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## Carbon spot prices in equilibrium frameworks associated with climate change

 1 School of Mathematics and Statistics, Guangdong University of Foreign Studies, Guangzhou, 510006, China 2 Lingnan (University) College, Sun Yat-Sen University, Guangzhou, 510275, China 3 Guangzhou Institute of International Finance, Guangzhou University, Guangzhou, 510405, China

* Corresponding author: Zhehao Huang

Received  July 2021 Revised  October 2021 Early access December 2021

Fund Project: The work was supported by the National Natural Science Foundation (No.12101622)

At present, it is believed that the best approach to mitigate global warming is the market-based formulation of carbon emission pricing. Thus, in this paper, we work on determining the carbon spot prices in a stochastic equilibrium framework associated with climate change. Two circumstances, differentiated by whether taking carbon trading in the market, are considered. We construct optimization problems and solve them by using dynamic programming principle. The Fourier transform and its properties are fully made use of to return the explicit formulas of carbon prices. In addition, some surprising but interesting properties of the carbon prices are also found. First, the carbon prices happen jumps at the end of the abatement period. Second, the return rates of carbon prices are completely dependent on the climate elements. Finally, we present some numeric results in response to our theoretical results.

Citation: Zhenzhen Wang, Hao Dong, Zhehao Huang. Carbon spot prices in equilibrium frameworks associated with climate change. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021214
##### References:
 [1] L. M. Abadie and J. M. Chamorro, European CO$_{2}$ prices and carbon capture investments, Energy Economics, 30 (2008), 2992-3015. [2] M. E. Arouri, F. Jawadi and D. K. Nguyen, Nonlinearities in carbon spot-futures price relationships during Phase II of the EU ETS, Economic Modelling, 29 (2012), 884-892.  doi: 10.1016/j.econmod.2011.11.003. [3] G. S. Atsalakis, Using computational intelligence to forecast carbon prices, Applied Soft Computing, 43 (2016), 107-116.  doi: 10.1016/j.asoc.2016.02.029. [4] E. Benz and S. Truck, Modeling the price dynamics of CO2 emission allowances, Energy Economics, 31 (2009), 4-15. [5] M. Burke, S. M. Hsiang and E. Miguel, Global non-linear effect of temperature on economic production, Nature, 527 (2015), 235-239.  doi: 10.1038/nature15725. [6] Y. Cai and T. S. Lontzek, The social cost of carbon with economic and climate risks, Journal of Political Economy, 127 (2019), 2684-2734.  doi: 10.1086/701890. [7] R. Carmona, M. Fehr and J. Hinz, Optimal stochastic control and carbon price formation, SIAM Journal on Control and Optimization, 48 (2009), 2168-2190. [8] R. Carmona and J. Hinz, Risk-neutral models for emission allowance prices and option valuation, Management Science, 57 (2011), 1453-1468.  doi: 10.1287/mnsc.1110.1358. [9] U. Çetin and M. Verschuere, Pricing and hedging in carbon emissions markets, International Journal of Theoretical and Applied Finance, 12 (2009), 949-967.  doi: 10.1142/S0219024909005531. [10] M. Chamon and P. Mauro, Pricing growth-indexed bonds, Journal of Banking and Finance, 30 (2006), 3349-3366. [11] P. Chen, Understanding economic complexity and coherence: Market crash, excess capacity, and technology wavelets, Working Paper, 2009. [12] P. Chen, A biological perspective of macro dynamics and division of labor: Persistent cycles, disruptive technology, and the trade-off between stability and complexity, Working Paper, 2003. [13] M. Chesney and L. Taschini, The endogenous price dynamics of emission allowances and an application to CO$_{2}$ option pricing, Applied Mathematical Finance, 19 (2012), 447-475.  