Emission Allowance as a Derivative on Commodity-Spread

Abstract

We provide a valuation formula for emission allowance. Assuming that the value of emission allowance on the last day of a trading phase is equal to a spread of commodity prices (e.g. electricity and natural gas) when the spread is positive and less than the penalty, we show that the emission allowance price is equal to the value of a portfolio of European call options on the spread of the commodities. Using the formula, we obtain a hedging strategy for emission allowance trading. We also empirically analyze option value embedded in emission allowance, and find by numerical analysis that the option value is relatively large.

Appendix

1.1 The Solutions of Spot Prices

The closed formulae for (3) and (4) are derived as follows. Let

$$\begin{aligned} d \ln S_i(t)&\equiv ( \beta _{S_i 0} (t) + \beta _{S_i \delta _i} \delta _i(t) ) dt + \sigma _{S_i} dW_{S_i}(t) \\ d\delta _i(t)&\equiv (\beta _{\delta _i 0} + \beta _{\delta _i \delta _i} \delta _i(t)) dt + \sigma _{\delta _i} dW_{\delta _i}(t) \end{aligned}$$

where

$$\begin{aligned}&\beta _{S_i 0} (t) = r - \frac{\sigma _{S_i}^2}{2} \\&\beta _{S_i \delta _i} = -1 \\&\beta _{\delta _i 0} = \kappa _i \hat{\alpha }_i \\&\beta _{\delta _i \delta _i} = - \kappa _i. \end{aligned}$$

This equation can be solved as follows.¹⁰

$$\begin{aligned} {{\varvec{X}}}(T) = e^{T {\varvec{\beta }}} \left\{ e^{-t {\varvec{\beta }}} {{\varvec{X}}}(t) + \int \limits _t^T e^{- s {\varvec{\beta }}} {\varvec{\beta }}_0(s) ds + \int \limits _t^T e^{- s {\varvec{\beta }}} d{{\varvec{W}}}_0(s) \right\} \end{aligned}$$

(8)

where

$$\begin{aligned}&{{\varvec{X}}}(t) = [ \ln S_1 (t), \ln S_2(t), \delta _1(t), \delta _2(t) ]^\top \nonumber \\&{\varvec{\beta }}_0 (t) = [ \beta _{S_1 0}(t), \beta _{S_2 0}(t), \beta _{\delta _1 0}, \beta _{\delta _2 0} ]^\top \nonumber \\&{\varvec{\beta }} = \left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l}&0&0&\beta _{S_1 \delta _1}&0 \\&0&0&0&\beta _{S_2 \delta _2} \\&0&0&\beta _{\delta _1 \delta _1}&0 \\&0&0&0&\beta _{\delta _2 \delta _2} \end{array} \right] \end{aligned}$$

and ${\varvec{W}}_0(t) = [\sigma _{S_1} W_{S_1}(t), \ldots , \sigma _{S_n} W_{S_n}(t), \sigma _{\delta _1} W_{\delta _1}(t), \ldots , \sigma _{\delta _n} W_{\delta _n}(t) ]^\top$ is a scaled Brownian motion vector.

Denote by $E[\cdot ]$ the expectation under the risk-neutral probability. The mean and covariances of $\ln S_i(T)$ are

$$\begin{aligned} \mu _{X_i}(t, T)&= E_t [ \ln S_i(T) ] \\&= \Bigg [ e^{T {\varvec{\beta }}} \Bigg \{ e^{-t {\varvec{\beta }}} {{\varvec{X}}}(t) + \int \limits _t^T e^{- s {\varvec{\beta }}} {\varvec{\beta }}_0 (s) ds \Bigg \} \Bigg ]_i \\ \sigma _{X_i X_j} (t, T)&= E_t[ (\ln S_i(T) - \mu _{X_i}(t, T)) (\ln S_j(T) - \mu _{X_i}(t, T))] \\&= \Bigg [\int \limits _t^T (e^{(T-s) {\varvec{\beta }}}) {\varvec{\Sigma }} (e^{(T-s) {\varvec{\beta }}})^\top ds \Bigg ]_{i j} \end{aligned}$$

where $[\cdot ]_{i}$ and $[\cdot ]_{ij}$ are $i$th element of vector and $[i,j]$ th element of matrix, respectively, and the covariance matrix

$$\begin{aligned} {\varvec{\Sigma }} = \left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l} \sigma _{S_1}^2&\rho _{S_1 S_2} \sigma _{S_1} \sigma _{S_2}&\rho _{S_1} \sigma _{S_1} \sigma _{\delta _1}&\rho _{S_1 \delta _2} \sigma _{S_1} \sigma _{\delta _2} \\ \rho _{S_2 S_1} \sigma _{S_2} \sigma _{S_1}&\sigma _{S_2}^2&\rho _{S_2 \delta _1} \sigma _{S_2} \sigma _{\delta _1}&\rho _{S_2 \delta _2} \sigma _{S_2} \sigma _{\delta _2} \\ \rho _{\delta _1 S_1} \sigma _{\delta _1} \sigma _{S_1}&\rho _{\delta _1 S_2} \sigma _{\delta _1} \sigma _{S_2}&\sigma _{\delta _1}^2&\rho _{\delta _1 \delta _2} \sigma _{\delta _1} \sigma _{\delta _2} \\ \rho _{\delta _2 S_1} \sigma _{\delta _2} \sigma _{S_1}&\rho _{\delta _2 S_2} \sigma _{\delta _2} \sigma _{S_2}&\rho _{\delta _2 \delta _1} \sigma _{\delta _2} \sigma _{\delta _1}&\sigma _{\delta _2}^2 \end{array} \right]. \end{aligned}$$

We use notations

$$\begin{aligned}&{\varvec{\mu }}_{X}(t, T) \triangleq \left( \begin{array}{l} \mu _{X_1}(t, T) \\ \mu _{X_2}(t, T) \end{array} \right) \\&{\varvec{\Sigma }}_{X}(t, T) \triangleq \left( \begin{array}{ll} \sigma _{X_1 X_1}(t, T)&\sigma _{X_1 X_2} (t, T) \\ \sigma _{X_2 X_1}(t, T)&\sigma _{X_2 X_2} (t, T) \end{array} \right) \end{aligned}$$

1.2 Proof of Proposition 2.1

Let us use the notation $\Phi (\cdot )$ and $\phi (\cdot )$ as the standard normal distribution and density function, respectively, and also $N( \cdot | \mu , \sigma ^2)$ and $n( \cdot | \mu , \sigma ^2)$ as normal distribution and density function with $\mu$ and $\sigma ^2$ as mean and variance, respectively. We calculate the following equation in this subsection.

$$\begin{aligned} S_e(t) = e^{-r(T-t)} E_t [ ( (H_1 S_1(T) - H_2 S_2(T)) \wedge Z ) \vee 0 ]. \end{aligned}$$

For notational convenience, we will omit the time parameters such as $\mu _{X_i} = \mu _{X_i}(t, T)$. The expectation is

$$\begin{aligned}&E_t [ ( (H_1 S_1(T) - H_2 S_2(T)) \wedge Z ) \vee 0 ] = H_1 \int \limits _{D_1} \exp \{x_1 \} n( {{\varvec{x}}} | {\varvec{\mu }}_X, \Sigma _X ) d{{\varvec{x}}} \\&\quad - H_2 \int \limits _{D_1} \exp \{ x_2 \} n( {{\varvec{x}}} | {\varvec{\mu }}_X, \Sigma _X ) d{{\varvec{x}}} + Z \int \limits _{D_2} n( {{\varvec{x}}} | {\varvec{\mu }}_X, \Sigma _X ) d{{\varvec{x}}} \end{aligned}$$

where

$$\begin{aligned} d(x_2, Z)&= \ln (H_2 \exp \{ x_2 \} + Z) - \ln H_1 \\ D_1&= \{ {{\varvec{x}}} = [x_1, x_2]^\top | d(x_2, 0) \le x_1 \le d(x_2, Z) \} \\ D_2&= \{ {{\varvec{x}}} = [x_1, x_2]^\top | x_1 > d(x_2, Z) \}. \end{aligned}$$

