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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References
Alaton P, Djehiche B, Stillberger D (2002) On modelling and pricing weather derivatives. Appl Math Finance 9(1):1–20
Alexandridis A, Zapranis A (2015) Weather derivatives: modeling and pricing weather-related risk. Springer, New York
Bellini F (2005) The weather derivatives market: modelling and pricing temperature. PhD thesis, University of Lugano, Switzerland
Benth F, Šaltytė-Benth J (2005) Stochastic modelling of temperature variations with a view towards weather derivatives. Appl Math Finance 12(1):53–85
Benth F, Šaltytė-Benth J (2011) Weather derivatives and stochastic modelling of temperature. Int J Stoch Anal. doi:10.1155/2011/576791
Benth F, Hardle W, Cabrera B (2011) Pricing of asian temperature risk, in anonymous Berlin. Springer, Heidelberg, pp 163–199
Cox DR, Miller HD (1965) The theory of stochastic processes. Methuen, London
Campbell S, Diebold F (2005) Weather forecasting for weather derivatives. J Am Stat Assoc 100(469):6–16
Cao M, Wei J (2004) Weather derivatives valuation and market price of weather risk. J Futures Mark 24(11):1065–1089
Considine G (2016) Introduction to weather derivatives. CME Group Web. http://web.math.pmf.unizg.hr/~amimica/pub/lapl.pdf
Date P, Mamon R, Tenyakov A (2013) Filtering and forecasting commodity futures prices under an HMM framework. Energy Econ 40:1001–1013
Davis M (2001) Pricing weather derivatives by marginal value. Quant Finance 1(3):305–308
Dischel B (1998) At least: a model for weather risk. Energy power risk management, March issue, weather risk special report, p 3032
Dorfleitner G, Wimmer M (2010) The pricing of temperature futures at the Chicago Mercantile Exchange. J Bank Finance 34(6):1360–1370
Dutton J (2002) Opportunities and priorities in a new era for weather and climate services. Bull Am Meteorol Soc 83(9):1303–1313
Elias R, Wahab M, Fang L (2014) A comparison of regime-switching temperature modeling approaches for applications in weather derivatives. Eur J Oper Res 232(3):549–560
Elliott R (1994) Exact adaptive filters for Markov chains observed in Gaussian noise. Automatica 30:1399–1408
Elliott R, Krishnamurthy V (1999) New finite-dimensional filters for parameter estimation of discrete-time linear Gaussian models. IEEE Trans Autom Control 44(5):938–951
Elliott RJ, Aggoun L, Moore J (1995) Hidden Markov models: estimation and control. Springer, New York
Elliott R, Siu T, Badescu A (2011) Bond valuation under a discrete-time regime switching term-structure model and its continuous-time extension. Manag Finance 37(11):1025–1047
Erlwein C, Mamon R (2009) An online estimation scheme for a HullWhite model with HMM-driven parameters. Stat Methods Appl 18(1):87–107
Erlwein C, Benth F, Mamon R (2010) HMM filtering and parameter estimation of an electricity spot price model. Energy Econ 32(5):1034–1043
Erlwein C, Mamon R, Davison M (2011) An examination of HMM-based investment strategies for asset allocation. Appl Stoch Models Bus Ind 27(3):204–221
Gao H, Mamon R, Liu X (2017) Risk measurement of a guaranteed annuity option under a stochastic modelling framework. Math Comput Simul 132:100–119
Geman H, Leonardi M (2005) Alternative approaches to weather derivatives pricing. Manag Finance 31(6):46–72
Grimmett G, Stirzaker D (2001) Probability and random processes. Oxford University Press, Oxford
Jewson S (2004) Introduction to weather derivative pricing. J Altern Invest 7(2):57–64
Jewson S, Brix A, Ziehmann C (2005) Weather derivative valuation: the meteorological, financial and mathematical foundations. Cambridge University Press, Cambridge
Moreno M (2000) Riding the temp. Weather derivatives, FOW Special Supplement, December
Richards T, Manfredo M, Sanders D (2004) Pricing weather derivatives. Am J Agric Econ 86(4):1005–1017
Siu T, Erlwein C, Mamon R (2008) The pricing of credit default swaps under a Markov-modulated Merton’s structural model. N Am Actuar J 12(1):19–46
van der Vaart A (1998) Asymptotic statistics. Cambridge University Press, Cambridge
Weron R (2006) Modeling and forecasting electricity loads and prices: a statistical approach. Wiley, Hoboken
Xi X, Mamon R (2011) Parameter estimation of an asset price model driven by a weak hidden Markov chain. Econ Model 28(1):36–46
Xiong H, Mamon R (2016) A self-updating model driven by a higher-order hidden Markov chain for temperature dynamics. J Comput Sci 17:47–61
Xu W, Odening M, Musshoff O (2008) Indifference pricing of weather derivatives. Am J Agric Econ 90(4):979–993
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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Appendices
Appendix A: Derivations of the model’s optimal parameter estimates
Optimal estimate for \(\delta \)
Define a new measure \(P^{\widehat{\upsilon }^*}\) via
where
Therefore, the log likelihood for \(\tilde{\Psi }^\delta _k\) is
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
We differentiate \( L (\widehat{\delta }_i)\) with respect to \(\widehat{\delta }_i\) and set the result to zero giving
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
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
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
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
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
Differentiating \( L (\widehat{\epsilon }^{2})\) with respect to \(\widehat{\epsilon }^{2}\) and equating the result to 0 yield
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
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
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
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
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
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
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
\(\square \)
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,
\(\square \)
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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DOI: https://doi.org/10.1007/s10287-017-0291-8