doi: 10.1080/1350486X.2011.639948. [14] G. Daskalakis, D. Psychoyios and R. N. Markellos, Modeling CO2 emission allowance prices and derivatives: Evidence from the European trading scheme, Journal of Banking & Finance, 33 (2009), 1230-1241.  doi: 10.1016/j.jbankfin.2009.01.001. [15] M. Dell, B. F. Jones and B. A. Olken, Temperature and income: Reconciling new cross-sectional and panel estimates, American Economic Review, 99 (2009), 198-204.  doi: 10.3386/w14680. [16] M. Dell, B. F. Jones and B. A. Olken, Temperature shocks and economic growth: Evidence from the last half century, American Economic Journal: Macroeconomics, 4 (2012), 66-95.  doi: 10.1257/mac.4.3.66. [17] T. Dietz and E. A. Rosa, Rethinking the environmental impacts of population, affluence and technology, Human Ecology Review, 1 (1994), 277-300. [18] A. Ellerman and B. K. Buchner, The european union emissions trading scheme: Origins, allocation, and early results, Review of Environmental Economics and Policy, 1 (2007), 66-87.  doi: 10.1093/reep/rem003. [19] X. Fan, S. Li and L. Tian, Chaotic characteristic identification for carbon price and an multi-layer perceptron network prediction model, Expert Systems with Applications, 42 (2015), 3945-3952.  doi: 10.1016/j.eswa.2014.12.047. [20] I. S. Farouq, N. U. Sambo, A. U. Ahmad, A. H. Jakada and I. A. Danmaraya, Does financial globalization uncertainty affect CO2 emissions? Empirical evidence from some selected SSA countries, Quantitative Finance and Economics, 5 (2021), 247-263. [21] Y. Fu and Z. Zheng, Volatility modeling and the asymmetric effect for China's carbon trading pilot market, Physica A, 542 (2020), 123401.  doi: 10.1016/j.physa.2019.123401. [22] C. García-Martos, J. Rodríguez and M. J. Sánchez, Modelling and forecasting fossil fuels, CO$_{2}$ and electricity prices and their volatilities, Applied Energy, 101 (2013), 363-375. [23] M. Golosov, J. Hassler, P. Krusell and A. Tsyvinski, Optimal taxes on fossil fuel in general equilibrium, Econometrica, 82 (2014), 41-88.  doi: 10.3982/ECTA10217. [24] H. Guo and J. Liang, An optimal control model for reducing and trading of carbon emissions, Physica A: Statistical Mechanics and its Applications, 446 (2016), 11-21.  doi: 10.1016/j.physa.2015.10.076. [25] H. Guo and J. Liang, An optimal control model of carbon reduction and trading, Mathematical Control and Related Fields, 6 (2016), 535-550.  doi: 10.3934/mcrf.2016015. [26] C. Hambel, H. Kraft and E. Schwartz, Optimal carbon abatement in a stochastic equilibrium model with climate change, European Economic Review, 132 (2021), 103642.  doi: 10.3386/w21044. [27] S. Hitzemann and M. Uhrig-Homburg, Equilibrium price dynamics of emission permits, Journal of Financial and Quantitative Analysis, 53 (2018), 1653-1678.  doi: 10.1017/S0022109018000297. [28] S. Hitzemann and M. Uhrig-Homburg, Empirical performance of reduced-formmodels for emission permit prices, Review of Derivatives Research, 22 (2019), 389-418. [29] G. Klepper and S. Peterson, Marginal abatement cost curves in general equilibrium: The influence of world energy prices, Resource and Energy Economics, 28 (2006), 1-23. [30] S. Kruse, M. Meitner and M. Schröder, On the pricing of GDP-linked financial products, Applied Financial Econonmics, 15 (2005), 1125-1133.  doi: 10.1080/09603100500359260. [31] Y. Li, Forecasting Chinese carbon emissions based on a novel time series prediction method, Energy Science & Engineering, 8 (2020), 2274-2285. [32] A. Mardani, D. Streimikiene, F. Cavallaro, N. Loganathan and M. Khoshnoudi, Carbon dioxide (CO$_{2}$) emissions and economic growth: A systematic review of two decades of research from 1995 to 2017, Science of the Total Environment, 649 (2019), 31-49. [33] I. Matei, Is financial development good for economic growth? Empirical insights from emerging European countries, Quantitative Finance and Economics, 4 (2020), 653-678.  