_1, G_2, s)+ \int _t^s 1_{\{\tau _u=u\}} e^{-r(u-t)} (r (h_1 G_1(u, T_1) \\&- h_2 G_2(u, T_2) - K)^+ du - dL_K(u, G)). \end{aligned}$$

This is a martingale which can be proved by the following argument. Since $C^A(G_1, G_2, \tau _t) = E_t[ (h_1 G_1(\tau _t, T_1) - h_2 G_2(\tau _t, T_2) - K)^+]$,

$$\begin{aligned} \int _t^{T_0} 1_{\{\tau _u=u\}} dC^A(G_1, G_2, u) = \int _t^{T_0} 1_{\{\tau _u=u\}} dD(u). \end{aligned}$$

Rearraging this equation, we have

$$\begin{aligned} \int _t^{T_0} 1_{\{\tau _u=u\}} dM(u)= & {} \int _t^{T_0} 1_{\{\tau _u=u\}} dC^A(G_1, G_2, u)\\&- \int _t^{T_0} 1_{\{\tau _u=u\}} e^{-r(u-t)} (- r (h_1 G_1(u, T_1)\\&-\,h_2 G_2(u, T_2) - K)^+ du + dL_K(u, G)). \end{aligned}$$

Applying Ito’s lemma, we have

$$\begin{aligned} N(s)= & {} C^A(G_1, G_2, s)\\&+ \int _t^s 1_{\{\tau _u=u\}} e^{-r(u-t)} r (h_1 G_1(u, T_1) - h_2 G_2(u, T_2) - K)^+ du\\= & {} C^A(G_1, G_2, t) + \int _t^{s} 1_{\{\tau _u=u\}} dC^A(G_1, G_2, u)\\&+ \int _t^{s} 1_{\{\tau _u>u\}} dC^A(G_1, G_2, u)\\&+ \int _t^s 1_{\{\tau _u=u\}} e^{-r(u-t)} r (h_1 G_1(u, T_1) - h_2 G_2(u, T_2) - K)^+ du\\= & {} C^A(G_1, G_2, t) + \int _t^{s} 1_{\{\tau _u>u\}} dC^A(G_1, G_2, u) + \int _t^{T} 1_{\{\tau _u=u\}} dM(u). \end{aligned}$$

Therefore, N(t) is a martingale. By definition

$$\begin{aligned} C^A(G_1, G_2, T_0)= & {} \sup _{\tau \in \mathcal {S}_{T_0, T_0}} E_{T_0} [ D(\tau ) ] = E_{T_0} [ D_{T_0} ] = D_{T_0} \end{aligned}$$

and

$$\begin{aligned} C^A(G_1, G_2, t)= & {} \sup _{\tau \in \mathcal {S}_{t, T_0}} E_{t} [ D_{\tau } ] = E_t [ D_{\tau _t} ] \end{aligned}$$

Using that N(t) is a martingale and the above two equalities, we have

$$\begin{aligned}&E_{t} \left[ e^{-r(\tau _t-t)} (h_1 G_1(\tau _t, T_1) - h_2 G_2(\tau _t, T_2) - K)^+ \right] \\&\quad = C^A(G_1, G_2, t) = N_t = E_t [ N_{T_0} ]\\&\quad =E_t \Bigg [ C^A(G_1, G_2, T_0)+ \int _t^{T_0} 1_{\{\tau _u=u\}} e^{-r(u-t)} (r (h_1 G_1(u, T_1) \\&\quad \qquad - h_2 G_2(u, T_2) - K)^+ du - dL_K(u, G)) \Bigg ]\\&\quad =E_t \Bigg [ e^{-r(T_0-t)} (h_1 G_1(T_0, T_1) - h_2 G_2(T_0, T_2) - K)^+\\&\quad \qquad + \int _t^{T_0} 1_{\{\tau _u=u\}} e^{-r(u-t)} (r (h_1 G_1(u, T_1) \\&\quad \qquad - h_2 G_2(u, T_2) - K)^+ du - dL_K(u, G)) \Bigg ]. \end{aligned}$$

