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Valuation of Spark-Spread Option Written on Electricity and Gas Forward Contracts Under Two-Factor Models with Non-Gaussian Lévy Processes

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Abstract

In energy markets, especially electricity and gas, one experience rather large and dramatic spikes in spot prices, but they are quickly reverting back. Hence it is appropriate to take into account the factors for the spike behavior observed in the spot price series, while other factors induce price evolution when the market is stable. To illustrate this issue, we analyze the dynamics of spikes and seasonality through a normal probability test for returns of spot prices. We propose two-factor model separately for gas and electricity markets, such that in both market model the logarithmic spot price is a stationary Ornstein-Uhlenbeck process and the long-term variations are a drifted Brownian motion. We derive the forward price and its dynamic under proposed model and prove the uniqueness of solution of the stochastic differential equation related to forward price. Then we price the spark-spread option written on electricity and gas forward contracts. In the following, we derive the spark-spread option price under a hybrid geometric Brownian motion and prove that it converges to the sparks-spread option price under the proposed two-factor model. Since the market model is incomplete, we apply the quadratic hedging strategy which minimizes the hedging error. We also investigate the convergence of this strategy through the spark-spread option under the hybrid geometric Brownian motion. Numerical results confirm the achievement of all these convergences. In order to estimate the model parameters, we consider the Lévy processes as the independent normal inverse Gaussian processes and independent compound Poisson processes which have the jump size with the exponential distribution.

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Acknowledgements

The author would like to thank the referees for very constructive suggestions which helped him to improve the paper.

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Correspondence to Farshid Mehrdoust.

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Appendix

Appendix

Proof of Proposation 1

1) First, notice that

$$\begin{aligned} X(T)&=X(t)+\mu (T-t)+\sigma B_1(T-t),\\ Y(T)&=e^{-\beta (T-t)}Y(t)+\int _t^T e^{-\beta (T-s)} dL_1(s),\\ U(T)&=U(t)+\alpha (T-t)+\xi B_2(T-t), \end{aligned}$$

and

$$\begin{aligned} V(T)=e^{-\kappa (T-t)}V(t)+\int _t^T e^{-\kappa (T-s)} dL_2(s). \end{aligned}$$

Due to the \({\mathcal {F}}(t)\)-adaptedness of X(t), Y(t), U(t), and V(t), the independent increment property of Lévy processes and the independence between \(B_i\) and \(L_i\), for \(i=1, 2\), we obtain

$$\begin{aligned} F(t,T)&=F^{EL}(t,T)-h_R F^{GA}(t,T)\\&={\mathbb {E}}^{{\mathbb {Q}}}[S^{EL}(T)|{\mathcal {F}}(t)]-h_R{\mathbb {E}}^{{\mathbb {Q}}}[S^{GA}(T)|{\mathcal {F}}(t)]\\&=\varLambda ^{EL}(T){\mathbb {E}}^{{\mathbb {Q}}}[\exp \{X(T)+Y(T)\}|{\mathcal {F}}(t)]\\&\quad -h_R\varLambda ^{GA}(T){\mathbb {E}} ^{{\mathbb {Q}}}[\exp \{U(T)+V(T)\}|{\mathcal {F}}(t)]\\&=\varLambda ^{EL}(T) \exp \{X(t)+\mu (T-t)\\&\quad +e^{-\beta (T-t)}Y(t)\}{\mathbb {E}}^{{\mathbb {Q}}}\big [\exp \{\sigma B_1(T-t)\}\big ] {\mathbb {E}}^{{\mathbb {Q}}} \big [\exp \{\int _t^T e^{-\beta (T-s)} dL_1(s)\}\big ] \\&\quad -h_R\varLambda ^{GA}(T)\exp \{U(t)+\alpha (T-t)\\&\quad +e^{-\kappa (T-t)}V(t)\}{\mathbb {E}}^{{\mathbb {Q}}}\big [\exp \{\xi B_2(T-t)\}\big ] {\mathbb {E}}^{{\mathbb {Q}}}\big [\exp \{\int _t^T e^{-\kappa (T-s)} dL_2(s)\}\big ]\\&=\varLambda ^{EL}(T)\exp \Big \{\mu (T-t)+\frac{1}{2}\sigma ^2(T-t)\\&\quad +\int _t^T\phi _1(e^{-\beta (T-s)})ds+X(t)+e^{-\beta (T-t)}Y(t)\Big \}\\&\quad -h_R\varLambda ^{GA}(T)\exp \Big \{\alpha (T-t)\\&\quad +\frac{1}{2}\xi ^2(T-t)+\int _t^T\phi _2(e^{-\kappa (T-s)})ds+U(t)+e^{-\kappa (T-t)}V(t)\Big \}. \end{aligned}$$

This proves the result of first part.

II) Applying Itô’s formula to\(F^{EL}(t,T)\) and \(F^{GA}(t,T)\), gives

$$\begin{aligned} \frac{dF^{EL}(t,T)}{F^{EL}(t^-,T)}&=-\phi _1(e^{-\beta (T-t)})dt+\sigma dB_1(t)+\int _{{\mathbb {R}}}\big (\exp \{e^{-\beta (T-t)} z_1\}-1\big ){\bar{N}}_1(dt,dz_1)\\&\qquad +\int _{|z_1|\le 1} \big (\exp \{e^{-\beta (T-t)} z_1\}-1-e^{-\beta (T-t)}z_1\big )\ell _1(dz_1) dt \Big ),\\ \frac{dF^{GA}(t,T)}{F^{GA}(t^-,T)}&=-\phi _2(e^{-\kappa (T-t)})dt+\xi dB_2(t)+\int _{{\mathbb {R}}}\big (\exp \{e^{-\kappa (T-t)} z_2\}-1\big ){\bar{N}}_2(dt,dz_2) \\&\qquad +\int _{|z_2|\le 1} \big (\exp \{e^{-\kappa (T-t)} z_2\}-1-e^{-\kappa (T-t)}z_2\big )\ell _2(dz_2) dt. \end{aligned}$$

According to \(F(t^-,T)=F^{EL}(t^-,T)-h_R F^{GA}(t^-,T)\), the second part of proposition is established. \(\square \)

Proof of Theorem 2

The uniqueness follows from the Local Lipschitz property (16) and Itô isometry. Let \(F_1(t,T)\) and \(F_2(t,T)\) be solutions with initial values \(Z, {\bar{Z}}\) respectively. For our purposes here we only need the case \(Z={\bar{Z}}\).

Put \({\mathscr {A}}_i(s,T)=A_i(s,F_1(s,T))-A_i(s,F_2(s,T)), {\mathscr {C}}_i(s,T)=C_i(s,F_1(s,T))-C_i(s,F_2(s,T))\) and \({\mathscr {D}}_i(s,z_i,T)=D_i(s,z_i,F_1(s,T))-D_i(s,z_i,F_2(s,T))\), for \(i=1,2\). Then

