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Putting a price tag on temperature

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Abstract

A model for the evolution of daily average temperatures (DATs) is put forward to support the analysis of weather derivatives. The goal is to capture simultaneously the stochasticity, mean-reversion, and seasonality patterns of the DATs process. An Ornstein–Uhlenbeck (OU) process modulated by a hidden Markov chain (HMC) is proposed to model both the mean-reversion and stochasticity of a deseasonalised component. The seasonality part is modelled by a combination of linear and sinusoidal functions. Modified and more efficient OU–HMM filtering algorithms relative to the current ones in the literature are presented for the evolution of adaptive and switching model parameter estimates. Numerical implementation of the estimation technique using a 4-year Toronto temperature data set compiled by the National Climatic Data Center was conducted. A sensitivity analysis of the option prices with respect to model parameters is included.

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Acknowledgements

The authors are grateful for the helpful comments of two anonymous referees. This work is supported by the Natural Sciences and Engineering Research Council of Canada through R. Mamon’s Discovery Grant (RGPIN-2017-04235). R. Mamon expresses his sincere appreciation for the hospitality of the Division of Physical Sciences and Mathematics, University of the Philippines Visayas, where certain revisions of this paper were made during an academic visit both as an Adjunct Professor and a DOST-PCIEERD Balik Scientist for the Philippine government.

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Correspondence to Rogemar Mamon.

Appendices

Appendix A: Derivations of the model’s optimal parameter estimates

Optimal estimate for \(\delta \)

Define a new measure \(P^{\widehat{\upsilon }^*}\) via

$$\begin{aligned} \left. \frac{\mathrm {d}P^{\widehat{\upsilon }^*}}{\mathrm {d}P^{\upsilon ^*}} \right| _{\mathcal{X}_k} = {\Psi }^\delta _k=\prod ^k_{l=1}{\varphi }^\delta _l, \end{aligned}$$

where

$$\begin{aligned} {\varphi }_{l}^{\delta }=\frac{\text {exp}\left( -\frac{1}{2\epsilon ^{2} \left( \mathbf {y}_{l}\right) }\left( X_{l+1}-\widehat{\delta } \left( \mathbf {y}_{l}\right) X_{l}-\eta \left( \mathbf {y}_{l}\right) \right) ^{2}\right) }{\text {exp}\left( -\frac{1}{2\epsilon ^{2}\left( \mathbf {y}_{l}\right) } \left( X_{l+1}-\delta \left( \mathbf {y}_{l}\right) X_{l}-\eta \left( \mathbf {y}_{l}\right) \right) ^{2}\right) }. \end{aligned}$$

Therefore, the log likelihood for \(\tilde{\Psi }^\delta _k\) is

$$\begin{aligned} \log {\Psi }^\delta _k&=\sum _{l=1}^{k}\Bigg (-\frac{\widehat{\delta }^{2} \left( \mathbf {y}_{l}\right) X_{l}^{2}-2X_{l+1}\widehat{\delta } \left( \mathbf {y}_{l}\right) X_{l}+2\eta \left( \mathbf {y}_{l}\right) \widehat{\delta } \left( \mathbf {y}_{l}\right) X_{l}}{2\epsilon ^{2}\left( \mathbf {y}_{l}\right) } +\mathrm {R}\left( {\delta }\left( \mathbf {y}_{l}\right) \right) \Bigg )\\&=\sum _{l=1}^{k}Bigg(\sum _{i=1}^{n}\langle \mathbf {y}_{l},\mathbf {e}_{i}\rangle \Bigg (-\frac{\widehat{\delta }_{i}^{2}X_{l}^{2}-2X_{l+1}\widehat{\delta }_{i}X_{l}+2\eta _{i} \widehat{\delta }_{i}X_{l}}{2\epsilon _{i}^{2}}+\mathrm {R}(\delta _{i})\Bigg )Bigg). \end{aligned}$$