doi: 10.3934/QFE.2020030. [34] W. D. Nordhaus, A Question of Balance: Weighing the Options on Global Warming Policies, Yale University Press, New Haven, 2008. [35] W. D. Nordhaus, Revisiting the social cost of carbon, Proceedings of the National Academy of Sciences of the United States of America, 114 (2017), 1518-1523.  doi: 10.1073/pnas.1609244114. [36] W. D. Nordhaus and P. Sztorc, DICE 2013R: Inreoduction and Users Manual, Technical Report, Yale University, 2013. [37] J. Patel, S. Shah, P. Thakkar and K. Kotecha, Predicting stockmarket index using fusion of machine learning techniques, Expert Systems with Applications, 42 (2015), 2162-2172. [38] K. Rana, S. R. Singh, N. Saxena and S. S. Sana, Growing items inventory model for carbon emission under the permissible delay in payment with partially backlogging, Green Finance, 3 (2021), 153-174.  doi: 10.3934/GF.2021009. [39] A. M. Rather, A. Agarwal and V. N. Sastry, Recurrent neural network and a hybrid model for prediction of stock returns, Expert Systems with Applications, 42 (2015), 3234-3241.  doi: 10.1016/j.eswa.2014.12.003. [40] M. E. Sanin, F. Violante and M. Mansanet-Bataller, Understanding volatility dynamics in the EU-ETS market, Energy Policy, 82 (2015), 321-331. [41] J. Seifert, M. Uhrig-Homburg and M. Wagner, Dynamic behavior of CO$_{2}$ spot prices, Journal of Environmental Economics and Management, 56 (2008), 180-194. [42] J. Solana, Climate change litigation as financial risk, Green Finance, 2 (2020), 344-372.  doi: 10.3934/GF.2020019. [43] K. Z. Tong and A. Liu, Modeling temperature and pricing weather derivatives based on subordinate Ornstein-Uhlenbeck processes, Green Finance, 2 (2020), 1-19.  doi: 10.3934/GF.2020001. [44] F. van der Ploeg and A. de Zeeuw, Pricing carbon and adjusting capital to fend off climate catastrophes, Environmental & Resource Economics, 72 (2018), 29-50. [45] L. L. Viguier, M. H. Babiker and J. M. Reilly, Carbon emissions and the Kyoto commitment in the European Union, Report No.70, MIT Joint Program on the Science and Policy of Global Change, 2001. [46] P. E. Waggoner and J. H. Ausubel, A framework for sustainability science: A renovated IPAT identity, Proceedings of the National Academy of Sciences of the United States of America, 99 (2002), 7860-7865.  doi: 10.1073/pnas.122235999. [47] J. Wang, X. Sun, Q. Cheng and and Q. Cui, An innovative random forest-based nonlinear ensemble paradigm of improved feature extraction and deep learning for carbon price forecasting, Science of the Total Environment, 762 (2021), 143099.  doi: 10.1016/j.scitotenv.2020.143099. [48] M. L. Weitzman, GHG Targets and Insurance against Catastrophic Climate Damages, Journal of Public Economic Theory, 14 (2012), 221-244.  doi: 10.3386/w16136. [49] X. Yang and J. Liang, Minimization of carbon abatement cost: modeling analysis and simulation, Discrete and Continuous Dynamical System B, 22 (2017), 2939-2969.  doi: 10.3934/dcdsb.2017158. [50] E. Zagheni and F. C. Billari, A cost valuation model based on a stochastic representation of the IPAT equation, Population & Environment, 29 (2007), 68-82.  doi: 10.1007/s11111-008-0061-1. [51] Y. Zhang and Y. M. Wei, An overview of current research on EU ETS: Evidencefrom its operating mechanism and economic effect, Applied Energy, 87 (2010), 1804-1814.  doi: 10.1016/j.apenergy.2009.12.019. [52] Z. Zheng, R. Xiao, H. Shi, G. Li and X. Zhou, Statistical regularities of Carbon emission trading market: Evidence from European Union allowances, Physica A, 426 (2015), 9-15. [53] P. Zhou, L. Zhang, D. Q. Zhou and W. J. Xia, Modeling economic performance of interprovincial CO$_{2}$ emission reduction quota trading in China, Applied Energy, 112 (2013), 1518-1528. [54] B. Zhu, X. Shi, J. Chevallier, P. Wang and Y. Wei, An adaptive multiscale ensemble learning paradigm for nonstationary and nonlinear energy price time series forecasting, Journal of Forecasting, 35 (2016), 633-651.  doi: 10.1002/for.2395.