$$\begin{aligned}&{\mathbb {E}}^{{\mathbb {Q}}}[|F_1(t,T)-F_2(t,T)|^2]\\&\quad ={\mathbb {E}}^{{\mathbb {Q}}}\Big [\Big (Z-{\bar{Z}}+ \sum _{i=1}^{2}\int _0^t {\mathscr {A}}_i ds+\sum _{i=1}^{2}\int _0^t {\mathscr {C}}_i dB_i(s)+\sum _{i=1}^{2}\int _0^t {\mathscr {D}}_i {\bar{N}}(ds,dz_i)\Big )^2\Big ]\\&\quad \le 4{\mathbb {E}}^{{\mathbb {Q}}}\big [\big |Z-{\bar{Z}}\big |^2\big ]+4{\mathbb {E}}^{{\mathbb {Q}}}\big [\big (\sum _{i=1}^2 \int _0^t{\mathscr {A}}_ids\big )^2\big ]\\&\qquad +4{\mathbb {E}}^{{\mathbb {Q}}}\big [\big (\sum _{i=1}^2 \int _0^t{\mathscr {C}}_idB_i(s)\big )^2\big ]+4{\mathbb {E}}^{{\mathbb {Q}}}\big [\big (\sum _{i=1}^2 \int _0^t\int _{{\mathbb {R}}}{\mathscr {D}}_i{\bar{N}}(ds,dz_i)\big )^2\big ]\\&\quad \le 4{\mathbb {E}}^{{\mathbb {Q}}}\big [\big |Z-{\bar{Z}}\big |^2\big ]+8\sum _{i=1}^2{\mathbb {E}}^{{\mathbb {Q}}}\big [\big ( \int _0^t{\mathscr {A}}_ids\big )^2\big ]\\&\qquad +8\sum _{i=1}^2{\mathbb {E}}^{{\mathbb {Q}}}\big [\big ( \int _0^t{\mathscr {C}}_idB_i(s)\big )^2\big ]+8\sum _{i=1}^2 {\mathbb {E}}^{{\mathbb {Q}}}\big [\big (\int _0^t\int _{{\mathbb {R}}}{\mathscr {D}}_i{\bar{N}}(ds,dz_i)\big )^2\big ]\\&\quad \le 4{\mathbb {E}}^{{\mathbb {Q}}}\big [\big |Z-{\bar{Z}}\big |^2\big ]+8t\sum _{i=1}^2{\mathbb {E}}^{{\mathbb {Q}}}\big [ \int _0^t{\mathscr {A}}_i^2ds\big ]\\&\qquad +8\sum _{i=1}^2{\mathbb {E}}^{{\mathbb {Q}}}\big [ \int _0^t{\mathscr {C}}_i^2ds\big ]+8\sum _{i=1}^2 {\mathbb {E}}^{{\mathbb {Q}}}\big [\int _0^t\int _{{\mathbb {R}}}{\mathscr {D}}_i^2\ell _i(dz_i)ds\big ]\\&\quad \le 4{\mathbb {E}}^{{\mathbb {Q}}}\big [\big |Z-{\bar{Z}}\big |^2\big ]+8(t+2)K^2\int _0^t {\mathbb {E}}^{{\mathbb {Q}}}\big [|F_1(s,T)-F_2(s,T)|^2\big ]ds. \end{aligned}$$

Then, the following function

$$\begin{aligned} \nu (t)={\mathbb {E}}^{{\mathbb {Q}}}\big [\big |F_1(t,T)-F_2(t,T)\big |^2\big ], \quad 0\le t \le T, \end{aligned}$$

satisfies

$$\begin{aligned} \nu (t)\le \gamma +\chi \int _0^t\nu (s)ds, \end{aligned}$$

where

$$\begin{aligned} \gamma =4{\mathbb {E}}^{{\mathbb {Q}}}\big [\big |Z-{\bar{Z}}\big |^2\big ] ~~~\textit{and}~~~ \chi =8(t+2)K^2. \end{aligned}$$

Applying Gronwall inequality, gives

$$\begin{aligned} \nu (t)\le \gamma e^{\chi t} \end{aligned}$$

By continuity of \(t\longrightarrow \big |F_1(t,T)-F_2(t,T)\big |\) it follows that

$$\begin{aligned} {\mathbb {P}}^{\mathbb {Q}}\big (\big |F_1(t,T)-F_2(t,T)\big |=0~~~~\textit{for all}~~ t\in [0,T]\big )=1, \end{aligned}$$

and the uniqueness of solution is proved. \(\square \)

Proof of Theorem 3

First, from Proposition 1, we have

$$\begin{aligned} F(\tau ,T)&= \frac{F^{EL}(0,T)}{F^{EL}(0,T)}\varLambda ^{EL}(T)\exp \Big \{\mu (T-\tau )+\frac{1}{2}\sigma ^2(T-\tau )\\&\quad +\int _\tau ^T\phi _1(e^{-\beta (T-s)})ds+X(\tau )+e^{-\beta (T-\tau )}Y(\tau )\Big \} \\&\quad -h_R\frac{F^{GA}(0,T)}{F^{GA}(0,T)}\varLambda ^{GA}(T)\exp \Big \{\alpha (T-\tau )+\frac{1}{2}\xi ^2(T-\tau )\\&\quad +\int _\tau ^T\phi _2(e^{-\kappa (T-s)})ds+U(\tau )+e^{-\kappa (T-\tau )}V(\tau )\Big \}. \end{aligned}$$

But,

$$\begin{aligned} e^{-\beta (T-\tau )}Y(\tau )&= e^{-\beta T} y+\int _0^\tau e^{-\beta (T-s)} dL_1(s),\\ e^{-\kappa (T-\tau )}V(\tau )&= e^{-\kappa T} v+\int _0^\tau e^{-\kappa (T-s)} dL_2(s),\\ X(\tau )&=x+\mu \tau +\sigma B_1(\tau ),\\ U(\tau )&=u+\alpha \tau +\xi B_2(\tau ). \end{aligned}$$

Therefore, we can write

$$\begin{aligned} F(\tau ,T)&= Z_1(f^{EL})e^{\sigma B_1(\tau )-\frac{1}{2}\sigma ^2 \tau }-Z_2(f^{GA})e^{\xi B_2(\tau )-\frac{1}{2}\xi ^2 \tau }. \end{aligned}$$

Then the spark-spread option price is expressed as follows

$$\begin{aligned}&C(0,\tau ,T,f^{EL},f^{GA}) =e^{-r T}{\mathbb {E}}^{\mathbb {Q}}\Big [\max (F(\tau ,T),0)\Big ]\\&\quad =e^{-r T}{\mathbb {E}}^{\mathbb {Q}}\Big [{\mathbb {E}}^{\mathbb {Q}}\big [\max \Big (Z_1(f^{EL})e^{\sigma B_1(\tau )-\frac{1}{2}\sigma ^2 \tau }-Z_2(f^{GA})e^{\xi B_2(\tau )-\frac{1}{2}\xi ^2 \tau }, 0\Big )\big |{\mathcal {F}}^Z(f^{EL},f^{GA})\big ]\Big ] . \end{aligned}$$

where \({\mathcal {F}}^Z:={\mathcal {F}}^Z(f^{EL},f^{GA})=\sigma \big ((Z_1, Z_2)_t,~ t\le \tau \big )\). Since \(L =(L_1, L_2)'\) is independent of \(B=(B_1, B_2)'\), \({\mathcal {F}}^Z\) is independent of B. Hence the inner expectation can be computed as follows

$$\begin{aligned}&\frac{1}{\sqrt{2\pi }}\int _{-\infty }^{+\infty } \max \Big (Z_1(f^{EL})e^{\sigma \sqrt{\tau }g-\frac{1}{2}\sigma ^2\tau }-Z_2(f^{GA})e^{\xi \sqrt{\tau }g-\frac{1}{2}\xi ^2\tau }, 0\Big )e^{-\frac{1}{2}g^2}dg\\&=\frac{1}{\sqrt{2\pi }}\int _{-\infty }^{d(f^{EL},f^{GA},\int _0^\tau e^{-\beta (T-s)}dL_1(s),\int _0^\tau e^{-\kappa (T-s)}dL_2(s))}Z_1(f^{EL})e^{-\frac{1}{2}(g+\sigma \sqrt{\tau })^2}dg\\&\quad -\frac{1}{\sqrt{2\pi }}\int _{-\infty }^{d(f^{EL},f^{GA},\int _0^\tau e^{-\beta (T-s)}dL_1(s),\int _0^\tau e^{-\kappa (T-s)}dL_2(s))} Z_2(f^{GA})e^{-\frac{1}{2}(g+\xi \sqrt{\tau })^2}dg\\&=Z_1(f^{EL})\varPhi (d(f^{EL},f^{GA},\int _0^\tau e^{-\beta (T-s)}dL_1(s),\int _0^\tau e^{-\kappa (T-s)}dL_2(s))+\sigma \sqrt{\tau }) \\&\quad -Z_2(f^{GA})\varPhi (d(f^{EL},f^{GA},\int _0^\tau e^{-\beta (T-s)}dL_1(s),\int _0^\tau e^{-\kappa (T-s)}dL_2(s))+\xi \sqrt{\tau }), \end{aligned}$$

where

$$\begin{aligned}&d(f^{EL},f^{GA},\omega _1,\omega _2)=\Big (\ln \big (\frac{f^{EL}}{h_R f^{GA}}\big )-\frac{1}{2}(\sigma ^2-\xi ^2)\tau \\&\quad -\int _0^{\tau } \big (\phi _1(e^{-\beta (T-s)})-\phi _2(e^{-\kappa (T-s)})\big )ds+\omega _1-\omega _2\Big )\Big ((\xi -\sigma )\sqrt{\tau }\Big )^{-1}. \end{aligned}$$