Since \(\mathrm {R}(\delta _i)\) does not contain \(\widehat{\delta }_i\), such remainder has no affect on the result of the derivation. From Eqs. (20) and (21), and considering \(\widehat{U}_l = \mathbb {E}(U_l | \mathcal{X}_k)\), we have

$$\begin{aligned} L (\widehat{\delta _i})&= \sum _{i=1}^{N}\mathbb {E}\left[ -\frac{1}{2\epsilon _{i}^{2}} \left( \widehat{\delta }_{i}^{2}\mathscr {T}_{k}^{i} \left( X_{k}^{2}\right) -2\widehat{\delta }_{i}\mathscr {T}_{k}^{i}\left( X_{k}, X_{k+1}\right) +2\eta _{i}\widehat{\delta }_{i}\mathscr {T}_{k}^{i}\left( X_{k}\right) \right) \bigg |\mathcal{X}_{k}\right] +\mathrm {R}\left( \delta _{i}\right) \\&=\sum _{i=1}^{N}-\frac{1}{2\epsilon _{i}^{2}}\left( \widehat{\delta }_{i}^{2}\mathscr {T}_{k}^{i} \left( X_{k}^{2}\right) -2\widehat{\delta }_{i}\mathscr {T}_{k}^{i}\left( X_{k}, X_{k+1}\right) +2\eta _{i}\widehat{\delta }_{i}\mathscr {T}_{k}^{i} \left( X_{k}\right) \right) +\mathrm {R}\left( \delta _{i}\right) . \end{aligned}$$

We differentiate \( L (\widehat{\delta }_i)\) with respect to \(\widehat{\delta }_i\) and set the result to zero giving

$$\begin{aligned} \widehat{\delta }_{i}=\frac{\widehat{\mathscr {T}}_{k}^{i}(X_{k},X_{k+1}) -\eta _{i}\widehat{\mathscr {T}}_{k}^{i}(X_{k})}{\widehat{\mathscr {T}}_{k}^{i}(X_{k}^{2})}. \end{aligned}$$

Optimal estimate for \(\eta \)

Define a new measure \(P^{\widehat{\upsilon }^*}\) through \(\left. \frac{\mathrm {d}P^{\widehat{\upsilon }^*}}{\mathrm {d}P^{\upsilon ^*}}\right| _{\mathcal{X}_k}={\Psi }_{k}^{\eta }=\prod _{l=1}^{k}{\varphi }_{l}^{\eta }, \) where \({\varphi }_{l}^{\eta }=\frac{\text {exp}\left( -\frac{1}{2\epsilon ^{2}\left( \mathbf {y}_{l}\right) }\left( X_{l+1}-{\delta }\left( \mathbf {y}_{l}\right) X_{l}-\widehat{\eta }\left( \mathbf {y}_{l}\right) \right) ^{2}\right) }{\text {exp}\left( -\frac{1}{2\epsilon ^{2}\left( \mathbf {y}_{l}\right) }\left( X_{l+1}-\delta \left( \mathbf {y}_{l}\right) X_{l}-\eta \left( \mathbf {y}_{l}\right) \right) ^{2}\right) } \) leading to the log likelihood

$$\begin{aligned} \log {\Psi }_{k}^{\eta } =\sum _{l=1}^{k}\left( -\frac{\widehat{\eta }^{2}\left( \mathbf {y}_{l}\right) -2X_{l+1}\widehat{\eta }\left( \mathbf {y}_{l}\right) +2\widehat{\eta }\left( \mathbf {y}_{l}\right) \delta \left( \mathbf {y}_{l}\right) X_{l}}{2\epsilon ^{2}\left( \mathbf {y}_{l}\right) }+\mathrm {R}\left( \eta \left( \mathbf {y}_{l}\right) \right) \right) . \end{aligned}$$

Invoking Eqs. (20) and (21) and then taking expectation of the log likelihood involving \(X_k\), we obtain \( L (\widehat{\eta })=\mathbb {E}\left[ \log \Psi _{k}^{\eta }\mid \mathcal{X}_{k}\right] \) with

$$\begin{aligned} \mathbb {E}\left[ \log \Psi _{k}^{\eta }\mid \mathcal{X}_{k}\right] =\sum _{l=1}^{k}\mathbb {E}\left[ \left( \sum _{i=1}^{N}\left( -\frac{\langle \mathbf {y}_{l},\mathbf {e}_{i}\rangle }{2\epsilon _{i}^{2}}\left( \widehat{\eta }_{i}^{2}-2X_{l+1}\widehat{\eta }_{i}+2\widehat{\eta }_{i}\delta _{i}X_{l}\right) \right) +\mathrm {R}\left( \eta _{i}\right) \right) \bigg |\mathcal{X}_{k}\right] . \end{aligned}$$

Differentiation of \( L (\widehat{\eta })\) and setting the result to 0, we get \(\displaystyle \widehat{\eta }_i=\frac{\widehat{\mathscr {T}}_{k+1}^{i}\left( X_{k+1}\right) -\delta _{i}\widehat{\mathscr {T}}_{k}^{i}\left( X_{k}\right) }{\widehat{\mathscr {O}}_{k}^{i}}. \)