show all references

##### References:
 [1] L. M. Abadie and J. M. Chamorro, European CO$_{2}$ prices and carbon capture investments, Energy Economics, 30 (2008), 2992-3015. [2] M. E. Arouri, F. Jawadi and D. K. Nguyen, Nonlinearities in carbon spot-futures price relationships during Phase II of the EU ETS, Economic Modelling, 29 (2012), 884-892.  doi: 10.1016/j.econmod.2011.11.003. [3] G. S. Atsalakis, Using computational intelligence to forecast carbon prices, Applied Soft Computing, 43 (2016), 107-116.  doi: 10.1016/j.asoc.2016.02.029. [4] E. Benz and S. Truck, Modeling the price dynamics of CO2 emission allowances, Energy Economics, 31 (2009), 4-15. [5] M. Burke, S. M. Hsiang and E. Miguel, Global non-linear effect of temperature on economic production, Nature, 527 (2015), 235-239.  doi: 10.1038/nature15725. [6] Y. Cai and T. S. Lontzek, The social cost of carbon with economic and climate risks, Journal of Political Economy, 127 (2019), 2684-2734.  doi: 10.1086/701890. [7] R. Carmona, M. Fehr and J. Hinz, Optimal stochastic control and carbon price formation, SIAM Journal on Control and Optimization, 48 (2009), 2168-2190. [8] R. Carmona and J. Hinz, Risk-neutral models for emission allowance prices and option valuation, Management Science, 57 (2011), 1453-1468.  doi: 10.1287/mnsc.1110.1358. [9] U. Çetin and M. Verschuere, Pricing and hedging in carbon emissions markets, International Journal of Theoretical and Applied Finance, 12 (2009), 949-967.  doi: 10.1142/S0219024909005531. [10] M. Chamon and P. Mauro, Pricing growth-indexed bonds, Journal of Banking and Finance, 30 (2006), 3349-3366. [11] P. Chen, Understanding economic complexity and coherence: Market crash, excess capacity, and technology wavelets, Working Paper, 2009. [12] P. Chen, A biological perspective of macro dynamics and division of labor: Persistent cycles, disruptive technology, and the trade-off between stability and complexity, Working Paper, 2003. [13] M. Chesney and L. Taschini, The endogenous price dynamics of emission allowances and an application to CO$_{2}$ option pricing, Applied Mathematical Finance, 19 (2012), 447-475.  doi: 10.1080/1350486X.2011.639948. [14] G. Daskalakis, D. Psychoyios and R. N. Markellos, Modeling CO2 emission allowance prices and derivatives: Evidence from the European trading scheme, Journal of Banking & Finance, 33 (2009), 1230-1241.  doi: 10.1016/j.jbankfin.2009.01.001. [15] M. Dell, B. F. Jones and B. A. Olken, Temperature and income: Reconciling new cross-sectional and panel estimates, American Economic Review, 99 (2009), 198-204.  doi: 10.3386/w14680. [16] M. Dell, B. F. Jones and B. A. Olken, Temperature shocks and economic growth: Evidence from the last half century, American Economic Journal: Macroeconomics, 4 (2012), 66-95.  doi: 10.1257/mac.4.3.66. [17] T. Dietz and E. A. Rosa, Rethinking the environmental impacts of population, affluence and technology, Human Ecology Review, 1 (1994), 277-300. [18] A. Ellerman and B. K. Buchner, The european union emissions trading scheme: Origins, allocation, and early results, Review of Environmental Economics and Policy, 1 (2007), 66-87.  doi: 10.1093/reep/rem003. [19] X. Fan, S. Li and L. Tian, Chaotic characteristic identification for carbon price and an multi-layer perceptron network prediction model, Expert Systems with Applications, 42 (2015), 3945-3952.  