\(\square \)

Proof of Theorem 4

We have

$$\begin{aligned}&\complement (0,\tau ,T,f^{EL},f^{GA})=e^{-rT}{\mathbb {E}}^{\mathbb {Q}}\big [\max (F(\tau ,T),0)\big ]\\&\quad =\frac{e^{-rT}}{\sqrt{2\pi }}\int _{-\infty }^\infty \big (\max (f^{EL} e^{\sigma \sqrt{\tau }g-\frac{1}{2}\sigma ^2\tau }-h_R f^{GA} e^{\xi \sqrt{\tau }g-\frac{1}{2}\xi ^2\tau },0)\big )e^{-\frac{1}{2}g^2}\\&\quad =\frac{e^{-rT}}{\sqrt{2\pi }}\int _{-\infty }^{d(f^{EL},f^{GA})}\big (f^{EL} e^{-\frac{1}{2}(g+\sigma \sqrt{\tau })^2}-h_R f^{GA} e^{-\frac{1}{2}(g+\xi \sqrt{\tau })^2}\big )dg\\&\quad =e^{-rT}\Big (f^{EL}\varPhi (d(f^{EL},f^{GA})+\sigma \sqrt{\tau }))-h_Rf^{GA}\varPhi (d(f^{EL},f^{GA})+\xi \sqrt{\tau }))\Big ), \end{aligned}$$

where

$$\begin{aligned} d(f^{EL},f^{GA})=\frac{\ln (\frac{f^{EL}}{h_R f^{GA}})-\frac{1}{2}\tau (\sigma ^2-\xi ^2)}{(\xi -\sigma )\sqrt{\tau }}. \end{aligned}$$

\(\square \)

Proof of Lemma 1

We have

$$\begin{aligned} d_1(f^{EL},f^{GA},0,0)&= d_1(f^{EL},f^{GA})+\frac{1}{(\sigma -\xi )\sqrt{\tau }}\int _0^\tau (\phi _1(e^{-\beta (T-s)}-\phi _2(e^{-\kappa (T-s)}))ds. \end{aligned}$$

Using the mean value theorem, there is a random variable g such that

$$\begin{aligned} \varPhi (d_1(f^{EL},f^{GA},0,0))-\varPhi (d_1(f^{EL},f^{GA}))&= \varPhi '(g)(d_1(f^{EL},f^{GA},0,0)- d_1(f^{EL},f^{GA})). \end{aligned}$$

But,

$$\begin{aligned} \varPhi '(g)&=\frac{1}{\sqrt{2\pi }}e^{-\frac{1}{2}g^2}\le \frac{1}{\sqrt{2\pi }}<1. \end{aligned}$$

Then

$$\begin{aligned}&\varPhi (d_1(f^{EL},f^{GA},0,0))-\varPhi (d_1(f^{EL},f^{GA}))<d_1(f^{EL},f^{GA},0,0)-d_1(f^{EL},f^{GA})\nonumber \\&\quad =\frac{1}{(\sigma -\xi )\sqrt{\tau }}\int _0^\tau (\phi _1(e^{-\beta (T-s)}-\phi _2(e^{-\kappa (T-s)}))ds. \end{aligned}$$
(26)

Now, we evaluate the log-moment generating function \(\phi _1(e^{-\beta (T-s)}\). For notational simplicity, let \(\gamma (s) = e^{-\beta (T - s)}\). From Eq. (7), we have

$$\begin{aligned} \phi _1(e^{-\beta (T-s)})=\ln {\mathbb {E}}[e^{\gamma (s)L_1(1)}]. \end{aligned}$$

Applying Itô’s formula to \(e^{\gamma (s)L_1(1)}\), gives

$$\begin{aligned} e^{\gamma (s)L_1(1)}&=\int _{|z_1|<1}(e^{\gamma (s)z_1}-1-z_1\gamma (s))\ell _1(dz_1)+\int _{{\mathbb {R}}}(e^{\gamma (s)z_1}-1){\bar{N}}_1(dz_1)\\&=\int _{|z_1|<1}(e^{\gamma (s)z_1}-1-z_1\gamma (s))\ell _1(dz_1)\\&\qquad +\int _{|z_1|<1}(e^{\gamma (s)z_1}-1)(N_1(dz_1)-\ell _1(dz_1))+\int _{|z_1|>1}(e^{\gamma (s)z_1}-1)N_1(dz_1)\\&=\int _{|z_1|<1}-z_1\gamma (s)\ell _1(dz_1)+\int _{{\mathbb {R}}}(e^{\gamma (s)z_1}-1)N_1(dz_1)\\&=\int _{|z_1|<1}-z_1\gamma (s)\ell _1(dz_1)+\int _{{\mathbb {R}}}(e^{\gamma (s)z_1}-1)(\overset{\sim }{N}_1(dz_1)+\ell _1(dz_1))\\&=\int _{{\mathbb {R}}}(e^{\gamma (s)z_1}-1-z_1\gamma (s)1_{|z_1|<1})\ell _1(dz_1)+\int _{{\mathbb {R}}}(e^{\gamma (s)z_1}-1)\overset{\sim }{N}_1(dz_1) \end{aligned}$$

Note that \(\overset{\sim }{N}_1\) is a martingale process. Thus we have

$$\begin{aligned} {\mathbb {E}}[e^{\gamma (s)L_1(1)}]=\int _{{\mathbb {R}}}(e^{\gamma (s)z_1}-1-z_1\gamma (s)1_{|z_1|<1})\ell _1(dz_1). \end{aligned}$$

By definition of the log-moment generating function

$$\begin{aligned} \phi _1(e^{-\beta (T-s)})=\ln \big (\int _{{\mathbb {R}}}(e^{\gamma (s)z_1}-1-z_1\gamma (s)1_{|z_1|<1})\ell _1(dz_1)\big ). \end{aligned}$$
(27)

For \(|z_1| \ge 1\), we obtain

$$\begin{aligned} |e^{\gamma (s)z_1}-1|&\le \sum _{n=1}^{\infty }\frac{(\gamma (s)|z_1|)^n}{n!}\\&=\gamma (s)|z_1| \sum _{n=2}^{\infty }\frac{(\gamma (s)|z_1|)^n}{n!}\\&\le \gamma (s)|z_1| \sum _{n=0}^{\infty }\frac{(\gamma (s)|z_1|)^n}{n!}\\&=\gamma (s)|z_1| e^{\gamma (s)|z_1|}. \end{aligned}$$

If \(|z_1| < 1\), the series representation of the exponential function gives

$$\begin{aligned} |e^{\gamma (s)z_1}-1-z_1\gamma (s)|&\le \sum _{n=2}^{\infty }\frac{(\gamma (s)|z_1|)^n}{n!}\\&\le \gamma ^2(s)|z_1|^2\sum _{n=3}^{\infty }\frac{(\gamma (s)|z_1|)^n}{n!}\\&\le \gamma ^2(s)|z_1|^2\sum _{n=0}^{\infty }\frac{(\gamma (s)|z_1|)^n}{n!}\\&\le \gamma ^2(s)|z_1|^2\sum _{n=0}^{\infty }\frac{(\gamma (s))^n}{n!}\\&= \gamma ^2(s)|z_1|^2e^{\gamma (s)}. \end{aligned}$$