Optimal estimate for \(\epsilon ^2\)

Construct a new measure \(P^{\widehat{\upsilon }^*}\) by setting

$$\begin{aligned} \left. \frac{\mathrm {d}P^{\widehat{\upsilon }^{*}}}{\mathrm {d}P^{\upsilon ^*}}\right| _{\mathcal{X}_k}={\Psi }_{k}^{\epsilon ^{2}}=\prod _{l=1}^{k}{\varphi }_{l}^{\epsilon ^{2}}, \end{aligned}$$

where \({\varphi }_{l}^{\epsilon ^{2}}=\frac{\epsilon (\mathbf {y}_{l})\text {exp}\left( -\frac{1}{2\widehat{\epsilon }^{2}(\mathbf {y}_{l})}\left( X_{l+1}-\delta \left( \mathbf {y}_{l}\right) X_{l}-\eta \left( \mathbf {y}_{l}\right) \right) ^{2}\right) }{\widehat{\epsilon }(\mathbf {y_{l}})\text {exp}\left( -\frac{1}{2\epsilon ^{2}(\mathbf {y_{l}})}\left( X_{l+1}-\delta \left( \mathbf {y}_{l}\right) X_{l}-\eta \left( \mathbf {y}_{l}\right) \right) ^{2}\right) }. \) The log likelihood of \({\Psi }_{k}^{\epsilon ^{2}}\) is calculated as

$$\begin{aligned} \log \Psi _{k}^{\epsilon ^{2}} =\sum _{l=1}^{k}\left( \log \left( \frac{1}{\widehat{\epsilon } (\mathbf {y}_{l})}\right) +\left( -\frac{1}{2\widehat{\epsilon }^{2} (\mathbf {y_{k}})}\right) \left( X_{l+1}-\delta \left( \mathbf {y}_{l}\right) X_{l} -\eta \left( \mathbf {y}_{l}\right) \right) ^{2}+\mathrm {R} \left( \epsilon (\mathbf {y}_{l})\right) \right) , \end{aligned}$$

where \(\mathrm {R}\left( \epsilon ^{2}\right) \) does not have \(\widehat{\epsilon }\). From Eqs. (20)–(21), we have the expectation of the log likelihood as a function of \(X_k\), denoted by \( L (\widehat{\epsilon }^{2})\), which is given by

$$\begin{aligned} \mathbb {E}\left[ \log \Psi _{k}^{\epsilon ^{2}}\mid \mathcal{X}_{k}\right]&=\sum _{l=1}^{k}\mathbb {E}Bigg[\sum _{i=1}^{N}\langle \mathbf {y}_{l}, \mathbf {e}_{i}\rangle \Bigg (\log \left( \frac{1}{\widehat{\epsilon }(\mathbf {y}_{l})}\right) +\left( -\frac{1}{2\widehat{\epsilon }^{2} (\mathbf {y}_{l})}\right) \left( X_{l+1}-\delta \left( \mathbf {y}_{l}\right) X_{l} -\eta \left( \mathbf {y}_{l}\right) \right) ^{2}\Bigg )Bigg]\quad +\mathrm {R}\left( \epsilon _{i}\right) . \end{aligned}$$

Differentiating \( L (\widehat{\epsilon }^{2})\) with respect to \(\widehat{\epsilon }^{2}\) and equating the result to 0 yield

$$\begin{aligned} \widehat{\epsilon }^2_{i}=\frac{\widehat{\mathscr {T}}_{k+1}^{i}\left( X_{k+1}^{2}\right) +\delta _{i}^{2}\widehat{\mathscr {T}}_{k}^{i}\left( X_{k+1}^{2}\right) +\eta _{i}^{2}\widehat{\mathscr {B}}_{k}^{i}+2\eta _{i}^{2}\delta _{i} \widehat{\mathscr {T}}_{k}^{i}\left( X_{k}\right) -2\delta _{i} \widehat{\mathscr {T}}_{k}^{i}(X_{k},X_{k+1})-2\eta _{i}\widehat{\mathscr {T}}_{k}^{i} \left( X_{k+1}\right) }{\widehat{\mathscr {O}}_{k+1}^{t}}. \end{aligned}$$

Optimal estimate for \(\pi _{ji}\)