We calculate each integral. Let us use ${{\varvec{e}}}_i$ to be the unit vector which $i$-th element is one. For the integrals of the first and second term, we have

$$\begin{aligned}&\int \limits _{D_1} \exp \{ x_i \} n( {{\varvec{x}}} | {\varvec{\mu }}_X, \Sigma _X ) d{{\varvec{x}}} = \exp \left\{ \mu _{X_i} + \frac{1}{2} \sigma _{X_i}^2 \right\} \int \limits _{D_1} (2 \pi )^{-1} | \Sigma _X |^{-\frac{1}{2}} \\&\quad \times \exp \Bigg \{ -\frac{1}{2} ({{\varvec{x}}} - {\varvec{\mu }}_X - \Sigma _X {{\varvec{e}}}_i)^\top \Sigma _X^{-1} ({{\varvec{x}}} - {\varvec{\mu }}_X - \Sigma _X {{\varvec{e}}}_i) \Bigg \} d{{\varvec{x}}} \end{aligned}$$

where we completed the squares. Furthermore, the integral can be expanded by changing the variables.

$$\begin{aligned}&\int \limits _{D_1} (2 \pi )^{-1} | \Sigma _X |^{-\frac{1}{2}} \exp \left\{ -\frac{1}{2} ({{\varvec{x}}} - {\varvec{\mu }}_X - \Sigma _X {{\varvec{e}}}_1)^\top \Sigma _X^{-1} ({{\varvec{x}}} - {\varvec{\mu }}_X - \Sigma _X {{\varvec{e}}}_1) \right\} d{{\varvec{x}}} \\&\quad = \int \limits _{-\infty }^\infty \int \limits _{d_1(x_2, 0)}^{d_1(x_2, Z)} (2 \pi (1 - \rho _{X_1 X_2}^2) )^{-\frac{1}{2}} \sigma _{X_1}^{-1} \exp \left\{ - \frac{y^2}{2} \right\} (1 - \rho _{X_1 X_2}^2)^{\frac{1}{2}} \sigma _{X_1} dy \\&\qquad \times (2 \pi )^{-\frac{1}{2}} \sigma _{X_2}^{-1} \exp \Bigg \{ -\frac{1}{2} \Bigg ( \frac{x_2 - \mu _{X_2} - \sigma _{X_1 X_2}}{\sigma _{X_2}} \Bigg )^2 \Bigg \} dx_2 \\&\quad = \int \limits _{-\infty }^\infty (\Phi (d_1(x_2, Z)) - \Phi (d_1(x_2, 0))) n(x_2 | \mu _{X_2} + \sigma _{X_1 X_2}, \sigma _{X_2}^2) dx_2 \end{aligned}$$

where

$$\begin{aligned} d_1(x, z)&= \frac{\ln (H_2 \exp \{ x \} + z) - \ln H_1 - \mu _{X_1} - \sigma _{X_1}^2}{\sigma _{X_1} \sqrt{1 - \rho _{X_1 X_2}^2}} \\&- \frac{\rho _{X_1 X_2} \sigma _{X_1} \frac{x - \mu _{X_2} - \sigma _{X_1 X_2}}{\sigma _{X_2}}}{\sigma _{X_1} \sqrt{1 - \rho _{X_1 X_2}^2}} \end{aligned}$$

In addition, we can simplify the second part of the integration. Generally it is known that,

$$\begin{aligned} \Phi (d_1)&= P(X_1 \le d_1) = P (X_1 \le d_1, X_2 \le \infty ) \nonumber \\&= \int \limits _{-\infty }^\infty \Phi \left(\frac{d_1 - \rho _{12} x_2}{\sqrt{1 - \rho _{12}^2}} \right) \phi (x_2) dx_2 \end{aligned}$$

(9)

where

$$\begin{aligned}&[X_1, X_2] \sim N(\mathbf{0 }, {\varvec{\Sigma }}) \\&{\varvec{\Sigma }} = \left[ \begin{array}{ll} 1&\rho _{12} \\ \rho _{12}&1 \\ \end{array} \right] \end{aligned}$$

Notice that,

$$\begin{aligned} d_1(x_2, 0)&= \frac{\ln (H_2 / H_1) - \mu _{X_1} - \sigma _{X_1}^2 + \rho _{X_1 X_2} \sigma _{X_1} \frac{\mu _{X_2} + \sigma _{X_1 X_2}}{\sigma _{X_2}}}{\sigma _{X_1} \sqrt{1 - \rho _{X_1 X_2}^2}} \\&- \frac{\left( \frac{\rho _{X_1 X_2} \sigma _{X_1}}{\sigma _{X_2}} - 1 \right) (\sigma _{X_2} \hat{x}_1 + \mu _{X_2} + \sigma _{X_1 X_2})}{\sigma _{X_1} \sqrt{1 - \rho _{X_1 X_2}^2}} \\&= \frac{\hat{\mu }_1 - \hat{\rho } \hat{x}_1}{\sqrt{1 - \hat{\rho }^2}} \end{aligned}$$

where we defined $\hat{x}_1 = (x_2 - \mu _{X_2} - \sigma _{X_1 X_2})/\sigma _{X_2}$ and used the following facts.

$$\begin{aligned} \frac{\hat{\rho }}{\sqrt{1 - \hat{\rho }^2}}&\equiv \frac{\left( \frac{\rho _{X_1 X_2} \sigma _{X_1}}{\sigma _{X_2}} - 1 \right) \sigma _{X_2}}{\sigma _{X_1} \sqrt{1 - \rho _{X_1 X_2}^2}} = \frac{\rho _{X_1 X_2} \sigma _{X_1} - \sigma _{X_2}}{\sigma _{X_1} \sqrt{1 - \rho _{X_1 X_2}^2}} \\ \Rightarrow \hat{\rho }&= \frac{\rho _{X_1 X_2} \sigma _{X_1} - \sigma _{X_2}}{\sqrt{\sigma _{X_1}^2 - 2 \rho _{X_1 X_2} \sigma _{X_1} \sigma _{X_2} + \sigma _{X_2}^2}} \\ \hat{\mu }_1&\equiv \sqrt{1 - \hat{\rho }^2} \Bigg ( \frac{\ln (H_2 / H_1) - \mu _{X_1} - \sigma _{X_1}^2 + \rho _{X_1 X_2} \sigma _{X_1} \frac{\mu _{X_2} + \sigma _{X_1 X_2}}{\sigma _{X_2}}}{\sigma _{X_1} \sqrt{1 - \rho _{X_1 X_2}^2}} \\&- \frac{\left( \frac{\rho _{X_1 X_2} \sigma _{X_1}}{\sigma _{X_2}} - 1 \right) (\mu _{X_2} + \sigma _{X_1 X_2}) }{\sigma _{X_1} \sqrt{1 - \rho _{X_1 X_2}^2}} \Bigg ) \\&= \frac{\ln (H_2 / H_1) - \mu _{X_1} + \mu _{X_2} - \sigma _{X_1}^2 + \sigma _{X_1 X_2}}{\sqrt{\sigma _{X_1}^2 - 2 \sigma _{X_1 X_2} + \sigma _{X_2}^2}} \end{aligned}$$

Now, we have

$$\begin{aligned}&- \int \limits _{-\infty }^\infty \Phi (d_1(x_2, 0)) n(x_2 | \mu _{X_2} + \sigma _{X_1 X_2}, \sigma _{X_2}^2) dx_2 \\&\quad =- \int \limits _{-\infty }^\infty \Phi \left( \frac{\hat{\mu }_1 - \hat{\rho } \hat{x}_1}{\sqrt{1 - \hat{\rho }^2}} \right) \phi (\hat{x}_1) d\hat{x}_1 = - \Phi ( \hat{\mu }_1 ) \end{aligned}$$

where we used $n(x | \mu _{X_2} + \sigma _{X_1 X_2}, \sigma _{X_2}^2) = \frac{1}{\sigma _{X_2}} \phi (\frac{x - \mu _{X_2} - \sigma _{X_1 X_2}}{\sigma _{X_2}})$, changed the variables and (9).