Hence, using the definition of \(\gamma (s),\)

$$\begin{aligned}&\int _{{\mathbb {R}}}\Big |e^{\gamma (s)z_1}-1-z_1\gamma (s)1_{|z_1|<1}\Big |\ell _1(dz_1)\\&\quad =\int _{|z_1|\ge 1}\Big |e^{\gamma (s)z_1}-1\Big |\ell _1(dz_1)+\int _{|z_1|<1}\Big |e^{\gamma (s)z_1}-1-z_1\gamma (s)1_{|z_1|<1}\Big |\ell _1(dz_1)\\&\quad \le \gamma (s)\int _{|z_1|\ge 1}|z_1|e^{\gamma (s)|z_1|}\ell _1(dz_1)+\gamma ^2(s)e^{\gamma (s)}\int _{|z_1|<1}z_1^2\ell _1(dz_1)\\&\quad =e^{-\beta (T-s)}\int _{|z_1|\ge 1}|z_1|e^{e^{-\beta (T-s)}|z_1|}\ell _1(dz_1)+e^{-2\beta (T-s)}e^{e^{-\beta (T-s)}}\int _{|z_1|<1}z_1^2\ell _1(dz_1)\\&\quad \le e^{-\beta (T-s)}\int _{|z_1|\ge 1}e^{2|z_1|}\ell _1(dz_1)+e^{-2\beta (T-s)+1}\int _{|z_1|<1}z_1^2\ell _1(dz_1)\\&\quad \le e^{-\beta (T-s)}\Big (\int _{|z_1|\ge 1}e^{2|z_1|}\ell _1(dz_1)+e^{1}\int _{|z_1|<1}z_1^2\ell _1(dz_1)\Big ). \end{aligned}$$

From Eq. (27), we have

$$ \begin{gathered} \phi _{1} \left( {e^{{ - \beta (T - s)}} } \right) \le \ln \left( {\int_{\mathbb{R}} | e^{{\gamma (s)z_{1} }} - 1 - z_{1} \gamma (s)1_{{|z_{1} | < 1}} |\ell \left( {dz_{1} } \right)} \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\; \le \ln \left( {e^{{ - \beta (T - s)}} \left( {\int_{{|z_{1} | \ge 1}} {e^{{2|z_{1} |}} } \ell _{1} (dz_{1} ) + e^{1} \int_{{|z_{1} | < 1}} {_{1}^{2} \ell \left( {dz_{1} } \right)} } \right)} \right)\, \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\; \le e^{{ - \beta (T - s)}} \left( {\int_{{|z_{1} | \ge 1}} {e^{{2|z_{1} |}} } \ell _{1} (dz_{1} ) + e^{1} \int_{{|z_{1} | < 1}} {z_{1}^{2} \ell (dz_{1} )} } \right) \hfill \\ \end{gathered} $$

Therefore, there exists a constant \(C > 0\), such that

$$\begin{aligned} \phi _1(e^{-\beta (T-s)})\le C e^{-\beta (T-s)}. \end{aligned}$$

Similarly, we get

$$\begin{aligned} \phi _2(e^{-\kappa (T-s)})\le C e^{-\kappa (T-s)}. \end{aligned}$$

By absolute value inequalities

$$\begin{aligned} \Big |\phi _1(e^{-\beta (T-s)})-\phi _2(e^{-\kappa (T-s)})\Big |&\le \Big | C e^{-\beta (T-s)}- C e^{-\kappa (T-s)}\Big |\\&\le C e^{-\beta (T-s)}+ C e^{-\kappa (T-s)}. \end{aligned}$$

Using the inequality (26), we obtain

$$\begin{aligned}&\Big |\varPhi (d_1(f^{EL},f^{GA},0,0))-\varPhi (d_1(f^{EL},f^{GA}))\Big |\\&\quad \le \frac{1}{|\sigma -\xi |\sqrt{\tau }}\int _0^\tau (C e^{-\beta (T-s)}+C e^{-\kappa (T-s)})ds\\&\quad =\frac{1}{|\sigma -\xi |\sqrt{\tau }}\Big (\frac{C e^{-\beta (T-\tau )}(1-e^{-\beta \tau })}{\beta }+\frac{C e^{-\kappa (T-\tau )}(1-e^{-\kappa \tau })}{\kappa }\Big ). \end{aligned}$$

\(\square \)

Proof of Lemma 2

Consider the new probability measure \(\overset{\sim }{{\mathbb {Q}}}\) with Radon-Nikodym derivative

$$\begin{aligned} \frac{d\overset{\sim }{{\mathbb {Q}}}}{d{\mathbb {Q}}}=\exp \Big \{\int _0^\tau e^{-\beta (T-s)}dL_1(s)-\int _0^\tau \phi _1(e^{-\beta (T-s)})ds\Big \}. \end{aligned}$$

This is an Esscher transform, turning the Lévy process \(L_1\) into a independent increment process. Then the logarithmic-moment generating function of \(\int _0^\tau e^{-\beta (T-s)}ds\) under \(\overset{\sim }{{\mathbb {Q}}}\) is as follows

$$\begin{aligned} \phi _1^{\beta ,\overset{\sim }{{\mathbb {Q}}}}(\theta _1)&=\ln {\mathbb {E}}^{\overset{\sim }{{\mathbb {Q}}}}\Big [\exp \Big \{\theta _1\int _0^\tau e^{-\beta (T-s)}dL_1(s)\Big \}\Big ]\\&=\ln \Big ({\mathbb {E}}^{{\mathbb {Q}}}\Big [\exp \big \{\theta _1\int _0^\tau e^{-\beta (T-s)}dL_1(s)\big \}\\&\quad \exp \big \{\int _0^\tau e^{-\beta (T-s)}dL_1(s)\big \}\Big ]\exp \big \{-\int _0^\tau \phi _1(e^{-\beta (T-s)})ds\big \}\Big )\\&=\ln \Big ({\mathbb {E}}^{{\mathbb {Q}}}\Big [\exp \big \{(\theta _1+1)\int _0^\tau e^{-\beta (T-s)}dL_1(s)\big \}\Big ]\Big )-\int _0^\tau \phi _1(e^{-\beta (T-s)})ds\\&=\int _0^\tau \big (\phi _1((\theta _1+1)e^{-\beta (T-s)})- \phi _1(e^{-\beta (T-s)})\big )ds. \end{aligned}$$

Applying Cauchy-Schwarz inequality, yields

$$\begin{aligned}&\Big |{\mathbb {E}}^{{\mathbb {Q}}}\Big [\frac{Z_1(f^{EL})}{f^{EL}}\varPhi (d_1(f^{EL},f^{GA}, \int _0^\tau e^{-\beta (T-s)}dL_1(s),\int _0^\tau e^{-\kappa (T-s)}dL_2(s)))\Big ]\\&\qquad -\varPhi (d_1(f^{EL},f^{GA},0,0))\Big |\\&\quad =\Big |{\mathbb {E}}^{\overset{\sim }{{\mathbb {Q}}}}\Big [\varPhi (d_1(f^{EL},f^{GA},\int _0^\tau e^{-\beta (T-s)}dL_1(s),\int _0^\tau e^{-\kappa (T-s)}dL_2(s)))\Big ]\\&\qquad -\varPhi (d_1(f^{EL},f^{GA},0,0))\Big |\\&\quad \le {\mathbb {E}}^{\overset{\sim }{{\mathbb {Q}}}}\Big [\Big |\varPhi (d_1(f^{EL},f^{GA},\int _0^\tau e^{-\beta (T-s)}dL_1(s),\int _0^\tau e^{-\kappa (T-s)}dL_2(s)))\\&\qquad -\varPhi (d_1(f^{EL},f^{GA},0,0))\Big |\Big ]. \end{aligned}$$

Using the mean value theorem, there is a random variable g such that

$$\begin{aligned}&\varPhi (d_1(f^{EL},f^{GA},\int _0^\tau e^{-\beta (T-s)}dL_1(s),\int _0^\tau e^{-\kappa (T-s)}dL_2(s)))\\&\quad -\varPhi (d_1(f^{EL},f^{GA},0,0))\\&\quad =\varPhi '(g)\big (d_1(f^{EL},f^{GA},\int _0^\tau e^{-\beta (T-s)}dL_1(s),\\&\quad \int _0^\tau e^{-\kappa (T-s)}dL_2(s)))-d_1(f^{EL},f^{GA},0,0)\big ). \end{aligned}$$