Define the Radon–Nikodym derivative of \(P^{\widehat{\upsilon }^*}\) with respect to \(P^{{\upsilon }^*}\) as \(\left. \frac{\mathrm {d}P^{\widehat{\upsilon }^*}}{\mathrm {d}P^{\upsilon ^*}}\right| _{\mathcal{X}_k}={\Psi }_{k}^{\pi }=\prod _{l=1}^{k}{\varphi }_{l}^{\pi } \)   where \({\varphi }_{l}^{\pi }=\prod _{j,i=1}^{n}\left( \frac{\widehat{\pi }_{ji}}{\pi _{ji}}\right) ^{\langle \mathbf {y}_{l-1},\mathbf {e}_{i}\rangle \langle \mathbf {y}_{l}^{w},\mathbf {e}_{j}\rangle }. \) Taking expectation of the log likelihood in conjunction with Eq. (19), we have

$$\begin{aligned} \mathbb {E}\left[ \log {\varphi }_{l}^{\pi }\mid \mathcal{X}_{k}\right]&=\left. \mathbb {E}\left[ \sum _{l=1}^{k}\sum _{j,i=1}^{n}\log \left( \frac{\widehat{\pi }_{ji}}{\pi _{ji}}\right) ^{\langle \mathbf {y}_{l-1},\mathbf {e}_{j}\rangle \langle \mathbf {y}_{l},\mathbf {e}_{i}\rangle }\right| \mathcal{X}_{k}\right] \\&=\left. \mathbb {E}\left[ \left( \sum _{l=1}^{k}\sum _{j,i=1}^{n}\left( \log \widehat{\pi }_{ji}-\log \pi _{ji}\right) \langle \mathbf {y}_{l-1},\mathbf {e}_{j}\rangle \langle \mathbf {y}_{l},\mathbf {e}_{i}\rangle \right) \right| \mathcal{X}_{k}\right] \\&=\mathbb {E}\left[ \sum _{j,i=1}^{n}\log \widehat{\pi }_{ji}\mathscr {J}_{k}^{tsr}\right] +\mathrm {R}\left( \pi _{ji}\right) , \end{aligned}$$

where \(\mathrm {R}\left( \pi _{ji}\right) \) is free of \(\widehat{\pi }_{ji}\). Noting \(\sum _{t=1}^{n}\widehat{\pi }_{ji}=1\) and introducing a Lagrange multiplier \(\varsigma \), we maximise

$$\begin{aligned} L \left( \widehat{\pi }_{ji},\varsigma \right)&=\sum _{j,i=1}^{n}\log \widehat{\pi }_{ji}\mathscr {J}_{k}^{ji}+\varsigma \left( \sum _{j=1}^{n}\widehat{\pi }_{ji}-1\right) +\mathrm {R}\left( \pi _{ji}\right) . \end{aligned}$$

Differentiation of \( L \left( \widehat{\pi }_{ji},\varsigma \right) \) with respect to \(\widehat{\pi }_{ji}\) and \(\varsigma \) and then setting each partial derivative to 0 would result to

$$\begin{aligned} \frac{1}{\widehat{\pi }_{ji}}\widehat{\mathscr {J}}_{k}^{ji}&+\varsigma =0. \end{aligned}$$

Since \(\sum _{j=1}^{n}\widehat{\pi }_{ji}=1\) and \(\sum _{j=1}^{n}\mathscr {J}_k^{ji}=\mathscr {O}_k^{i}\), we have \(\sum _{j=1}^{n}\widehat{\pi }_{ji}=\frac{\sum _{j=1}^{n}\widehat{\mathscr {J}}_{k}^{ji}}{-\varsigma }=\frac{\mathscr {O}_{k}^{i}}{-\varsigma }=1, \) which can be re-expressed as \(\sum _{j=1}^{n}\widehat{\pi }_{ji}=\frac{\sum _{j=1}^{n}\widehat{\mathscr {J}}_{k}^{ji}}{\widehat{\mathscr {O}}{}_{k}^{i}} \) Therefore, the optimal estimate for p is given by \(\widehat{\pi }_{ji} = \frac{\widehat{\mathscr {J}}^{ji}_k}{\widehat{\mathscr {O}}^i_k}. \)