The other integrals are calculated in similar manner.

$$\begin{aligned}&\int \limits _{D_1} (2 \pi )^{-1} | \Sigma _X |^{-\frac{1}{2}} \exp \left\{ -\frac{1}{2} ({{\varvec{x}}} - {\varvec{\mu }}_X - \Sigma _X {{\varvec{e}}}_2)^\top \Sigma _X^{-1} ({{\varvec{x}}} - {\varvec{\mu }}_X - \Sigma _X {{\varvec{e}}}_2) \right\} d{{\varvec{x}}} \\&= \int \limits _{-\infty }^\infty \Phi (d_2(x_2, Z)) n(x_2 | \mu _{X_2} + \sigma _{X_2}^2, \sigma _{X_2}^2) dx_2 - \Phi (\hat{\mu }_2) \\ \end{aligned}$$

where

$$\begin{aligned} d_2(x, z)&= \frac{\ln (H_2 \exp \{ x \} + z) - \ln H_1 - \mu _{X_1} - \rho _{X_1 X_2} \sigma _{X_1} \frac{x - \mu _{X_2}}{\sigma _{X_2}}}{\sigma _{X_1} \sqrt{1 - \rho _{X_1 X_2}^2}} \\ \hat{\mu }_2&= \frac{\ln (H_2 / H_1) - \mu _{X_1} + \mu _{X_2} - \sigma _{X_1 X_2} + \sigma _{X_2}^2}{\sqrt{\sigma _{X_1}^2 - 2 \sigma _{X_1 X_2} + \sigma _{X_2}^2}} \end{aligned}$$

The integral of the last term is

$$\begin{aligned}&\int \limits _{D_2} (2 \pi )^{-1} | \Sigma _X |^{-\frac{1}{2}} \exp \left\{ -\frac{1}{2} ({{\varvec{x}}} - {\varvec{\mu }}_X)^\top \Sigma _X^{-1} ({{\varvec{x}}} - {\varvec{\mu }}_X) \right\} d{{\varvec{x}}} \\&= \int \limits _{-\infty }^\infty \int \limits _{d(x_2, Z)}^{\infty } (2 \pi )^{-1} \left(\sigma _{X_1} \sigma _{X_2} \sqrt{1 - \rho _{X_1 X_2}^2} \right)^{-1} \exp \Bigg \{ - \frac{1}{2 \left(1 - \rho _{X_1 X_2}^2\right)} \\&\quad \times \Bigg ( \Bigg ( \frac{x_1 - \mu _{X_1}}{\sigma _{X_2}} \Bigg )^2 - 2 \rho _{X_1 X_2} \Bigg (\frac{x_1 - \mu _{X_1}}{\sigma _{X_1}} \Bigg ) \Bigg ( \frac{x_2 - \mu _{X_2}}{\sigma _{X_2}} \Bigg ) \\&\quad + \Bigg ( \frac{x_2 - \mu _{X_2}}{\sigma _{X_2}} \Bigg )^2 \Bigg ) \Bigg \} dx_2 dx_1 = \int \limits _{-\infty }^\infty \int \limits _{d(x_2, Z)}^{\infty } (2 \pi (1 - \rho _{X_1 X_2}^2) )^{-\frac{1}{2}} \sigma _{X_1}^{-1} \\&\quad \times \exp \Bigg \{ - \frac{\Big ( x_1 - \mu _{X_1} - \rho _{X_1 X_2} \sigma _{X_1} \frac{x_2 - \mu _{X_2}}{\sigma _{X_2}} \Big )^2}{2 \left(1 - \rho _{X_1 X_2}^2\right) \sigma _{X_1}^2} \Bigg \} dx_1 \\&\quad \times (2 \pi )^{-\frac{1}{2}} \sigma _{X_2}^{-1} \exp \Bigg \{ -\frac{1}{2} \Bigg ( \frac{x_2 - \mu _{X_2}}{\sigma _{X_2}} \Bigg )^2 \Bigg \} dx_2 \\&= \int \limits _{-\infty }^\infty \int \limits _{d_2(x_2, Z)}^{\infty } (2 \pi (1 - \rho _{X_1 X_2}^2) )^{-\frac{1}{2}} \sigma _{X_1}^{-1} \exp \left\{ - \frac{y^2}{2} \right\} \left(1 - \rho _{X_1 X_2}^2\right)^{\frac{1}{2}} \sigma _{X_1} dy \\&\quad \times (2 \pi )^{-\frac{1}{2}} \sigma _{X_2}^{-1} \exp \Bigg \{ -\frac{1}{2} \Bigg ( \frac{x_2 - \mu _{X_2}}{\sigma _{X_2}} \Bigg )^2 \Bigg \} dx_2 \\&= \int \limits _{-\infty }^\infty (1 - \Phi (d_2(x_2, Z))) n(x_2 | \mu _{X_2}, \sigma _{X_2}^2) dx_2 \\&= \int \limits _{-\infty }^\infty \Phi (-d_2(x_2, Z)) n(x_2 | \mu _{X_2}, \sigma _{X_2}^2) dx_2 \end{aligned}$$

Collecting all terms, we have the valuation formula.

$$\begin{aligned} S_e(t) =\hat{H}_1(t, T) - \hat{H}_2(t, T) + \hat{H}_3(t, T) Z \end{aligned}$$

(10)

where

$$\begin{aligned} \hat{H}_1(t, T)&= H_1 \exp \Bigg \{ -r(T-t) + \mu _{X_1}(t, T) + \frac{1}{2}\sigma _{X_1}^2(t, T) \Bigg \} \\&\times \Bigg ( \int \limits _{-\infty }^\infty \Phi (d_1(x_2, Z)) n( x_2 | \mu _{X_2}(t, T) + \sigma _{X_1 X_2}(t, T), \sigma _{X_2}^2(t, T)) dx_2 \\&- \Phi (\hat{\mu }_1(t, T)) \Bigg ) \\ \hat{H}_2(t, T)&= H_2 \exp \Bigg \{ -r(T-t) + \mu _{X_2}(t, T) + \frac{1}{2}\sigma _{X_2}^2(t, T) \Bigg \} \\&\times \Bigg ( \int \limits _{-\infty }^\infty \Phi (d_2(x_2, Z)) n( x_2 | \mu _{X_2}(t, T) + \sigma _{X_2}^2(t, T), \sigma _{X_2}^2(t, T)) dx_2 \\&- \Phi (\hat{\mu }_2(t, T)) \Bigg ) \\ \hat{H}_3(t, T)&= \exp (-r(T-t)) \int \limits _{-\infty }^\infty \Phi (-d_2(x_2, Z)) n( x_2 | \mu _{X_2}(t, T), \sigma _{X_2}^2(t, T) ) dx_2 \\ d_1(x, z)&= d_2(x, z) - \sigma _{X_1}(t, T) \sqrt{1 - \rho _{X_1 X_2}^2(t, T)} \\ d_2(x, z)&= \frac{\ln (H_2 \exp \{ x \} + z) - \ln H_1 - \mu _{X_1}(t, T)}{\sigma _{X_1}(t, T) \sqrt{1 - \rho _{X_1 X_2}^2(t, T)}} \\&- \frac{\rho _{X_1 X_2}(t, T) \sigma _{X_1}(t, T) \frac{x - \mu _{X_2}(t, T)}{\sigma _{X_2}(t, T)}}{\sigma _{X_1}(t, T) \sqrt{1 - \rho _{X_1 X_2}^2(t, T)}} \\ \hat{\mu }_1(t, T)&= \frac{\ln (H_2 / H_1) - \mu _{X_1}(t, T) + \mu _{X_2}(t, T) - \sigma _{X_1}^2(t, T) + \sigma _{X_1 X_2}(t, T)}{\sqrt{\sigma _{X_1}^2(t, T) - 2 \sigma _{X_1 X_2}(t, T) + \sigma _{X_2}^2(t, T)}} \\ \hat{\mu }_2(t, T)&= \hat{\mu }_1(t, T) + \sqrt{\sigma _{X_1}^2(t, T) - 2 \sigma _{X_1 X_2}(t, T) + \sigma _{X_2}^2(t, T)} \end{aligned}$$

1.3 Proof of Proposition 3.1

In this subsection, we derive the hedging strategy for emission allowance using futures commodities. First, we use the future commodity prices equation written in terms of spot commodity prices and derive the future price process using Ito’s lemma. This price process can be explicitly written in terms of futures price levels. Then, we calculate the expectation and covariance of stochastic terms of futures price using properties of stochastic calculus.