Note that

$$\begin{aligned} \varPhi '(g)<1. \end{aligned}$$

Thus

$$\begin{aligned}&\varPhi (d_1(f^{EL},f^{GA},\int _0^\tau e^{-\beta (T-s)}dL_1(s),\int _0^\tau e^{-\kappa (T-s)}dL_2(s)))-\varPhi (d_1(f^{EL},f^{GA},0,0))\\&\quad <d_1(f^{EL},f^{GA},\int _0^\tau e^{-\beta (T-s)}dL_1(s),\int _0^\tau e^{-\kappa (T-s)}dL_2(s))-d_1(f^{EL},f^{GA},0,0). \end{aligned}$$

Since

$$\begin{aligned}&d_1(f^{EL},f^{GA},\int _0^\tau e^{-\beta (T-s)}dL_1(s),\int _0^\tau e^{-\kappa (T-s)}dL_2(s))=d_1(f^{EL},f^{GA},0,0)\\&\quad +\frac{1}{(\xi -\sigma )\sqrt{\tau }}\Big (\int _0^\tau e^{-\beta (T-s)}dL_1(s)\\&\quad -\int _0^\tau e^{-\kappa (T-s)}dL_2(s)\Big ). \end{aligned}$$

From Cauchy-Schwarz inequality we have

$$\begin{aligned}&{\mathbb {E}}^{\overset{\sim }{{\mathbb {Q}}}}\Big |\varPhi (d_1(f^{EL},f^{GA},\int _0^\tau e^{-\beta (T-s)}dL_1(s),\int _0^\tau e^{-\kappa (T-s)}dL_2(s)))-\varPhi (d_1(f^{EL},f^{GA},0,0))\Big |\\&\quad \le \frac{1}{|\xi -\sigma |\sqrt{\tau }}\Big ({\mathbb {E}}^{\overset{\sim }{{\mathbb {Q}}}}\Big |\int _0^\tau e^{-\beta (T-s)}dL_1(s)-\int _0^\tau e^{-\kappa (T-s)}dL_2(s)\Big |\Big )\\&\quad \le \frac{1}{|\xi -\sigma |\sqrt{\tau }}\Big ({\mathbb {E}}^{\overset{\sim }{{\mathbb {Q}}}}\Big |\int _0^\tau e^{-\beta (T-s)}dL_1(s)-\int _0^\tau e^{-\kappa (T-s)}dL_2(s)\Big |^2\Big )^{\frac{1}{2}}. \end{aligned}$$

Jensen’s inequality implies

$$\begin{aligned}&{\mathbb {E}}^{\overset{\sim }{{\mathbb {Q}}}}\Big |\int _0^\tau e^{-\beta (T-s)}dL_1(s)-\int _0^\tau e^{-\kappa (T-s)}dL_2(s)\Big |^2\nonumber \\&\quad \le 2\Big ({\mathbb {E}}^{\overset{\sim }{{\mathbb {Q}}}}\big [(\int _0^\tau e^{-\beta (T-s)}dL_1(s))^2\big ]+{\mathbb {E}}^{\overset{\sim }{{\mathbb {Q}}}}\big [(\int _0^\tau e^{-\kappa (T-s)}dL_2(s))^2\big ]\Big ). \end{aligned}$$
(28)

From basic probability theory

$$\begin{aligned}&{\mathbb {E}}^{\overset{\sim }{{\mathbb {Q}}}}\big [(\int _0^\tau e^{-\beta (T-s)}dL_1(s))^2\big ]=\frac{\partial ^2}{\partial \theta _1^2}e^{\phi _1^{\beta ,\overset{\sim }{{\mathbb {Q}}}}(\theta _1)}\Big |_{\theta _1=0}\nonumber \\&\quad =(\phi _1^{\beta ,\overset{\sim }{{\mathbb {Q}}}})^{''}(0)e^{\phi _1^{\beta ,\overset{\sim }{{\mathbb {Q}}}}(0)}+(\phi _1^{\beta ,\overset{\sim }{{\mathbb {Q}}}})^{'}(0))^2e^{\phi _1^{\beta ,\overset{\sim }{{\mathbb {Q}}}}(0)}\nonumber \\&\quad =(\phi _1^{\beta ,\overset{\sim }{{\mathbb {Q}}}})^{''}(0)+((\phi _1^{\beta ,\overset{\sim }{{\mathbb {Q}}}})^{'}(0))^2. \end{aligned}$$
(29)

Denoting \(\gamma (s)=e^{-\beta (T-s)}\), it follows,

$$\begin{aligned} (\phi _1^{\beta ,\overset{\sim }{{\mathbb {Q}}}})^{'}(0)=\int _0^\tau e^{-\beta (T-s)} \phi ^{'}_1(e^{-\beta (T-s)})ds. \end{aligned}$$
(30)

We have

$$\begin{aligned} \phi _1(\gamma (s))=\int _{{\mathbb {R}}}(e^{\gamma (s)z_1}-1-\gamma (s)z_1 1_{|z_1|<1})\ell _1(dz_1). \end{aligned}$$

Thus

$$\begin{aligned} \phi '_1(\gamma (s))&=\frac{\partial }{\partial h}\int _{{\mathbb {R}}}(e^{h z_1}-1-h z_1 1_{|z_1|<1})\ell _1(dz_1)\Big |_{h=\gamma (s)}\\&=\int _{{\mathbb {R}}}(z_1 e^{\gamma (s)z_1}-z_1 1_{|z_1|<1})\ell _1(dz_1)\\&=\int _{|z_1|<1}z_1 (e^{\gamma (s)z_1}-1)\ell _1(dz_1)+\int _{|z_1|\ge 1}z_1 e^{\gamma (s)z_1}\ell _1(dz_1). \end{aligned}$$

As

$$\begin{aligned} \big |e^{\gamma (s)z_1}-1\big |&\le \gamma (s)|z_1|e^{\gamma (s)|z_1|}\\&\le |z_1|e^1, \end{aligned}$$

for \(|z_1| < 1\), while for \(|z_1| \ge 1\)

$$\begin{aligned} |z_1|e^{\gamma (s)|z_1|}\le e^{2|z_1|}, \end{aligned}$$

it follows that

$$\begin{aligned} |\phi '_1(\gamma (s))|\le e^1\int _{|z_1|<1}z_1^2\ell (dz_1)+\int _{|z_1|\ge 1}e^{2|z_1|}\ell _1(dz_1)\le C, \end{aligned}$$

where C is a positive constant.

Therefore, from expression (30) we have

$$\begin{aligned} |(\phi _1^{\beta ,\overset{\sim }{{\mathbb {Q}}}})^{'}(0)|\le C\int _0^\tau e^{-\beta (T-s)}ds=\frac{C}{\beta }(1-e^{-\beta (T-s)}) \end{aligned}$$
(31)

On the other hand

$$\begin{aligned} (\phi _1^{\beta ,\overset{\sim }{{\mathbb {Q}}}})^{''}(0)=\int _0^\tau e^{-2\beta (T-s)}\phi _1^{''}(e^{-\beta (T-s)})ds. \end{aligned}$$

Then

$$\begin{aligned} \phi {''}_1(\gamma (s))&=\frac{\partial }{\partial h}\int _{{\mathbb {R}}}(z_1e^{h z_1}- z_1 1_{|z_1|<1})\ell _1(dz_1)\Big |_{h=\gamma (s)} \\&=\int _{{\mathbb {R}}} z_1^2 e^{\gamma (s)z_1}\ell _1(dz_1) \\&=\int _{{\mathbb {R}}}(z_1^2 e^{\gamma (s)z_1} 1_{|z_1|<1}+z_1^2 e^{\gamma (s)z_1} 1_{|z_1|\ge 1}) \ell _1(dz_1). \end{aligned}$$

For \(|z_1|<1\), the series representation of the exponential function gives

$$\begin{aligned} |e^{\gamma (s)z_1}| \le \sum _{n=0}^{\infty }\frac{(\gamma (s)|z_1|)^n}{n!}\le \sum _{n=0}^{\infty }\frac{(\gamma (s))^n}{n!}=e^{\gamma (s)}\le e^1. \end{aligned}$$

While if \(|z_1 | \ge 1 \), we have

$$\begin{aligned} z_1^2e^{\gamma (s)z_1}\le z_1^2 e^{z_1}\le e^{3z_1}. \end{aligned}$$