doi: 10.1016/j.eswa.2014.12.047. [20] I. S. Farouq, N. U. Sambo, A. U. Ahmad, A. H. Jakada and I. A. Danmaraya, Does financial globalization uncertainty affect CO2 emissions? Empirical evidence from some selected SSA countries, Quantitative Finance and Economics, 5 (2021), 247-263. [21] Y. Fu and Z. Zheng, Volatility modeling and the asymmetric effect for China's carbon trading pilot market, Physica A, 542 (2020), 123401.  doi: 10.1016/j.physa.2019.123401. [22] C. García-Martos, J. Rodríguez and M. J. Sánchez, Modelling and forecasting fossil fuels, CO$_{2}$ and electricity prices and their volatilities, Applied Energy, 101 (2013), 363-375. [23] M. Golosov, J. Hassler, P. Krusell and A. Tsyvinski, Optimal taxes on fossil fuel in general equilibrium, Econometrica, 82 (2014), 41-88.  doi: 10.3982/ECTA10217. [24] H. Guo and J. Liang, An optimal control model for reducing and trading of carbon emissions, Physica A: Statistical Mechanics and its Applications, 446 (2016), 11-21.  doi: 10.1016/j.physa.2015.10.076. [25] H. Guo and J. Liang, An optimal control model of carbon reduction and trading, Mathematical Control and Related Fields, 6 (2016), 535-550.  doi: 10.3934/mcrf.2016015. [26] C. Hambel, H. Kraft and E. Schwartz, Optimal carbon abatement in a stochastic equilibrium model with climate change, European Economic Review, 132 (2021), 103642.  doi: 10.3386/w21044. [27] S. Hitzemann and M. Uhrig-Homburg, Equilibrium price dynamics of emission permits, Journal of Financial and Quantitative Analysis, 53 (2018), 1653-1678.  doi: 10.1017/S0022109018000297. [28] S. Hitzemann and M. Uhrig-Homburg, Empirical performance of reduced-formmodels for emission permit prices, Review of Derivatives Research, 22 (2019), 389-418. [29] G. Klepper and S. Peterson, Marginal abatement cost curves in general equilibrium: The influence of world energy prices, Resource and Energy Economics, 28 (2006), 1-23. [30] S. Kruse, M. Meitner and M. Schröder, On the pricing of GDP-linked financial products, Applied Financial Econonmics, 15 (2005), 1125-1133.  doi: 10.1080/09603100500359260. [31] Y. Li, Forecasting Chinese carbon emissions based on a novel time series prediction method, Energy Science & Engineering, 8 (2020), 2274-2285. [32] A. Mardani, D. Streimikiene, F. Cavallaro, N. Loganathan and M. Khoshnoudi, Carbon dioxide (CO$_{2}$) emissions and economic growth: A systematic review of two decades of research from 1995 to 2017, Science of the Total Environment, 649 (2019), 31-49. [33] I. Matei, Is financial development good for economic growth? Empirical insights from emerging European countries, Quantitative Finance and Economics, 4 (2020), 653-678.  doi: 10.3934/QFE.2020030. [34] W. D. Nordhaus, A Question of Balance: Weighing the Options on Global Warming Policies, Yale University Press, New Haven, 2008. [35] W. D. Nordhaus, Revisiting the social cost of carbon, Proceedings of the National Academy of Sciences of the United States of America, 114 (2017), 1518-1523.  doi: 10.1073/pnas.1609244114. [36] W. D. Nordhaus and P. Sztorc, DICE 2013R: Inreoduction and Users Manual, Technical Report, Yale University, 2013. [37] J. Patel, S. Shah, P. Thakkar and K. Kotecha, Predicting stockmarket index using fusion of machine learning techniques, Expert Systems with Applications, 42 (2015), 2162-2172. [38] K. Rana, S. R. Singh, N. Saxena and S. S. Sana, Growing items inventory model for carbon emission under the permissible delay in payment with partially backlogging, Green Finance, 3 (2021), 153-174.  