First,

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

The partial derivatives are

$$\begin{aligned} \frac{\partial G_i(t, T)}{\partial S_i(t)}&= \frac{\left[e^{(T-t) {\varvec{\beta }}}\right]_{i,i}}{S_i(t)} G_i(t, T) \\ \frac{\partial G_i(t, T)}{\partial \delta _i(t)}&= \left[e^{(T-t) {\varvec{\beta }}}\right]_{i,n+i} G_i(t, T) \end{aligned}$$

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

$$\begin{aligned} dG_i(t, T) = \sigma _{S_i} S_i(t) \frac{\partial G_i}{\partial S_i} dW_{S_i}(t) + \sigma _{\delta _i} \frac{\partial G_i}{\partial \delta _i} dW_{\delta _i}(t) \end{aligned}$$

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,

$$\begin{aligned}&d \log G_i(t, T) = - \frac{1}{2} \Bigg \{ \sigma _{S_i}^2 \left[e^{(T-t) {\varvec{\beta }}}\right]_{i,i}^2 + 2 \sigma _{S_i \delta _i} \left[e^{(T-t) {\varvec{\beta }}}\right]_{i,i} \left[e^{(T-t) {\varvec{\beta }}}\right]_{i,n+i} \\&\quad +\, \sigma _{\delta _i}^2 \left[e^{(T-t) {\varvec{\beta }}}\right]_{i,n+i}^2 \Bigg \} dt + \sigma _{S_i} \left[e^{(T-t) {\varvec{\beta }}}\right]_{i,i} dW_{S_i}(t) + \sigma _{\delta _i} \left[e^{(T-t) {\varvec{\beta }}}\right]_{i,n+i} dW_{\delta _i}(t) \\ \end{aligned}$$

The futures price can be expressed as follows.

$$\begin{aligned} G_i(T_0, T_i) = G_i(t, T_i) e^{\hat{X}_{G_i}(t, T_0, T_i)}, t \le T_0 \le T_i \end{aligned}$$

where

$$\begin{aligned} \hat{X}_{G_i}(t, T_0, T_i)&\equiv \mu _{\hat{X}_{G_i}}(t, T_0, T_i) \\&+ \int \limits _t^{T_0} \sigma _{S_i} \left[e^{(T_i-t) {\varvec{\beta }}}\right]_{i,i} dW_{S_i}(u) + \int \limits _t^{T_0} \sigma _{\delta _i} \left[e^{(T_i-t) {\varvec{\beta }}}\right]_{i,n+i} dW_{\delta _i}(u) \\ \end{aligned}$$

The expectation value is

$$\begin{aligned}&\mu _{\hat{X}_{G_i}}(t, T_0, T_i) \equiv E_t[\hat{X}_{G_i}(t, T_0, T_i)] \\&\quad = - \frac{1}{2} \int \limits _t^{T_0} \sigma _{S_i}^2 \left[e^{(T_i-u) {\varvec{\beta }}}\right]_{i,i}^2 du - \int \limits _t^{T_0} \sigma _{S_i \delta _i} \left[e^{(T_i-u) {\varvec{\beta }}}\right]_{i,i} \left[e^{(T_i-u) {\varvec{\beta }}}\right]_{i,n+i} du \\&\qquad - \frac{1}{2} \int \limits _t^{T_0} \sigma _{\delta _i}^2 \left[e^{(T_i-u) {\varvec{\beta }}}\right]_{i,n+i}^2 du \end{aligned}$$

The covariance of $\hat{X}_{G_i}(t, T_0, T_i)$ and $\hat{X}_{G_j}(t, T_0, T_j)$ is

$$\begin{aligned} \sigma _{\hat{X}_{G_i} \hat{X}_{G_j}}(t, T_0, T_i, T_j)&\equiv {\text{ cov}}_t [\hat{X}_{G_i}(t, T_0, T_i), \hat{X}_{G_i}(t, T_0, T_j)] \\&= \int \limits _t^{T_0} \sigma _{S_i S_j} \left[e^{(T_i-u) {\varvec{\beta }}}\right]_{i,i} \left[e^{(T_j-u) {\varvec{\beta }}}\right]_{j,j} du \\&\quad + \int \limits _t^{T_0} \sigma _{S_i \delta _j} \left[e^{(T_i-u) {\varvec{\beta }}}\right]_{i,i} \left[e^{(T_j-u) {\varvec{\beta }}}\right]_{j,n+j} du \\&\quad + \int \limits _t^{T_0} \sigma _{S_j \delta _i} \left[e^{(T_j-u) {\varvec{\beta }}}\right]_{j,j} \left[e^{(T_i-u) {\varvec{\beta }}}\right]_{i,n+i} du \\&\quad + \int \limits _t^{T_0} \sigma _{\delta _i \delta _j} \left[e^{(T_i-u) {\varvec{\beta }}}\right]_{i,n+j} \left[e^{(T_j-u) {\varvec{\beta }}}\right]_{j,n+j} du. \end{aligned}$$

Now we derive the emission allowance futures price using commodity future prices.

$$\begin{aligned} G_e(t, T)&= E_t [ S_e(T) ] = E_t [ ((H_1 S_1(T) - H_2 S_2(T)) \wedge Z) \vee 0 ] \\&= E_t [ ((H_1 G_1(t, T) e^{\hat{X}_{G_1}(t, T, T)} - H_2 G_2(t, T) e^{\hat{X}_{G_2}(t, T, T)}) \wedge Z) \vee 0 ] \end{aligned}$$

With the same argument as in the proof of Proposition , we have

$$\begin{aligned} G_e(t, T) = \hat{\hat{H}}_1(t, T) G_1(t, T) - \hat{\hat{H}}_2(t, T) G_2(t, T) + \hat{\hat{H}}_3(t, T) Z \end{aligned}$$

where

$$\begin{aligned} \hat{\hat{H}}_1(t, T)&= H_1 \exp \Bigg \{ \mu _{\hat{X}_{G_1}}(t, T, T) + \frac{1}{2} \sigma _{\hat{X}_{G_1}}^2 (t, T, T, T) \Bigg \} \Bigg \{ \int \limits _{-\infty }^\infty \Phi ( d_{G_1}(x_2, Z) ) \\&\times n\left(x_2 | \mu _{\hat{X}_{G_2}}(t, T, T) + \sigma _{\hat{X}_{G_1} \hat{X}_{G_2}} (t, T, T, T), \sigma _{\hat{X}_{G_2}}^2 (t, T, T, T) \right) dx_2 \\&- \Phi ( \hat{\mu }_{G_1}(t, T) ) \Bigg \}\\ \hat{\hat{H}}_2(t, T)&= H_2 \exp \Bigg \{ \mu _{\hat{X}_{G_2}}(t, T, T) + \frac{1}{2} \sigma _{\hat{X}_{G_2}}^2 (t, T, T, T) \Bigg \} \Bigg \{ \int \limits _{-\infty }^\infty \Phi ( d_{G_2}(x_2, Z) ) \\&\times n\left(x_2 | \mu _{\hat{X}_{G_2}}(t, T, T) + \sigma _{\hat{X}_{G_2}}^2 (t, T, T, T), \sigma _{\hat{X}_{G_2}}^2 (t, T, T, T) \right) dx_2 \\&- \Phi ( \hat{\mu }_{G_2}(t, T) ) \Bigg \} \\ \hat{\hat{H}}_3(t, T)&= \int \limits _{-\infty }^\infty \Phi ( - d_{G_2}(x_2, Z) ) n\left(x_2 | \mu _{\hat{X}_{G_2}}(t, T, T), \sigma _{\hat{X}_{G_2}}^2 (t, T, T, T) \right) dx_2 \\ d_{G_1} (x, Z)&= d_{G_2} (x, Z) - \sigma _{\hat{X}_{G_1}} (t, T, T, T) \sqrt{1 - \rho _{\hat{X}_{G_1} \hat{X}_{G_2}}^2 (t, T, T, T)} \\ d_{G_2} (x, Z)&= \frac{\ln (H_2 G_2(t, T) \exp (x) + Z) - \ln (H_1 G_1(t, T)) - \mu _{\hat{X}_{G_1}} (t, T, T)}{\sigma _{\hat{X}_{G_1}} (t, T, T, T) \sqrt{1 - \rho _{\hat{X}_{G_1} \hat{X}_{G_2}}^2 (t, T, T, T)}} \\&- \frac{\rho _{\hat{X}_{G_1} \hat{X}_{G_2}} (t, T, T, T) \sigma _{\hat{X}_{G_1}} (t, T, T, T) \frac{x - \mu _{\hat{X}_{G_2}} (t, T, T)}{\sigma _{\hat{X}_{G_2} }(t, T, T, T)}}{\sigma _{\hat{X}_{G_1}} (t, T, T, T) \sqrt{1 - \rho _{\hat{X}_{G_1} \hat{X}_{G_2}}^2 (t, T, T, T)}} \\ \hat{\mu }_{G_1} (t, T)&= \frac{\ln (H_2 G_2(t, T) / H_1 G_1(t, T)) - \mu _{\hat{X}_{G_1}} (t, T, T) + \mu _{\hat{X}_{G_2}} (t, T, T)}{\sqrt{\sigma _{\hat{X}_{G_1}}^2 (t, T, T, T) - 2 \sigma _{\hat{X}_{G_1} \hat{X}_{G_2}} (t, T, T, T) + \sigma _{\hat{X}_{G_2}}^2 (t, T, T, T)}} \\&+ \frac{- \sigma _{\hat{X}_{G_1}}^2 (t, T, T, T) + \sigma _{\hat{X}_{G_1} \hat{X}_{G_2}} (t, T, T, T)}{\sqrt{\sigma _{\hat{X}_{G_1}}^2 (t, T, T, T) - 2 \sigma _{\hat{X}_{G_1} \hat{X}_{G_2}} (t, T, T, T) + \sigma _{\hat{X}_{G_2}}^2 (t, T, T, T)}} \\ \hat{\mu }_{G_2} (t, T)&= \hat{\mu }_{G_1} (t, T) \\&+ \sqrt{\sigma _{\hat{X}_{G_1}}^2 (t, T, T, T) - 2 \sigma _{\hat{X}_{G_1} \hat{X}_{G_2}} (t, T, T, T) + \sigma _{\hat{X}_{G_2}}^2 (t, T, T, T)} \end{aligned}$$