Therefore from the condition on the Lévy measure in inequality (6), we get

$$\begin{aligned} |\phi {''}_1(e^{-\beta (T-s)})|\le C, \end{aligned}$$

where C is a positive constant and obtain

$$\begin{aligned} |(\phi _1^{\beta ,\overset{\sim }{{\mathbb {Q}}}})^{''}(0)|\le C\int _0^\tau e^{-2\beta (T-s)}ds=\frac{C}{2\beta }(1-e^{-2\beta (T-\tau )}). \end{aligned}$$
(32)

According to expression (29) and inequalities (31), (32) we have

$$\begin{aligned} {\mathbb {E}}^{\overset{\sim }{{\mathbb {Q}}}}\Big [\big (\int _0^\tau e^{-\beta (T-s)}dL_1(s)\big )^2\Big ]\le \frac{C^2}{\beta ^2}(1-e^{-\beta (T-\tau )})^2+\frac{C}{2\beta }(1-e^{-2\beta (T-\tau )}). \end{aligned}$$
(33)

Similarly, we get

$$\begin{aligned} {\mathbb {E}}^{\overset{\sim }{{\mathbb {Q}}}}\Big [\big (\int _0^\tau e^{-\kappa (T-s)}dL_1(s)\big )^2\Big ]\le \frac{C^2}{\kappa ^2}(1-e^{-2\kappa (T-\tau )})^2+\frac{C}{2\kappa }(1-e^{-2\kappa (T-\tau )}). \end{aligned}$$
(34)

Combining inequalities (28), (33), and (34) get

$$\begin{aligned}&{\mathbb {E}}^{\overset{\sim }{{\mathbb {Q}}}}\Big [\big (\int _0^\tau e^{-\beta (T-s)}dL_2(s)-\int _0^\tau e^{-\kappa (T-s)}dL_2(s)\big )^2\Big ]\\&\quad \le \frac{C^2}{\beta ^2}(1-e^{-\beta (T-\tau )})^2+\frac{C}{2\beta }(1-e^{-2\beta (T-\tau )})\\&\quad +\frac{C^2}{\kappa ^2}(1-e^{-\kappa (T-\tau )})^2+\frac{C}{2\kappa }(1-e^{-2\kappa (T-\tau )}). \end{aligned}$$

Thus

$$\begin{aligned}&\Big |{\mathbb {E}}^{{\mathbb {Q}}}\Big [\frac{Z_1(f^{EL})}{f^{EL}}\varPhi (d_1(f^{EL},f^{GA},\int _0^\tau e^{-\beta (T-s)}dL_1(s),\\&\quad \int _0^\tau e^{-\kappa (T-s)}dL_2(s)))\Big ]-\varPhi (d_1(f^{EL},f^{GA},0,0))\Big |\\&\quad \le {\mathbb {E}}^{\overset{\sim }{{\mathbb {Q}}}}\Big |\varPhi (d_1(f^{EL},f^{GA},\int _0^\tau e^{-\beta (T-s)}dL_1(s),\\&\quad \int _0^\tau e^{-\kappa (T-s)}dL_2(s)))-\varPhi (d_1(f^{EL},f^{GA},0,0))\Big |\\&\quad \le \frac{1}{|\xi -\sigma |\sqrt{\tau }}\Big ({\mathbb {E}}^{\overset{\sim }{{\mathbb {Q}}}}\Big |\int _0^\tau e^{-\beta (T-s)}dL_1(s)\\&\quad -\int _0^\tau e^{-\kappa (T-s)}dL_2(s)\Big |^2\Big )^{\frac{1}{2}}\\&\quad \le \frac{1}{|\xi -\sigma |\sqrt{\tau }}\Big (\frac{C^2}{\beta ^2}(1-e^{-\beta (T-\tau )})^2+\frac{C}{2\beta }(1-e^{-2\beta (T-\tau )})\\&\quad +\frac{C^2}{\kappa ^2}(1-e^{-\kappa (T-\tau )})^2+\frac{C}{2\kappa }(1-e^{-2\kappa (T-\tau )})\Big )^{\frac{1}{2}}. \end{aligned}$$

\(\square \)

Proof of Theorem 5

Applying triangle inequality and Lemmas 1-3, yield

$$\begin{aligned}&|C(0,\tau ,T,f^{EL},f^{GA})-\complement (0,\tau ,T,f^{EL},f^{GA})|\\&\quad =e^{-rT}\Big |\Big ({\mathbb {E}}^{{\mathbb {Q}}}\Big [Z_1(f^{EL})\varPhi (d_1(f^{EL},f^{GA},\int _0^\tau e^{-\beta (T-s)}dL_1(s),\\&\quad \int _0^\tau e^{-\kappa (T-s)}dL_2(s)))\Big ]-f^{El}\varPhi (d_1(f^{EL},f^{GA}))\Big )\\&\qquad -\Big ({\mathbb {E}}^{{\mathbb {Q}}}\Big [Z_2(f^{GA})\varPhi (d_2(f^{EL},f^{GA},\int _0^\tau e^{-\beta (T-s)}dL_1(s),\int _0^\tau e^{-\kappa (T-s)}dL_2(s)))\Big ]\\&\qquad -h_Rf^{GA}\varPhi (d_2(f^{EL},f^{GA}))\Big )\Big |\\&\le e^{-rT}\Big (\Big |\Big ({\mathbb {E}}^{{\mathbb {Q}}}\Big [Z_1(f^{EL})\varPhi (d_1(f^{EL},f^{GA},\int _0^\tau e^{-\beta (T-s)}dL_1(s),\int _0^\tau e^{-\kappa (T-s)}dL_2(s)))\\&\qquad -f^{El}\varPhi (d_1(f^{EL},f^{GA}))\Big ]\Big |\\&\qquad +\Big |{\mathbb {E}}^{{\mathbb {Q}}}\Big [Z_2(f^{GA})\varPhi (d_2(f^{EL},f^{GA},\int _0^\tau e^{-\beta (T-s)}dL_1(s),\int _0^\tau e^{-\kappa (T-s)}dL_2(s)))\\&\qquad -h_Rf^{GA}\varPhi (d_2(f^{EL},f^{GA}))\Big ]\Big |\Big )\\&\quad \le e^{-rT}f^{EL}\Big (\Big |{\mathbb {E}}^{{\mathbb {Q}}}\Big [\frac{Z_1(f^{EL})}{f^{EL}}\varPhi (d_1(f^{EL},f^{GA},\int _0^\tau e^{-\beta (T-s)}dL_1(s),\int _0^\tau e^{-\kappa (T-s)}dL_2(s)))\\&\qquad -\varPhi (d_1(f^{EL},f^{GA},0,0))\Big ]\Big |\\&\qquad +\Big |{\mathbb {E}}^{{\mathbb {Q}}}\Big [\varPhi (d_1(f^{EL},f^{GA},0,0))-\varPhi (d_1(f^{EL},f^{GA}))\Big ]\Big |\Big )\\&\qquad +e^{-rT}h_Rf^{GA}\Big (\Big |{\mathbb {E}}^{{\mathbb {Q}}}\Big [\frac{Z_2(f^{GA})}{h_R f^{GA}}\varPhi (d_2(f^{EL},f^{GA},\\&\quad \int _0^\tau e^{-\beta (T-s)}dL_1(s),\int _0^\tau e^{-\kappa (T-s)}dL_2(s)))\\&\qquad -\varPhi (d_2(f^{EL},f^{GA},0,0))\Big ]\Big |\\&\qquad +\Big |{\mathbb {E}}^{{\mathbb {Q}}}\Big [\varPhi (d_2(f^{EL},f^{GA},0,0))-\varPhi (d_2(f^{EL},f^{GA}))\Big ]\Big |\Big ) \\&\quad \le \Big (\frac{e^{-rT}}{|\xi -\sigma |\sqrt{\tau }}\big (\frac{C^2}{\beta ^2}(1-e^{-\beta (T-\tau )})^2+\frac{C}{2\beta }(1-e^{-2\beta (T-\tau )})\\&\qquad +\frac{C^2}{\kappa ^2}(1-e^{-\kappa (T-\tau )})^2+\frac{C}{2\kappa }(1-e^{-2\kappa (T-\tau )})\big )^{\frac{1}{2}}\\&\qquad +\frac{1}{|\sigma -\xi |\sqrt{\tau }}\big (\frac{C e^{-\beta (T-\tau )}(1-e^{-\beta \tau })}{\beta }+\frac{C e^{-\kappa (T-\tau )}(1-e^{-\kappa \tau })}{\kappa }\big )\Big )\Big (f^{EL}+h_Rf^{GA}\Big ). \end{aligned}$$