As indicated by Rutkowski (1994),²⁸

$$\begin{aligned}&\int _t^{T_0} 1_{\{\tau _u=u\}} e^{-r(u-t)} dL_K(u, G)\\&\quad = \int _t^{T_0} 1_{\{\tau _u=u\}} 1_{\{h_1 G_1(u, T_1) - h_2 G_2(u, T_2) = K\}} e^{-r(u-t)} dL_K(u, G)\\&\quad = 0. \end{aligned}$$

where the first equality is due to the properties of the local time²⁹ and the second equality is from the fact $\{ \tau _u=u \} \subset \{h_1 G_1(u, T_1) > h_2 G_2(u, T_2) + K \}$ where Proposition 2.5 implies. Moreover, since

$$\begin{aligned} \tau _u=u \iff (G_1(u, T_1), G_2(u, T_2), u) \in \mathcal {E} \iff G_1(u, T_1) \ge B(G_2(u, T_2), u), \end{aligned}$$

we have the early exercise premium representation for American commodity spread options.

The boundary condition $C^A(B(G_2, t), G_2, t) = B(G_2, t) - G_2 - K$ imply the recursive Eq. (11). The boundary condition (12) is the value of spread option at maturity. The boundary condition (13) hold since the spread option is a standard option on a single asset when price $G_2$ is zero. $\square$

1.3.5 Proof of Proposition 2.9

Finally, we provide an analytical approximation pricing formula for American commodity spread option. The difficulty is the calculation of early exercise premium, more precisely the domain of integration or the condition inequality of exercise which are not analytically tractable. Therefore, we use the scheme of Bjerksund and Stensland (1994) to approximate the condition inequality which split spread option in to two call option that the first option has stochastic exercise price and then use Barone-Adesi and Whaley (1987) framework to approximate the two American option. The formula can be calculated as we did in European call option which derives the analytical approximated pricing formula for American commodity spread options.

Proof

As Bjerksund and Stensland (1994)³⁰ proposed, we approximate the early exercise premium as follows.

$$\begin{aligned}&E_t \left[ (h_1 G_1(u, T_1) - h_2 G_2(u, T_2) - K) 1_{ \{ G_1(u, T_1) \ge B(G_2(u, T_2), u) \} } \right] \\&\quad \approx E_t \left[ (h_1 G_1(u, T_1) - h_2 G_2(u, T_2) - K) 1_{ \{ h_1 G_1(u, T_1) \ge h_2 B_2 G_2(u, T_2) + B_K K \}} \right] , \end{aligned}$$

where

$$\begin{aligned} B_2= & {} B_0\left( T_0 - u, \sigma _{X_{G_1}}^2(u, T_0) - 2 \sigma _{X_{G_1} X_{G_2}} (u, T_0) + \sigma _{X_{G_2}}^2(u, T_0)\right) ,\\ B_K= & {} B_0\left( T_0 - u, \sigma _{X_{G_1}}^2 (u, T_0)\right) ,\\ B_0(t, \sigma ^2)= & {} e^{g(t, \sigma ^2)} + \left( 1 - e^{g(t, \sigma ^2)}\right) B_{\infty }(\sigma ^2),\\ B_{\infty }(\sigma ^2)= & {} \frac{\gamma (\sigma ^2)}{\gamma (\sigma ^2) - 1},\\ \gamma (\sigma ^2)= & {} \frac{1}{2} + \sqrt{\frac{1}{4} + \frac{2r}{\sigma ^2}}, \end{aligned}$$

and

$$\begin{aligned} g(t, \sigma ^2) = - 2 \sigma \sqrt{t} (\gamma (\sigma ^2) - 1). \end{aligned}$$

The approximation is constructed in two steps. The first step is to split the spread option into an exchange option and vanilla type option. And the second step is due to Barone-Adesi and Whaley (1987) framework of approximating American option.

