# On the Stochastic Properties of Carbon Futures Prices

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## Abstract

Pricing carbon is a central concern in environmental economics, due to the worldwide importance of emissions trading schemes to regulate pollution. This paper documents the presence of small and large jumps in the stochastic process of the CO\(_2\) futures price. The large jumps have a discrete origin, i.e. they can arise from various demand factors or institutional decisions on the tradable permits market. Contrary to the existing literature, we show that the stochastic process of carbon futures prices does not contain a continuous component (Brownian motion). The results are derived by using high-frequency data in the activity signature function framework (Todorov and Tauchen in J Econom 154:125–138, 2010; Todorov and Tauchen in J Bus Econ Stat 29:356–371, 2011). The implication is that the carbon futures price should be modeled as an appropriately sampled, centered Lévy or Poisson process. The pure-jump behavior of the carbon price might be explained by the lower volume of trades on this allowance market (compared to other highly liquid financial markets).

## Keywords

Carbon price Stochastic modeling Activity signature function## JEL Classification

C14 C32 G1 Q4## Notes

### Acknowledgments

We wish to thank the Editor Michael Finus as well as two anonymous referees for their advice, which led to a greatly improved version of the paper. Helpful comments were received from seminar participants at the 10th Envecon Applied Environmental Economics Conference in London (UKNEE). We wish to thank ECX for providing the data. The usual disclaimer applies.