The right hand side of last inequality tend to zero as \(T\rightarrow \infty \). \(\square \)

Proof of Theorem 6

Let us \(C(t, \tau , T, x, y)\) and V(t) are respectively the discounted spark-spread option and the discounted self-financing portfolio. such that the variation of self-financing portfolio is

$$\begin{aligned} dV (t) = \varPsi (t)e^{-rt} dF(t, T). \end{aligned}$$

From no-arbitrage pricing theory, \(dC(t, \tau , T, x, y)\) and dV(t) are martingale. Thus from Proposition 1, and applying Itô’s formula to \(C(t, \tau , T, x, y)\) and V(t), we have

$$\begin{aligned}&dC(t, \tau , T, x, y)=e^{-rt}\Big (\sigma F^{EL}(t,T) C_x(t,\tau ,T,F^{EL}(t,T),F^{GA}(t,T))d\overset{\sim }{B}_1(t)\\&\quad -h_R \xi F^{GA}(t,T)C_y(t,\tau ,T,F^{EL}(t,T),F^{GA}(t,T))d\overset{\sim }{B}_2(t)\\&\quad +\int _{\mathbb {R}}\big (C(t,\tau ,T,F^{EL}(t,T)\exp \{z_1e^{-\beta (T-t)}\},F^{GA}(t,T))\\&\quad -C(t,\tau ,T,F^{EL}(t,T),F^{GA}(t,T))\big )\overset{\sim }{N}_1(dt,dz_1)\\&\quad -h_R \int _{\mathbb {R}}\big (C(t,\tau ,T,F^{EL}(t,T),F^{GA}(t,T)\exp \{z_2e^{-\kappa (T-t)}\})\\&\quad -C(t,\tau ,T,F^{EL}(t,T),F^{GA}(t,T))\big )\overset{\sim }{N}_2(dt,dz_2)\Big ),\\&dV(t)=\varPsi (t) e^{-rt} \Big (F^{EL}(t,T)\big (\sigma d\overset{\sim }{B}_1(t)\\&\quad +\int _{\mathbb {R}}(\exp \{z_1e^{-\beta (T-s)}\}-1)\overset{\sim }{N}_1(dt,dz_1)\big )\\&\quad -h_R F^{GA}(t,T)\big (\xi d\overset{\sim }{B}_2(t)+\int _{\mathbb {R}}(\exp \{z_2e^{-\kappa (T-s)}\}-1)\overset{\sim }{N}_2(dt,dz_2)\big )\Big ), \end{aligned}$$

where \(\overset{\sim }{B_1}\) and \(\overset{\sim }{B_2}\) are two independent Brownian motions under \({\mathbb {Q}}\).

Suppose that \(V (0)= C(0, \tau , T, F^{EL}(0, T),F^{GA}(0,T))\). The hedging error is

$$\begin{aligned} {\mathcal {E}}(\varPsi ):= V(t, \tau , T, F^{EL}(t, T),F^{GA}(t,T))- C(t, \tau , T, F^{EL}(t, T),F^{GA}(t,T)), \end{aligned}$$

for \(\tau \le T\). Thus, according to the Itô isometry property for stochastic integration obtain

$$\begin{aligned}&{\mathbb {E}}^{{\mathbb {Q}}}\big [{\mathcal {E}}^2(\varPsi )\big ]=\int _0^\tau {\mathbb {E}}^{{\mathbb {Q}}}\\&\quad \big [(\sigma e^{-rs}F^{EL}(s,T))^2 \big (\varPsi (s)\\&\quad -C_x(s,\tau ,T,F^{EL}(s,T),F^{GA}(s,T))\big )^2\big ]ds\\&\quad + \int _0^\tau {\mathbb {E}}^{{\mathbb {Q}}}\big [(h_R \xi e^{-rs}F^{GA}(s,T))^2 \\&\quad \big (C_y(s,\tau ,T,F^{EL}(s,T),F^{GA}(s,T))-\varPsi (s)\big )^2\big ]ds\\&\quad +\int _0^\tau \int _{\mathbb {R}}{\mathbb {E}}^{{\mathbb {Q}}}\Big [e^{-2rs}\Big (\varPsi (s)F^{EL}(s,T)\\&\quad \big (\exp \{z_1 e^{-\beta (T-s)}\} -1\big ) +C(s,\tau ,T,F^{EL}(s,T),F^{GA}(s,T))\\&\quad -C(s,\tau ,T,F^{EL}(s,T)\exp \{z_1e^{-\beta (T-s)}\},F^{GA}(s,T))\Big )^2\ell _1(dz_1)\Big ]ds\\&\quad +\int _0^\tau \int _{\mathbb {R}}{\mathbb {E}}^{{\mathbb {Q}}}\Big [h_R^2e^{-2rs}\Big (-\varPsi (s)F^{GA}(s,T)\\&\quad \big (\exp \{z_2 e^{-\kappa (T-s)}\} -1\big ) -C(s,\tau ,T,F^{EL}(s,T),F^{GA}(s,T))\\&\quad +C(s,\tau ,T,F^{EL}(s,T),F^{GA}(s,T)\exp \{z_2e^{-\kappa (T-s)}\})\Big )^2\ell _2(dz_2)\Big ]ds. \end{aligned}$$

The first order condition for the minimizer of this quadratic expression solves

$$\begin{aligned}&\varPsi (t)\Big ((\sigma F^{EL}(t,T))^2+(h_R\xi F^{GA}(t,T))^2+(F^{EL}(t,T))^2\\&\quad \int _{\mathbb {R}}(\exp \{z_1e^{-\beta (T-t)}\}-1)^2\ell _1(dz_1)\\&\quad +(h_R F^{GA}(t,T))^2\int _{\mathbb {R}}(\exp \{z_2e^{-\kappa (T-t)}\}-1)^2\ell _2(dz_2)\Big )\\&\quad =(\sigma F^{EL}(t,T))^2C_x(t,\tau ,T,F^{EL}(t,T),F^{GA}(t,T))\\&\quad +(h_R\xi (F^{GA}(t,T))^2C_y(t,\tau ,T,F^{EL}(t,T),F^{GA}(t,T)) \\&\quad +F^{EL}(t,T)\int _{\mathbb {R}}\big (C(t,\tau ,T,F^{EL}(t,T)\\&\quad \exp \{z_1e^{-\beta (T-t)}\},F^{GA}(t,T))-C(t,\tau ,T,F^{EL}(t,T),F^{GA}(t,T))\big ) \\&\quad \big (\exp \{z_1e^{-\beta (T-t)}\}-1\big )\ell _1(dz_1)\\&\quad +h^2_R F^{GA}(t,T)\int _{\mathbb {R}}\big (C(t,\tau ,T,F^{EL}(t,T),F^{GA}(t,T)\exp \{z_2e^{-\kappa (T-t)}\})\\&\quad -C(t,\tau ,T,F^{EL}(t,T),F^{GA}(t,T))\big ) \\&\quad \big (\exp \{z_2e^{-\kappa (T-t)}\}-1\big )\ell _2(dz_2) \end{aligned}$$

and the proof is completed. \(\square \)