We now derive the hedging equation for emission allowance futures price using commodity futures prices. Using Ito’s lemma, the dynamics of emission allowance futures price $dG_e(t, T)$ is

$$\begin{aligned} dG_e(t, T)&= G_1(t, T) d\hat{\hat{H}}_1 (t, T) + \hat{\hat{H}}_1 (t, T) dG_1(t, T) + d\hat{\hat{H}}_1(t, T) dG_1(t, T) \\&-\, G_2(t, T) d\hat{\hat{H}}_2 (t, T) - \hat{\hat{H}}_2 (t, T) dG_2(t, T) - d\hat{\hat{H}}_2 (t, T) dG_2(t, T) \\&+\, Z d\hat{\hat{H}}_3 (t, T) \end{aligned}$$

and $d \hat{\hat{H}}_i(t, T)$ is

$$\begin{aligned}&d \hat{\hat{H}}_i (t, T) = \frac{\partial \hat{\hat{H}}_i (t, T)}{\partial t} dt + \sum _{j=1}^2 \frac{\partial \hat{\hat{H}}_i (t, T)}{\partial G_j(t, T)} dG_j(t, T) \\&\quad +\, \frac{1}{2} \frac{\partial ^2 \hat{\hat{H}}_i (t, T)}{\partial G_1(t, T) \partial G_1(t, T)} G_1(t, T)^2 \Bigg ( \sigma _{S_1 S_1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,1}^2 \\&\quad -\, 2 \sigma _{S_1 \delta _1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,3} + \sigma _{\delta _1 \delta _1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,3}^2 \Bigg ) dt \\&\quad +\, \frac{1}{2} \frac{\partial ^2 \hat{\hat{H}}_i (t, T)}{\partial G_1(t, T) \partial G_2(t, T)} G_1(t, T) G_2(t, T) \Bigg ( \sigma _{S_1 S_2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,2} \\&\quad -\, \sigma _{S_1 \delta _2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,4} - \sigma _{S_2 \delta _1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,3} \\&\quad +\, \sigma _{\delta _1 \delta _2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,3} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,4} \Bigg ) dt +\, \frac{1}{2} \frac{\partial ^2 \hat{\hat{H}}_i (t, T)}{\partial G_2(t, T) \partial G_1(t, T)} G_1(t, T) G_2(t, T)\nonumber \\&\quad \times \Bigg ( \sigma _{S_2 S_1} -\, \sigma _{S_2 \delta _1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,3} - \sigma _{S_1 \delta _2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,4} \\&\quad +\, \sigma _{\delta _2 \delta _1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,3} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,4} \Bigg ) dt +\, \frac{1}{2} \frac{\partial ^2 \hat{\hat{H}}_i (t, T)}{\partial G_2(t, T) \partial G_2(t, T)} G_2(t, T)^2 \\&\quad \times \,\Bigg ( \sigma _{S_2 S_2}-\, 2 \sigma _{S_2 \delta _2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,4} + \sigma _{\delta _2 \delta _2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,4}^2 \Bigg ) dt \end{aligned}$$

Substituting $d \hat{\hat{H}}_i(t, T)$ to $dG_e(t, T)$, we have

$$\begin{aligned} dG_e(t, T)&= \Bigg \{ \frac{\partial \hat{\hat{H}}_1(t, T)}{\partial t} G_1(t, T) - \frac{\partial \hat{\hat{H}}_2 (t, T)}{\partial t} G_2(t, T) + Z \frac{\partial \hat{\hat{H}}_3(t, T)}{\partial t} \\&+ \Bigg ( \frac{G_1(t, T)}{2} \frac{\partial ^2 \hat{\hat{H}}_1(t, T)}{\partial G_1(t, T) \partial G_1(t, T)} - \frac{G_2(t, T)}{2} \frac{\partial ^2 \hat{\hat{H}}_2(t, T)}{\partial G_1(t, T) \partial G_1(t, T)} \\&+ \frac{Z}{2} \frac{\partial ^2 \hat{\hat{H}}_3(t, T)}{\partial G_1(t, T) \partial G_1(t, T)} \Bigg ) G_1(t, T)^2 \Bigg ( \sigma _{S_1 S_1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,1}^2 \\&- 2\sigma _{S_1 \delta _1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,3}+ \sigma _{\delta _1 \delta _1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,3}^2 \Bigg ) \\&+ \Bigg ( \frac{G_1(t, T)}{2} \frac{\partial ^2 \hat{\hat{H}}_1(t, T)}{\partial G_1(t, T) \partial G_2(t, T)} - \frac{G_2(t, T)}{2} \frac{\partial ^2 \hat{\hat{H}}_2(t, T)}{\partial G_1(t, T) \partial G_2(t, T)} \\&+ \frac{Z}{2} \frac{\partial ^2 \hat{\hat{H}}_3(t, T)}{\partial G_1(t, T) \partial G_2(t, T)} \Bigg ) G_1(t, T) G_2(t, T) \Bigg ( \sigma _{S_1 S_2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,2} \\&- \sigma _{S_1 \delta _2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,4} - \sigma _{S_2 \delta _1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,3} \\&+\sigma _{\delta _1 \delta _2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,3} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,4} \Bigg ) \\&+\Bigg ( \frac{G_1(t, T)}{2} \frac{\partial ^2 \hat{\hat{H}}_1(t, T)}{\partial G_2(t, T) \partial G_1(t, T)} - \frac{G_2(t, T)}{2} \frac{\partial ^2 \hat{\hat{H}}_2(t, T)}{\partial G_2(t, T) \partial G_1(t, T)} \\&+\frac{Z}{2} \frac{\partial ^2 \hat{\hat{H}}_3(t, T)}{\partial G_2(t, T) \partial G_1(t, T)} \Bigg ) G_1(t, T) G_2(t, T) \Bigg ( \sigma _{S_2 S_1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,2} \\&-\sigma _{S_2 \delta _1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,3} - \sigma _{S_1 \delta _2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,4} \\&+\sigma _{\delta _2 \delta _1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,3} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,4} \Bigg ) \\&+ \Bigg ( \frac{G_1(t, T)}{2} \frac{\partial ^2 \hat{\hat{H}}_1(t, T)}{\partial G_2(t, T) \partial G_2(t, T)} - \frac{G_2(t, T)}{2} \frac{\partial ^2 \hat{\hat{H}}_2(t, T)}{\partial G_2(t, T) \partial G_2(t, T)} \\&+ \frac{Z}{2} \frac{\partial ^2 \hat{\hat{H}}_3(t, T)}{\partial G_2(t, T) \partial G_2(t, T)} \Bigg ) G_2(t, T)^2 \Bigg ( \sigma _{S_2 S_2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,2}^2\\&- 2\sigma _{S_2 \delta _2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,4} + \sigma _{\delta _2 \delta _2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,4}^2 \Bigg ) \\&+ \frac{\partial \hat{\hat{H}}_1 (t, T)}{\partial G_1(t, T)} G_1(t, T)^2 \Bigg ( \sigma _{S_1}^2 \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,1}^2 - 2 \sigma _{S_1 \delta _1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,3} \\&+ \sigma _{\delta _1}^2 \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,3}^2 \Bigg ) \\&+ \frac{\partial \hat{\hat{H}}_1 (t, T)}{\partial G_2(t, T)} G_1(t, T) G_2(t, T) \Bigg ( \sigma _{S_1 S_2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,2} \\&- \sigma _{S_1 \delta _2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,4} - \sigma _{S_2 \delta _1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,3} \\&+ \sigma _{\delta _1 \delta _2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,3} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,4} \Bigg ) - \frac{\partial \hat{\hat{H}}_2 (t, T)}{\partial G_2(t, T)} G_2(t, T)\\&\qquad \times \Bigg ( \sigma _{S_2}^2 \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,2}^2 - 2 \sigma _{S_2 \delta _2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,4} + \sigma _{\delta _2}^2 \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,4}^2 \Bigg ) \\&- \frac{\partial \hat{\hat{H}}_2 (t, T)}{\partial G_1(t, T)} G_1(t, T) G_2(t, T) \Bigg ( \sigma _{S_1 S_2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,2} \\&- \sigma _{S_1 \delta _2} (T-t) \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,4} - \sigma _{S_2 \delta _1} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,3} \\&+ \sigma _{\delta _1 \delta _2} \left[e^{(T-t) {\varvec{\beta }}}\right]_{1,3} \left[e^{(T-t) {\varvec{\beta }}}\right]_{2,4} \Bigg ) \Bigg \} dt \\&+ \sum _{j=1}^2 \frac{\partial \hat{\hat{H}}_1 (t, T)}{\partial G_j(t, T)} G_1(t, T) dG_j(t, T) + \hat{\hat{H}}_1 (t, T) dG_1(t, T) \\&- \sum _{j=1}^2 \frac{\partial \hat{\hat{H}}_2 (t, T)}{\partial G_j(t, T)} G_2(t, T) dG_j(t, T) - \hat{\hat{H}}_2 (t, T) dG_2(t, T) \\&+ \sum _{j=1}^2 \frac{\partial \hat{\hat{H}}_3 (t, T)}{\partial G_j(t, T)} Z dG_j(t, T) \end{aligned}$$