doi: 10.3934/GF.2021009. [39] A. M. Rather, A. Agarwal and V. N. Sastry, Recurrent neural network and a hybrid model for prediction of stock returns, Expert Systems with Applications, 42 (2015), 3234-3241.  doi: 10.1016/j.eswa.2014.12.003. [40] M. E. Sanin, F. Violante and M. Mansanet-Bataller, Understanding volatility dynamics in the EU-ETS market, Energy Policy, 82 (2015), 321-331. [41] J. Seifert, M. Uhrig-Homburg and M. Wagner, Dynamic behavior of CO$_{2}$ spot prices, Journal of Environmental Economics and Management, 56 (2008), 180-194. [42] J. Solana, Climate change litigation as financial risk, Green Finance, 2 (2020), 344-372.  doi: 10.3934/GF.2020019. [43] K. Z. Tong and A. Liu, Modeling temperature and pricing weather derivatives based on subordinate Ornstein-Uhlenbeck processes, Green Finance, 2 (2020), 1-19.  doi: 10.3934/GF.2020001. [44] F. van der Ploeg and A. de Zeeuw, Pricing carbon and adjusting capital to fend off climate catastrophes, Environmental & Resource Economics, 72 (2018), 29-50. [45] L. L. Viguier, M. H. Babiker and J. M. Reilly, Carbon emissions and the Kyoto commitment in the European Union, Report No.70, MIT Joint Program on the Science and Policy of Global Change, 2001. [46] P. E. Waggoner and J. H. Ausubel, A framework for sustainability science: A renovated IPAT identity, Proceedings of the National Academy of Sciences of the United States of America, 99 (2002), 7860-7865.  doi: 10.1073/pnas.122235999. [47] J. Wang, X. Sun, Q. Cheng and and Q. Cui, An innovative random forest-based nonlinear ensemble paradigm of improved feature extraction and deep learning for carbon price forecasting, Science of the Total Environment, 762 (2021), 143099.  doi: 10.1016/j.scitotenv.2020.143099. [48] M. L. Weitzman, GHG Targets and Insurance against Catastrophic Climate Damages, Journal of Public Economic Theory, 14 (2012), 221-244.  doi: 10.3386/w16136. [49] X. Yang and J. Liang, Minimization of carbon abatement cost: modeling analysis and simulation, Discrete and Continuous Dynamical System B, 22 (2017), 2939-2969.  doi: 10.3934/dcdsb.2017158. [50] E. Zagheni and F. C. Billari, A cost valuation model based on a stochastic representation of the IPAT equation, Population & Environment, 29 (2007), 68-82.  doi: 10.1007/s11111-008-0061-1. [51] Y. Zhang and Y. M. Wei, An overview of current research on EU ETS: Evidencefrom its operating mechanism and economic effect, Applied Energy, 87 (2010), 1804-1814.  doi: 10.1016/j.apenergy.2009.12.019. [52] Z. Zheng, R. Xiao, H. Shi, G. Li and X. Zhou, Statistical regularities of Carbon emission trading market: Evidence from European Union allowances, Physica A, 426 (2015), 9-15. [53] P. Zhou, L. Zhang, D. Q. Zhou and W. J. Xia, Modeling economic performance of interprovincial CO$_{2}$ emission reduction quota trading in China, Applied Energy, 112 (2013), 1518-1528. [54] B. Zhu, X. Shi, J. Chevallier, P. Wang and Y. Wei, An adaptive multiscale ensemble learning paradigm for nonstationary and nonlinear energy price time series forecasting, Journal of Forecasting, 35 (2016), 633-651.  doi: 10.1002/for.2395.
Carbon price surface in the circumstance without considering carbon trading
Carbon price curves with respect to different carbon emissions in the circumstance without considering carbon trading
Carbon price surfaces in the circumstance with considering carbon trading with respect to different carbon market prices
Carbon price surfaces in the circumstance with considering carbon trading with respect to different carbon emissions
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