Proof of Theorem 7

Applying triangle inequality, gives

$$\begin{aligned}&|\varPsi (t)-\big ((\sigma x)^2\complement _x+(h_R\xi y)^2\complement _y\big )|\nonumber \\&\le \Big ((\sigma x)^2|C_x-\complement _x|+(h_R\xi y)^2 |C_y- \complement _y|\nonumber \\&\quad +x\int _{\mathbb {R}}\big |C^\beta -C\big |\big |\exp \{z_1e^{-\beta (T-t)}\}-1\big |\ell _1(dz_1)\nonumber \\&\quad +h^2_R y\int _{\mathbb {R}}\big |C^\kappa -C\big |\big |\exp \{z_2e^{-\kappa (T-t)}\}-1\big |\ell _2(dz_2)\Big )\times \nonumber \\&\quad \Big ((\sigma x)^2+(h_R\xi y)^2+x^2\int _{\mathbb {R}}(\exp \{z_1e^{-\beta (T-t)}\}-1)^2\ell _1(dz_1)\nonumber \\&\quad +(h_R y)^2\int _{\mathbb {R}}(\exp \{z_2e^{-\kappa (T-t)}\}-1)^2\ell _2(dz_2)\Big )^{-1}, \end{aligned}$$
(35)

where

$$\begin{aligned}&C:=C(t,\tau ,T,x,y),~ \complement :=\complement (t,\tau ,T,x,y),~ C^\beta :=C(t,\tau ,T,x\exp \{z_1e^{-\beta (T-t)}\},y),\\&\quad C^\kappa :=C(t,\tau ,T,x,y\exp \{z_2e^{-\kappa (T-t)}\}). \end{aligned}$$

We have

$$\begin{aligned}&(\sigma x)^2|C_x-\complement _x|\Big ((\sigma x)^2+(h_R\xi y)^2+x^2\int _{\mathbb {R}}(\exp \{z_1e^{-\beta (T-t)}\}-1)^2\ell _1(dz_1)\\&\quad +(h_R y)^2\int _{\mathbb {R}}(\exp \{z_2e^{-\kappa (T-t)}\}-1)^2\ell _2(dz_2)\Big )^{-1}\\&\le e^{-r(T-t)}\Big ( {\mathbb {E}}^{\mathbb {Q}}\Big |\frac{Z_1(x)}{x}\varPhi (d_1(x,y,\omega _1,\omega _2)) -\varPhi (d_1(x,y))\Big |\\&+\frac{1}{|\xi -\sigma |\sqrt{\tau -t}} {\mathbb {E}}^{\mathbb {Q}}\Big |\frac{Z_1(x)}{x}\varPhi '(d_1(x,y,\omega _1,\omega _2)) -\varPhi '(d_1(x,y))\Big |\\&+\frac{h_Ry}{x|\xi -\sigma |\sqrt{\tau -t}} {\mathbb {E}}^{\mathbb {Q}}\Big |\frac{Z_2(y)}{h_R y}\varPhi '(d_2(x,y,\omega _1,\omega _2)) -\varPhi '(d_2(x,y))\Big |\Big ). \end{aligned}$$

Similar to proof of Lemma 2 for each of the statements on right hand side, we obtain

$$\begin{aligned}&(\sigma x)^2|C_x-\complement _x|\Big ((\sigma x)^2+(h_R\xi y)^2+x^2\int _{\mathbb {R}}(\exp \{z_1e^{-\beta (T-t)}\}-1)^2\ell _1(dz_1) \nonumber \\&\quad +(h_R y)^2\int _{\mathbb {R}}(\exp \{z_2e^{-\kappa (T-t)}\}-1)^2\ell _2(dz_2)\Big )^{-1}=0\quad as \quad T\rightarrow 0. \end{aligned}$$
(36)

Similarly, the following expression holds

$$\begin{aligned}&(h_R \xi y)^2|C_y-\complement _y|\Big ((\sigma x)^2+(h_R\xi y)^2+x^2\int _{\mathbb {R}}(\exp \{z_1e^{-\beta (T-t)}\}-1)^2\ell _1(dz_1) \nonumber \\&\quad +(h_R y)^2\int _{\mathbb {R}}(\exp \{z_2e^{-\kappa (T-t)}\}-1)^2\ell _2(dz_2)\Big )^{-1}=0\quad as \quad T\rightarrow 0. \end{aligned}$$
(37)

By mean value theorem, we have

$$\begin{aligned}&x\int _{\mathbb {R}}\big |C^\beta -C\big |\big |\exp \{z_1e^{-\beta (T-t)}\}-1\big |\ell _1(dz_1)\Big ((\sigma x)^2+(h_R\xi y)^2\\&\quad +x^2\int _{\mathbb {R}}(\exp \{z_1e^{-\beta (T-t)}\}-1)^2\ell _1(dz_1) \\&\quad +(h_R y)^2\int _{\mathbb {R}}(\exp \{z_2e^{-\kappa (T-t)}\}-1)^2\ell _2(dz_2)\Big )^{-1}\\&=\frac{1}{\sigma ^2}\int _{\mathbb {R}}\big |C_x (t,\tau ,g,y)\big |\big |\exp \{z_1e^{-\beta (T-t)}\}-1\big |^2\ell _1(dz_1), \end{aligned}$$

for some \(g\ge 0\).

It can be easily shown that

$$\begin{aligned} |C_x(t,\tau ,T,g,y)|\le e^{-r(T-t)}\Big (1+\frac{g+h_R y}{g|\xi -\sigma |\sqrt{\tau -t}}\Big ). \end{aligned}$$

On the other hand, for \(K\le 1\),

$$\begin{aligned} |e^{Kz_1-1}|\le K|z_1|\sum _{n=1}^\infty \frac{K^{n-1}|z_1|^{n-1}}{n!}\le K|z_1|\sum _{n=0}^\infty \frac{|z_1|^n}{n!}=K|z_1|e^{|z_1|}. \end{aligned}$$

If K is replaced with \(e^{-\beta (T-t)}\), we have

$$\begin{aligned}&\int _{\mathbb {R}}\Big (e^{-\beta (T-t)}-1\Big )^2\ell _1(dz_1)\le e^{-\beta (T-t)}\int _{\mathbb {R}}|z_1|^2 e^{2|z_1|}\ell _1(dz_1)\\&\quad \le e^{-2\beta (T-t)}\Big (e^2\int _{|z_1|\le 1}|z_1|^2\ell _1(dz_1)+e^2\int _{|z_1|> 1}z_1^2e^{2z_1}\ell _1(dz_1)\Big ). \end{aligned}$$

From the exponential moment condition on \(\ell _1(dz_1)\) and the condition that \(\ell _1\) is a Lévy measure, following inequality is obtained

$$\begin{aligned} \int _{\mathbb {R}}\Big (e^{-\beta (T-t)}-1\Big )^2\ell _1(dz_1)\le C e^{-2\beta (T-t)}, \end{aligned}$$

for a constant \(C > 0\).

Thus

$$\begin{aligned}&x\int _{\mathbb {R}}\big |C^\beta -C\big |\big |\exp \{z_1e^{-\beta (T-t)}\}-1\big |\ell _1(dz_1) \nonumber \\&\quad \Big ((\sigma x)^2+(h_R\xi y)^2+x^2\int _{\mathbb {R}}(\exp \{z_1e^{-\beta (T-t)}\}-1)^2\ell _1(dz_1) \nonumber \\&\quad +(h_R y)^2\int _{\mathbb {R}}(\exp \{z_2e^{-\kappa (T-t)}\}-1)^2\ell _2(dz_2)\Big )^{-1}=0 \quad as \quad T\rightarrow \infty . \end{aligned}$$
(38)

Similarly

$$\begin{aligned}&h_r^2 y\int _{\mathbb {R}}\big |C^\kappa -C\big |\big |\exp \{z_2e^{-\kappa (T-t)}\}-1\big |\ell _2(dz_2)\Big ((\sigma x)^2+(h_R\xi y)^2 \nonumber \\&\quad +x^2\int _{\mathbb {R}}(\exp \{z_1e^{-\beta (T-t)}\}-1)^2\ell _1(dz_1) \nonumber \\&\quad +(h_R y)^2\int _{\mathbb {R}}(\exp \{z_2e^{-\kappa (T-t)}\}-1)^2\ell _2(dz_2)\Big )^{-1}=0 \quad as \quad T\rightarrow \infty . \end{aligned}$$
(39)

Combining Eqs. (35)–(39) gives the result. \(\square \)

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Mehrdoust, F., Noorani, I. Valuation of Spark-Spread Option Written on Electricity and Gas Forward Contracts Under Two-Factor Models with Non-Gaussian Lévy Processes. Comput Econ 61, 807–853 (2023). https://doi.org/10.1007/s10614-021-10232-4

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