Here after, we omit the time parameters. The partial derivatives are calculated as follows.

$$\begin{aligned} \frac{\partial \hat{\hat{H}}_i }{\partial G_j}&= H_i \exp \Bigg \{ \mu _{\hat{X}_{G_i}} + \frac{1}{2} \sigma _{\hat{X}_{G_i}}^2 \Bigg \} \\&\times \Bigg \{ \int \limits _{-\infty }^\infty \phi ( d_{G_i}(x_2, Z) ) \frac{\partial d_{G_i} (x_2, Z)}{\partial G_j} n \left(x_2 | \mu _{\hat{X}_{G_2}} + \sigma _{\hat{X}_{G_2} \hat{X}_{G_i}} , \sigma _{\hat{X}_{G_2}}^2 \right) dx_2 \\&- \phi ( \hat{\mu }_{G_i} ) \frac{\partial \hat{\mu }_{G_i} }{\partial G_j} \Bigg \}, \quad i, j = 1, 2\\ \frac{\partial ^2 \hat{\hat{H}}_i }{\partial G_j \partial G_k}&= H_i \exp \Bigg \{ \mu _{\hat{X}_{G_i}} + \frac{1}{2} \sigma _{\hat{X}_{G_i}}^2 \Bigg \} \\&\times \Bigg \{ \int \limits _{-\infty }^\infty \phi ^{\prime } (d_{G_i} (x_2, Z)) \frac{\partial d_{G_i} (x_2, Z)}{\partial G_k} \frac{\partial d_{G_i} (x_2, Z)}{\partial G_j} \\&\times n\left(x_2 | \mu _{\hat{X}_{G_2}} + \sigma _{\hat{X}_{G_2} \hat{X}_{G_i}} , \sigma _{\hat{X}_{G_2}}^2 \right) dx_2 \\&+ \int \limits _{-\infty }^\infty \phi ( d_{G_i} (x_2, Z) ) \frac{\partial ^2 d_{G_i} (x_2, Z)}{\partial G_j \partial G_k} n\left(x_2 | \mu _{\hat{X}_{G_2}} + \sigma _{\hat{X}_{G_2} \hat{X}_{G_i}} , \sigma _{\hat{X}_{G_2}}^2 \right) dx_2 \\&-\, \phi ^{\prime } ( \hat{\mu }_{G_i} ) \frac{\partial \hat{\mu }_{G_i} }{\partial G_k} \frac{\partial \hat{\mu }_{G_i} }{\partial G_j} - \phi ( \hat{\mu }_{G_i} ) \frac{\partial ^2 \hat{\mu }_{G_i} }{\partial G_j \partial G_k} \Bigg \} \\ \frac{\partial \hat{\hat{H}}_i }{\partial t}&= H_i \Bigg ( \frac{\partial \mu _{\hat{X}_{G_i}} }{\partial t} + \frac{1}{2} \frac{\partial \sigma _{\hat{X}_{G_i}}^2 }{\partial t} \Bigg ) \exp \Bigg \{ \mu _{\hat{X}_{G_i}} + \frac{1}{2} \sigma _{\hat{X}_{G_i}}^2 \Bigg \} \\&\times \Bigg \{ \int \limits _{-\infty }^\infty \Phi (d_{G_i}(x_2, Z)) n\left(x_2 | \mu _{\hat{X}_{G_2}} + \sigma _{\hat{X}_{G_2} \hat{X}_{G_i}} , \sigma _{\hat{X}_{G_2}}^2 \right) dx_2 - \Phi ( \hat{\mu }_{G_i} ) \Bigg \} \\&+ H_i \exp \Bigg \{ \mu _{\hat{X}_{G_i}} + \frac{1}{2} \sigma _{\hat{X}_{G_i}}^2 \Bigg \} \\&\times \Bigg \{ \int \limits _{-\infty }^\infty \phi ( d_{G_i} (x_2, Z) ) \frac{\partial d_{G_i} (x_2, Z)}{\partial t} n\left(x_2 | \mu _{\hat{X}_{G_2}} + \sigma _{\hat{X}_{G_2} \hat{X}_{G_i}}, \sigma _{\hat{X}_{G_2}}^2 \right) dx_2 \\&+ \int \limits _{-\infty }^\infty \Phi (d_{G_i} (x_2, Z)) \frac{\partial n\left(x_2 | \mu _{\hat{X}_{G_2}} + \sigma _{\hat{X}_{G_2} \hat{X}_{G_i}} , \sigma _{\hat{X}_{G_2}}^2 \right)}{\partial t} dx_2 \\&- \phi (\hat{\mu }_{G_i} ) \frac{\partial \hat{\mu }_{G_i} }{\partial t} \Bigg \},\quad i = 1, 2 \end{aligned}$$