Note that the exercise region is

$$\begin{aligned}&h_1 G_1 (u, T_1) \ge h_2 B_2 G_2(u, T_2) + B_K K \\&\quad \Leftrightarrow x_1 \le d_{eep0}(x_2, u) \equiv \ln (h_2 B_2 G_2 (t, T_2) e^{x_2} + B_K K) - \ln (h_1 G_1(t, T_1)). \end{aligned}$$

The integrals of early exercise premium can be calculated just as the integral of European commodity spread option which we now have

$$\begin{aligned}&a^A (G_1, G_2, t; B)\\&\quad = r \int _t^{T_0} e^{-r(u-t)} E_t \left[ (h_1 G_1(u, T_1) - h_2 G_2(u, T_2) - K)\right. \\&\left. \qquad \times 1_{ \{ G_1(u, T_1) \ge B(G_2(u, T_2), u) \} } \right] du\\&\quad \approx r \int _t^{T_0} e^{-r(u-t)} E_t \left[ (h_1 G_1(u, T_1) - h_2 G_2(u, T_2) - K)\right. \\&\left. \qquad \times 1_{ \{ x_1 \le d_{eep0}(x_2, u) \} } \right] du\\&\quad = r \Biggl [ h_1 G_1(t, T_1) \int _t^{T_0} \exp \Biggl \{ -r (u - t) + \mu _{X_{G_1}}(t, u, T_1) + \frac{1}{2} \sigma _{X_{G_1}}^2(t, u, T_1) \Biggr \}\\&\quad \qquad \times \int _{-\infty }^{\infty } \Phi (d_{eep2}(x_2, u))\\&\quad \qquad \times n\left( x_2 | \mu _{X_{G_2}}(t, u, T_2) + \sigma _{X_{G_1} X_{G_2}}(t, u, T_1, T_2), \sigma _{X_{G_2}}^2(t, u, T_2)\right) dx_2 du\\&\quad \qquad - h_2 G_2(t, T_2) \int _t^{T_0} \exp \Biggl \{ -r (u - t) + \mu _{X_{G_2}}(t, u, T_2) + \frac{1}{2} \sigma _{X_{G_2}}^2(t, u, T_2)\Biggr \}\\&\quad \qquad \times \int _{-\infty }^{\infty } \Phi (d_{eep1}(x_2, u))\\&\quad \qquad \times n\left( x_2 | \mu _{X_{G_2}}(t, u, T_2) + \sigma _{X_{G_2}}^2(t, u, T_2), \sigma _{X_{G_2}}^2(t, u, T_2)\right) dx_2 du\\&\quad \qquad - K \int _t^{T_0} e^{-r(u-t)} \int _{-\infty }^{\infty } \Phi (d_{eep1}(x_2, u))\\&\quad \qquad \times n\left( x_2 | \mu _{X_{G_2}}(t, u, T_2), \sigma _{X_{G_2}}^2(t, u, T_2)\right) dx_2 du \Biggr ], \end{aligned}$$

where

$$\begin{aligned} d_{eep1}(x_2, u)= & {} - \frac{\ln (h_2 B_2 G_2(t, T_2) e^{x_2} + B_K K) - \ln (h_1 G_1(t, T_1)) - \mu _{X_{G_1}}(t, u, T_1)}{\sigma _{X_{G_1}}(t, u, T_1) \sqrt{1 - \rho _{X_{G_1} X_{G_2}}^2(t, u, T_1, T_2)}} \\&+ \frac{\rho _{X_{G_1} X_{G_2}}(t, u, T_1, T_2) \sigma _{X_{G_1}}(t, u, T_1) \frac{x_2 - \mu _{X_{G_2}}(t, u, T_2)}{\sigma _{X_{G_2}}^2(t, u, T_2)}}{\sigma _{X_{G_1}}(t, u, T_1) \sqrt{1 - \rho _{X_{G_1} X_{G_2}}^2(t, u, T_1, T_2)}}, \\ \end{aligned}$$

and

$$\begin{aligned} d_{eep2}(x_2, u)= & {} d_{eep}(x_2, u) + \sigma _{X_{G_1}}(t, u, T_1) \sqrt{1 - \rho _{X_{G_1} X_{G_2}}^2(t, u, T_1, T_2)}. \end{aligned}$$

This is the approximation formula. $\square$