Appendix B: Proofs of Propositions

Proof of Proposition 6.1

Proof

The HDD futures price is calculated as the expected value of HDD over the contract period given the current value at time t under the risk-neutral measure Q. Based on the the Fubini–Tonelli theorem, expectation and integration can be interchanged. We firstly perform the proof for the case of \(0 \le t \le \tau _1 < \tau _2\) as follows

$$\begin{aligned} F_{H_{}}\left( t,\tau _{1},\tau _{2}\right)&=\mathbb {E}^Q\left[ {H}_{}\Big |\mathscr {F}_{t}\right] =\mathbb {E}^Q\left[ \int _{\tau _{1}}^{\tau _{2}}\text {max} \left( T_\text {base}-T_{v},\,0\right) \mathrm {d}v\Big |\mathscr {F}_{t}\right] \\&=\int _{\tau _{1}}^{\tau _{2}}\mathbb {E}^Q\left[ \text {max} \left( T_\text {base}-T_{v},\,0\right) \Big |\mathscr {F}_{t}\right] \mathrm {d}v. \end{aligned}$$

To solve for the futures price, it is necessary to compute the value of \(\mathbb {E}^Q\left[ \text {max}\left( T_\text {base}-T,\,0\right) \big |\mathscr {F}_{t}\right] \). We suppose \(\alpha (\mathbf {y}_{t})\) is deterministic so that \(\int _{s}^{t} e^{\alpha u}\mathrm {d}B_u\) in Eq. (7) is a Wiener process [cf. Benth and Šaltytė-Benth (2011)]. The integral \(\int _{s}^{t} e^{\alpha (\mathbf {y}_{u}) u}\mathrm {d}B_u^Q\) in Eq. (43), when evaluated using the optimal parameter estimates under the HMM settings, is then a Wiener process that incorporates deterministic regime-switching with filtration extended to time t. Under the normality assumption, \(T_\text {base}-T_v\) is normally distributed with \(\mathbb {E}^Q\left( \text {max}\left( T_\text {base}-T,\,0\right) \big |\mathscr {F}_{t}\right) =M\left( t,v, X_t \right) \), and \(\text {Var}\left( \text {max}\left( T_\text {base}-T,\,0\right) \big |\mathscr {F}_{t}\right) =A^{2}\left( t,v\right) \). Write \(M:=M\left( t,v, X_t \right) \) and \(A^2:=A^{2}\left( t,v\right) .\) Let \(T^*=\displaystyle \frac{T_\text {base}-T-M}{A^2}\), which has a standard normal distribution as well. Then

$$\begin{aligned} \mathbb {E}^Q\left[ \text {max}\left( T_\text {base}-T,\,0\right) \Big |\mathscr {F}_{t}\right]&=\int _{-\infty }^{T_\text {base}}-\left( T_\text {base}-T\right) \frac{1}{\sqrt{2\pi }}e^{-\frac{\left( T-\left( T_\text {base}-M\right) \right) ^{2}}{2A^{2}}}\mathrm {d}T\\&=\int _{-\frac{M}{A}}^{\infty }\left( AT^{*}+M\right) \frac{1}{\sqrt{2\pi }A}e^{-\frac{\left( T_\text {base}-AT^{*}-M-\left( T_\text {base}-M\right) \right) ^{2}}{2A^{2}}}A\mathrm {d}T^{*}\\&=\int _{-\frac{M}{A}}^{\infty }M\frac{1}{\sqrt{2\pi }}e^{-\frac{\left( T^{*}\right) ^{2}}{2}}\mathrm {d}T^{*}+\int _{-\frac{M}{A}}^{\infty }AT^{*}\frac{1}{\sqrt{2\pi }}e^{-\frac{\left( T^{*}\right) ^{2}}{2}}\mathrm {d}T^{*}\\&=M\left( 1-\int _{-\infty }^{-\frac{M}{A}}\frac{1}{\sqrt{2\pi }}e^{-\frac{\left( T^{*}\right) ^{2}}{2}}\mathrm {d}T^{*}\right) +A\left( -\frac{1}{\sqrt{2\pi }}e^{-\frac{\left( T^{*}\right) ^{2}}{2}}Bigg|_{-\frac{M}{A}}^{\infty }\right) \\&=M\left( 1-\Phi \left( -\frac{M}{A}\right) \right) +A\frac{1}{\sqrt{2\pi }}e^{-\frac{\left( \frac{M}{A}\right) ^{2}}{2}}\\&=M\Phi \left( \frac{M}{A}\right) +A\phi \left( \frac{M}{A}\right) . \end{aligned}$$