$$\begin{aligned} \frac{\partial \hat{\hat{H}}_3 }{\partial t}&= - \int \limits _{-\infty }^\infty \phi ( -d_{G_2} (x_2, Z) ) \frac{\partial d_{G_2} (x_2, Z)}{\partial t} n\left(x_2 | \mu _{\hat{X}_{G_2}} , \sigma _{\hat{X}_{G_2}}^2 \right) dx_2 \\&+ \int \limits _{-\infty }^\infty \Phi ( -d_{G_2} (x_2, Z)) \frac{\partial n\left(x_2 | \mu _{\hat{X}_{G_2}} , \sigma _{\hat{X}_{G_2}}^2 \right)}{\partial t} dx_2 \\ \frac{\partial \hat{\hat{H}}_3 }{\partial G_j}&= - \int \limits _{-\infty }^\infty \phi ( - d_{G_2} (x_2, Z) ) \frac{\partial d_{G_2} (x_2, Z)}{\partial G_j} n\left(x_2 | \mu _{\hat{X}_{G_2}} , \sigma _{\hat{X}_{G_2}}^2 \right) dx_2 \\ \frac{\partial \hat{\hat{H}}_3 }{\partial G_j \partial G_k}&= \int \limits _{-\infty }^\infty \phi ^{\prime } (- d_{G_2}(x_2, Z) ) \frac{\partial d_{G_2} (x_2, Z)}{\partial G_k} \frac{\partial d_{G_2} (x_2, Z)}{\partial G_j} n\left(x_2 | \mu _{\hat{X}_{G_2}} , \sigma _{\hat{X}_{G_2}}^2 \right) dx_2 \\&- \int \limits _{-\infty }^\infty \phi ( - d_{G_2} (x_2, Z) ) \frac{\partial ^2 d_{G_2} (x_2, Z)}{\partial G_j \partial G_k} n\left(x_2 | \mu _{\hat{X}_{G_2}} , \sigma _{\hat{X}_{G_2}}^2 \right) dx_2 \end{aligned}$$

where

$$\begin{aligned} \frac{\partial \mu _{\hat{X}_{G_i}} }{\partial t}&= \frac{1}{2} \sigma _{S_i}^2 \left[e^{(T_i-t) {\varvec{\beta }}}\right]_{i,i}^2 - \sigma _{S_i \delta _i} \left[e^{(T_i-t) {\varvec{\beta }}}\right]_{i,i} \left[e^{(T_i-t) {\varvec{\beta }}}\right]_{i,n+i} \\&+ \frac{1}{2} \sigma _{\delta _i}^2 \left[e^{(T_i-t) {\varvec{\beta }}}\right]_{i,n+i}^2 \\ \frac{\partial \sigma _{\hat{X}_{G_i}} }{\partial t}&= - \sigma _{S_i S_j} [e^{(T_i-t) {\varvec{\beta }}}]_{i,i} \left[e^{(T_j-t) {\varvec{\beta }}}\right]_{j,j} - \sigma _{S_i \delta _j} \left[e^{(T_i-t) {\varvec{\beta }}}\right]_{i,i} \left[e^{(T_j-t) {\varvec{\beta }}}\right]_{j,n+j} \\&- \sigma _{S_j \delta _i} \left[e^{(T_i-t) {\varvec{\beta }}}\right]_{i,n+i} \left[e^{(T_j-t) {\varvec{\beta }}}\right]_{j,j} - \sigma _{\delta _i \delta _j} [e^{(T_i-t) {\varvec{\beta }}}]_{i,n+i} \left[e^{(T_j-t) {\varvec{\beta }}}\right]_{j,n+j} \end{aligned}$$

$$\begin{aligned}&\frac{\partial \rho _{\hat{X}_{G_i} \hat{X}_{G_j}} }{\partial t} = \frac{\frac{\partial \sigma _{\hat{X}_{G_i} \hat{X}_{G_j}} }{\partial t} - \rho _{\hat{X}_{G_i} \hat{X}_{G_j}} \Bigg ( \frac{\partial \sigma _{\hat{X}_{G_i}} }{\partial t} \sigma _{\hat{X}_{G_j}} + \sigma _{\hat{X}_{G_i}} \frac{\partial \sigma _{\hat{X}_{G_j}}}{\partial t} \Bigg ) }{\sigma _{\hat{X}_{G_i}} \sigma _{\hat{X}_{G_j}}}, \quad i, j = 1, 2 \\&\frac{\partial \sigma _{\hat{X}_{G_i}} }{\partial t} = \frac{1}{2} \sigma _{\hat{X}_{G_i}}^{-1} \frac{\partial \sigma _{\hat{X}_{G_i}}^2 }{\partial t}, \quad i = 1, 2 \\&\frac{\partial d_{G_2} (x_2, Z)}{\partial t} = \left(\sigma _{\hat{X}_{G_1}}^2 \left(1 - \rho _{\hat{X}_{G_1} \hat{X}_{G_2}}^2 \right)\right)^{-1} \\&\quad \times \Bigg \{ \Bigg ( - \frac{\partial \mu _{\hat{X}_{G_1}} }{\partial t} + \frac{- \frac{\partial \rho _{\hat{X}_{G_1} \hat{X}_{G_2}} }{\partial t} \sigma _{\hat{X}_{G_1}} \sigma _{\hat{X}_{G_2}} \left(x_2 - \mu _{\hat{X}_{G_2}} \right)}{\sigma _{\hat{X}_{G_2}}^2 } \\&\quad - \frac{\frac{\partial \sigma _{\hat{X}_{G_1}} }{\partial t} \rho _{\hat{X}_{G_1} \hat{X}_{G_2}} \sigma _{\hat{X}_{G_2}} \left(x_2 - \mu _{\hat{X}_{G_2}}\right)}{\sigma _{\hat{X}_{G_2}}^2 } \\&\quad + \frac{\sigma _{\hat{X}_{G_1} \hat{X}_{G_2}} \frac{\partial \mu _{\hat{X}_{G_2}} }{\partial t}}{\sigma _{\hat{X}_{G_2}}^2 } + \rho _{\hat{X}_{G_1} \hat{X}_{G_2}} \sigma _{\hat{X}_{G_1}} \left(x_2 - \mu _{\hat{X}_{G_2}}\right) \frac{\frac{\partial \sigma _{\hat{X}_{G_2}}}{\partial t}}{\sigma _{\hat{X}_{G_2}}^2} \Bigg ) \\&\quad \times \sqrt{ \sigma _{\hat{X}_{G_1}}^2 \left(1 - \rho _{\hat{X}_{G_1} \hat{X}_{G_2}}^2 \right) } \\&\quad - \Bigg ( \ln (H_2 G_2 e^{x_2} + Z) - \ln ( H_1 G_1 ) - \mu _{\hat{X}_{G_1}} - \rho _{\hat{X}_{G_1} \hat{X}_{G_2}} \sigma _{\hat{X}_{G_1}} \frac{x_2 - \mu _{\hat{X}_{G_2}} }{\sigma _{\hat{X}_{G_2}} } \Bigg ) \\&\quad \times \Bigg ( \frac{\partial \sigma _{\hat{X}_{G_1}}}{\partial t} \sqrt{1 - \rho _{\hat{X}_{G_1} \hat{X}_{G_2}}^2 } - \rho _{\hat{X}_{G_1} \hat{X}_{G_2}} \sigma _{\hat{X}_{G_1}} \left(1 - \rho _{\hat{X}_{G_1} \hat{X}_{G_2}}^2\right)^{-\frac{1}{2}} \frac{\partial \rho _{\hat{X}_{G_1} \hat{X}_{G_2}} }{\partial t} \Bigg ) \Bigg \} \end{aligned}$$

$$\begin{aligned} \frac{\partial d_{G_1} (x_2, Z)}{\partial t}&= \frac{\partial d_{G_2}(x_2, Z)}{\partial t} - \frac{\partial \sigma _{\hat{X}_{G_1}} }{\partial t} \sqrt{1 - \rho _{\hat{X}_{G_1} \hat{X}_{G_2}}^2 } \\&+ \rho _{\hat{X}_{G_1} \hat{X}_{G_2}} \sigma _{\hat{X}_{G_1}} \left(1 - \rho _{\hat{X}_{G_1} \hat{X}_{G_2}}^2 \right)^{-\frac{1}{2}} \frac{\partial \rho _{\hat{X}_{G_1} \hat{X}_{G_2}} }{\partial t} \\ \frac{\partial \hat{\mu }_{G_1} }{\partial t}&= \left(\sigma _{\hat{X}_{G_1}}^2 - 2 \sigma _{\hat{X}_{G_1} \hat{X}_{G_2}} + \sigma _{\hat{X}_{G_2}}^2 \right)^{-1} \\&\times \Bigg \{ \Bigg ( - \frac{\partial \mu _{\hat{X}_{G_1}} }{\partial t} + \frac{\partial \mu _{\hat{X}_{G_2}} }{\partial t} - \frac{\partial \sigma _{\hat{X}_{G_1}}^2 }{\partial t} + \frac{\partial \sigma _{\hat{X}_{G_1} \hat{X}_{G_2}}}{\partial t} \Bigg ) \\&\times \sqrt{ \sigma _{\hat{X}_{G_1}}^2 - 2 \sigma _{\hat{X}_{G_1} \hat{X}_{G_2}} + \sigma _{\hat{X}_{G_2}}^2 } \\&- \left( \ln ( H_2 G_2 / H_1 G_1 ) - \mu _{\hat{X}_{G_1}} + \mu _{\hat{X}_{G_2}} - \sigma _{\hat{X}_{G_1}}^2 + \sigma _{\hat{X}_{G_1} \hat{X}_{G_2}} \right) \\&\times \frac{1}{2} \left(\sigma _{\hat{X}_{G_1}}^2 - 2 \sigma _{\hat{X}_{G_1} \hat{X}_{G_2}} + \sigma _{\hat{X}_{G_2}}^2 \right)^{-\frac{1}{2}} \\&\times \Bigg ( \frac{\partial \sigma _{\hat{X}_{G_1}}^2 }{\partial t} - 2 \frac{\partial \sigma _{\hat{X}_{G_1} \hat{X}_{G_2}} }{\partial t} + \frac{\partial \sigma _{\hat{X}_{G_2}}^2 }{\partial t} \Bigg ) \Bigg \} \\ \frac{\partial \hat{\mu }_{G_2} }{\partial t}&= (2 \pi )^{-\frac{1}{2}} \exp \Bigg \{ - \frac{1}{2} \Bigg ( \frac{x_2 - \mu _{\hat{X}_{G_2}} - \sigma _{\hat{X}_{G_2} \hat{X}_{G_i}} }{\sigma _{\hat{X}_{G_2}} } \Bigg )^2 \Bigg \} \sigma _{\hat{X}_{G_2}}^{-1} \\&\times \Bigg ( \frac{x_2 - \mu _{\hat{X}_{G_2}} - \sigma _{\hat{X}_{G_2} \hat{X}_{G_i}} }{\sigma _{\hat{X}_{G_2}} } \Bigg ) \\&\times \frac{\Bigg ( \frac{\partial \mu _{\hat{X}_{G_2}} }{\partial t} + \frac{\partial \sigma _{\hat{X}_{G_2} \hat{X}_{G_i}} }{\partial t} \Bigg ) \sigma _{\hat{X}_{G_2}} + \left(x_2 - \mu _{\hat{X}_{G_2}} - \sigma _{\hat{X}_{G_2} \hat{X}_{G_i}}\right) \frac{\partial \sigma _{\hat{X}_{G_2}} }{\partial t}}{\sigma _{\hat{X}_{G_2}}^2 } \\&- (2 \pi )^{-\frac{1}{2}} \exp \Bigg \{ -\frac{1}{2} \Bigg ( \frac{x_2 - \mu _{\hat{X}_{G_2}} - \sigma _{\hat{X}_{G_2} \hat{X}_{G_i}} }{\sigma _{\hat{X}_{G_2}} } \Bigg )^2 \Bigg \} \frac{\frac{\partial \sigma _{\hat{X}_{G_2}} }{\partial t}}{\sigma _{\hat{X}_{G_2}}^2 } , \quad i = 1, 2 \end{aligned}$$

$$\begin{aligned} \frac{\partial d_{G_i}(x_2, Z)}{\partial G_1}&= - \left( \sigma _{\hat{X}_{G_1}} \sqrt{1 - \rho _{\hat{X}_{G_1} \hat{X}_{G_2}}^2 } \right)^{-1} G_1^{-1}, \quad i = 1, 2 \\ \frac{\partial d_{G_i}(x_2, Z)}{\partial G_2}&= \left( \sigma _{\hat{X}_{G_1}} \sqrt{1 - \rho _{\hat{X}_{G_1} \hat{X}_{G_2}}^2 } \right)^{-1} \frac{H_2 e^{x_2}}{H_2 G_2 e^{x_2} + Z}, \quad i = 1, 2 \\ \frac{\partial \hat{\mu }_{G_i} }{\partial G_1}&= - \left(\sigma _{\hat{X}_{G_1}}^2 - 2 \sigma _{\hat{X}_{G_1} \hat{X}_{G_2}} + \sigma _{\hat{X}_{G_2}}^2 \right)^{-\frac{1}{2}} G_1^{-1},\quad i = 1, 2 \\ \frac{\partial \hat{\mu }_{G_i} }{\partial G_2}&= \left(\sigma _{\hat{X}_{G_1}}^2 - 2 \sigma _{\hat{X}_{G_1} \hat{X}_{G_2}} + \sigma _{\hat{X}_{G_2}}^2 \right)^{-\frac{1}{2}} G_2^{-1},\quad i = 1, 2 \\ \phi ^{\prime } (x)&= \frac{-x}{\sqrt{2\pi }} e^{-\frac{1}{2} x^2} \\ \frac{\partial ^2 d_{G_i} (x_2, Z)}{\partial G_1^2}&= \left(\sigma _{\hat{X}_{G_1}} \sqrt{1 - \rho _{\hat{X}_{G_1} \hat{X}_{G_2}}^2 }\right)^{-1} G_1^{-2},\quad i = 1, 2 \\ \frac{\partial ^2 d_{G_i} (x_2, Z)}{\partial G_2^2}&= - \left(\sigma _{\hat{X}_{G_1}} \sqrt{1 - \rho _{\hat{X}_{G_1} \hat{X}_{G_2}}^2 }\right)^{-1} \Bigg (\frac{H_2 e^{x_2}}{H_2 G_2 e^{x_2} + Z} \Bigg )^2,\quad i = 1, 2 \\ \frac{\partial ^2 d_{G_i} (x_2, Z)}{\partial G_1 \partial G_2}&= 0,\quad i = 1, 2 \\ \frac{\partial ^2 \hat{\mu }_{G_i} }{\partial G_1^2}&= \left( \sigma _{\hat{X}_{G_1}}^2 - 2 \sigma _{\hat{X}_{G_1} \hat{X}_{G_2}} + \sigma _{\hat{X}_{G_2}}^2 \right)^{-\frac{1}{2}} G_1^{-2},\quad i = 1, 2 \\ \frac{\partial ^2 \hat{\mu }_{G_i} }{\partial G_2^2}&= - \left( \sigma _{\hat{X}_{G_1}}^2 - 2 \sigma _{\hat{X}_{G_1} \hat{X}_{G_2}} + \sigma _{\hat{X}_{G_2}}^2 \right)^{-\frac{1}{2}} G_2^{-2},\quad i = 1, 2 \\ \frac{\partial ^2 \hat{\mu }_{G_i} }{\partial G_1 \partial G_2}&= 0,\quad i = 1, 2 \\ \frac{\partial n\left(x_2 | \mu _{\hat{X}_{G_2}} , \sigma _{\hat{X}_{G_2}}^2 \right)}{\partial t}&= (2 \pi )^{-\frac{1}{2}} \exp \Bigg \{ - \frac{1}{2} \Bigg ( \frac{x_2 - \mu _{\hat{X}_{G_2}} }{\sigma _{\hat{X}_{G_2}} } \Bigg )^2 \Bigg \} \sigma _{\hat{X}_{G_2}}^{-1} \\&\times \Bigg ( \frac{x_2 - \mu _{\hat{X}_{G_2}} }{\sigma _{\hat{X}_{G_2}} } \Bigg ) \frac{\frac{\partial \mu _{\hat{X}_{G_2}} }{\partial t} \sigma _{\hat{X}_{G_2}} + \left(x_2 - \mu _{\hat{X}_{G_2}} \right) \frac{\partial \sigma _{\hat{X}_{G_2}} }{\partial t}}{\sigma _{\hat{X}_{G_2}}^2} \\&- (2 \pi )^{-\frac{1}{2}} \exp \Bigg \{ - \frac{1}{2} \Bigg ( \frac{x_2 - \mu _{\hat{X}_{G_2}}}{\sigma _{\hat{X}_{G_2}} } \Bigg )^2 \Bigg \} \frac{\frac{\partial \sigma _{\hat{X}_{G_2}} }{\partial t}}{\sigma _{\hat{X}_{G_2}}^2 }. \end{aligned}$$