Therefore, in terms of \(M(t, v, X_t)\), \(A^2(t,v)\), and \(D\left( x\right) =x\Phi \left( x\right) +\phi \left( x\right) \), the HDD futures price for the period \(0\le t\le \tau _{1}<\tau _{2}\) is given by

$$\begin{aligned} F_{H_{f}}\left( t,\tau _{1},\tau _{2}\right)&=\int _{\tau _{1}}^{\tau _{2}}M\left( t,v,X_{t}\right) \Phi \left( \frac{M\left( t,v,X_{t}\right) }{A\left( t,v\right) }\right) +A\left( t,v\right) \phi \left( \frac{M\left( t,v,X_{t}\right) }{A\left( t,v\right) }\right) \mathrm {d}v \\&=\int _{\tau _{1}}^{\tau _{2}}A\left( t,v\right) \left( \frac{M\left( t,v,X_{t}\right) }{A\left( t,v\right) }\Phi \left( \frac{M\left( t,v,X_{t}\right) }{A\left( t,v\right) }\right) +\phi \left( \frac{M\left( t,v,X_{t}\right) }{A\left( t,v\right) }\right) \right) \mathrm {d}v\\&=\int _{\tau _{1}}^{\tau _{2}}A\left( t,v\right) D\left( \frac{M\left( t,v,X_{t}\right) }{A\left( t,v\right) }\right) \mathrm {d}v. \end{aligned}$$

For the scenario \(0\le \tau _{1}\le t<\tau _{2}\), based on the result from the first case and the fact that if Z is \(\mathscr {F}_{t}\)-measurable, \(\mathbb {E}^Q(Z \,|\mathscr {F}_{t})=Z\), we have

$$\begin{aligned} F_{H_{}}\left( t,\tau _{1},\tau _{2}\right)&=\mathbb {E}^Q\left[ \int _{\tau _{1}}^{\tau _{2}}\text {max} \left( T_\text {base}-T_{v},\,0\right) \mathrm {d}v\Big |\mathscr {F}_{t}\right] \\&=\mathbb {E}^Q\left[ \int _{\tau _{1}}^{t}\text{ max }\left( T_\text {base}-T_{v},\,0\right) \mathrm {d}v + \int _{t}^{\tau _{2}}\text {max}\left( T_\text {base}-T_{v},\,0\right) \mathrm {d}v\Big |\mathscr {F}_{t}\right] \\&=\int _{\tau _{1}}^{t}\text{ max }\left( T_\text {base}-T_{u},\,0\right) \mathrm {d}u +\mathbb {E}^Q\left[ \int _{t}^{\tau _{2}}\text {max}\left( T_\text {base}-T_{v},\,0\right) \mathrm {d}v\Big |\mathscr {F}_{t}\right] \\&=\int _{\tau _{1}}^{t}\text{ max }\left( T_\text {base}-T_{u},\,0\right) \mathrm {d}u+\int _{t}^{\tau _{2}}A\left( t,v\right) D \left( \frac{M\left( t,v,X_{t}\right) }{A\left( t,v\right) }\right) \mathrm {d}v. \end{aligned}$$

\(\square \)

Proof of Proposition 6.2

Proof

Since \(F_{H_{}}\) is independent of \(\mathscr {F}_{t}\), its conditional expectation given \(\mathscr {F}_{t}\) is equal to \(\mathbb {E}^Q[F_{H_{}}]\). Thus,

$$\begin{aligned} C_{F_{H_{}}}\left( t,K_{},\tau _{T}\right)&=\left. e^{-r\left( T-t\right) }\mathbb {E}^Q\left[ \text{ max }\left( \int _{\tau _{1}}^{\tau _{2}}A\left( t,v\right) D\left( \frac{M\left( t,v,X_{t}\right) }{A\left( t,v\right) }\right) \mathrm {d}v -K_{},\,0\right) \right| \mathscr {F}_{t}\right] \\&=\left. e^{-r\left( T-t\right) }\mathbb {E}^Q\left[ \text{ max }\left( F_{H_{}}-K_{},\,0\right) \right| \mathscr {F}_{t}\right] \\&=e^{-r\left( T-t\right) }\mathbb {E}^Q\left( \text{ max }\left( F_{H_{}}-K_{},\,0\right) \right) \\&=e^{-r\left( T-t\right) }\int _{K_{}}^{F_{m}}\left( F_{H_{}}-K_{F}\right) g\left( F_{H_{}}\right) \mathrm {d}F_{H_{}}. \end{aligned}$$

\(\square \)

Proof of Proposition 6.3

Proof

This can be straightforwardly derived from the HDD formula and employing similar arguments in the proof of Proposition 6.2. \(\square \)

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Xiong, H., Mamon, R. Putting a price tag on temperature. Comput Manag Sci 15, 259–296 (2018). https://doi.org/10.1007/s10287-017-0291-8

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