## References

- Aït-Sahalia Y, Jacod J (2009a) Testing for jumps in a discretely observed process. Ann Stat 37:184–222CrossRefGoogle Scholar
- Aït-Sahalia Y, Jacod J (2009b) Estimating the degree of activity of jumps in high frequency financial data. Ann Stat 37:2202–2244CrossRefGoogle Scholar
- Aït-Sahalia Y, Jacod J (2010) Is Brownian motion necessary to model high-frequency data? Ann Stat 38: 3093–3128Google Scholar
- Ballotta L (2005) A Lévy process-based framework for the fair valuation of participating life insurance contracts. Insur Math Econ 37:173–196CrossRefGoogle Scholar
- Barndorff-Nielsen O, Shephard N (2001) Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J R Stat Soc Ser B 63:167–241CrossRefGoogle Scholar
- Barndorff-Nielsen O, Shephard N (2004) Power and bipower variation with stochastic volatility and jumps. J Financial Econ 2:1–37Google Scholar
- Barndorff-Nielsen O, Shephard N (2006) Econometrics of testing for jumps in financial economics using bipower variation. J Financ Econ 4:1–30Google Scholar
- Barndorff-Nielsen O, Shephard N (2012) Lévy driven volatility models. Oxford University Press (forthcoming)Google Scholar
- Barndorff-Nielsen OE, Hansen PR, Lunde A, Shephard N (2009) Realized kernels in practice: trades and quotes. Econom J 12:C1–C32CrossRefGoogle Scholar
- Benz E, Trück S (2009) Modeling the price dynamics of \(\text{ CO }_2\) emission allowances. Energy Econ 31:4–15CrossRefGoogle Scholar
- Blumenthal R, Getoor R (1961) Sample functions of stochastic processes with independent increments. J Math Mech 10:493–516Google Scholar
- Borovkov K, Decrouez G, Hinz J (2011) Jump-diffusion modeling in emission markets. Stoch Model 27:50–76CrossRefGoogle Scholar
- Carmona R, Hinz J (2011) Risk-neutral models for emission allowance prices and option valuation. Manag Sci 57:1453–1468CrossRefGoogle Scholar
- Carmona R, Fehr M, Hinz J (2009) Optimal stochastic control and carbon price formation. SIAM J Control Optim 48:2168–2190CrossRefGoogle Scholar
- Carmona R, Fehr M, Hinz J, Porchet A (2010) Market designs for emissions trading schemes. SIAM Rev 52:403–452CrossRefGoogle Scholar
- Carr P, Madan DB (1999) Option valuation using the fast fourier transforms. J Comput Financ 2:61–73Google Scholar
- Carr P, Geman H, Madan DB, Yor M (2003) Stochastic volatility for Lévy processes. Math Financ 13:345–382CrossRefGoogle Scholar
- Çetin U, Verschuere M (2009) Pricing and hedging in carbon emissions markets. Int J Theor Appl Financ 12:949–967CrossRefGoogle Scholar
- Chesney M, Taschini L (2012) The endogenous price dynamics of emission allowance and an application to CO\(_2\) option pricing. Appl Math Financ, (forthcoming) doi: 10.1080/1350486X.2011.639948
- Chevallier J (2011) A model of carbon price interactions with macroeconomic and energy dynamics. Energy Econ 33:1295–1312CrossRefGoogle Scholar
- Chevallier J, Sévi B (2010) Jump-robust estimation of realized volatility in the EU emissions trading scheme. J Energy Mark 3:49–67Google Scholar
- Chevallier J, Sévi B (2011) On the realized volatility of the ECX CO\(_2\) emissions 2008 futures contract: distribution, dynamics, and forecasting. Ann Financ 7:1–29CrossRefGoogle Scholar
- Cont R, Tankov P (2004) Financial modelling with jump processes. Chapman & Hall/CRC, Boca RatonGoogle Scholar
- Cronshaw MB, Kruse JB (1996) Regulated firms in pollution permit markets with banking. J Regul Econ 9:179–189CrossRefGoogle Scholar
- Cvitanić J, Polimenis V, Zapatero F (2008) Optimal portfolio allocation with higher moments. Ann Financ 4:1–28CrossRefGoogle Scholar
- Dales JH (1968) Pollution, property and prices. Toronto University Press, CanadaGoogle Scholar
- Dannenberg H, Ehrenfeld W (2011) A model for the valuation of carbon price risk. In: Antes R, Hansjürgens B, Letmathe P, Pickl S (eds) Emissions trading: institutional design, decision making and corporate strategies. Springer, GermanyGoogle Scholar
- Daskalakis G, Psychoyios D, Markellos RN (2009) Modeling CO\(_2\) emission allowance prices and derivatives: evidence from the European trading scheme. J Banking Financ 33:1230–1241CrossRefGoogle Scholar
- Dixit AK, Pindyck RS (1994) Investment under uncertainty. Princeton University Press, PrincetonGoogle Scholar
- Ellerman AD (2005) A note on tradeable permits. Environ Resour Econ 31:123–131CrossRefGoogle Scholar
- Ellerman AD, Buchner BK (2008) Over-allocation or abatement? A preliminary analysis of the EU ETS based on the 2005–06 emissions data. Environ Resour Econ 41:267–287CrossRefGoogle Scholar
- Ellerman AD, Joskow PL, Schmalensee R, Montero JP, Bailey E (2000) Markets for clean air: the US acid rain program, 2nd edn. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Feldman RM, Valdez-Flores C (2010) Applied probability and stochastic processes. Springer, HeidelbergCrossRefGoogle Scholar
- Ghahramani S (2005) Fundamentals of probability with stochastic processes. Pearson Education, Prentice Hall, New JerseyGoogle Scholar
- Gillespie DT (1996) Exact numerical simulation of the Ornstein–Uhlenbeck process and its integral. Phys Rev E 54:2084–2091CrossRefGoogle Scholar
- Hanson FB (2007) Applied stochastic processes and control for jump-diffusions: modeling, analysis and computation. SIAM Press, Society for Industrial and Applied Mathematics, PhiladelphiaCrossRefGoogle Scholar
- Hanson FB, Westman JJ (2002) Portfolio optimization with jump-diffusions: estimation of time-dependent parameters and application. In: Proceedings of the 41st IEEE conference on decision and control, pp 377–381Google Scholar
- Hinz J, Novikov A (2010) On fair pricing of emission-related derivatives. Bernoulli 16(4):1240–1261CrossRefGoogle Scholar
- Jing B-Y, Kong X-B, Liu Z (2011) Estimating the jump activity index under noisy observations using high-frequency data. J Am Stat Assoc 106:558–568CrossRefGoogle Scholar
- Jing B-Y, Kong X-B, Liu Z (2012) Modeling high-frequency financial data by pure jump processes. Ann Stat 40:759–784CrossRefGoogle Scholar
- Kannan D (1979) An introduction to stochastic processes. North Holland Series in Probability and Applied Mathematics, New YorkGoogle Scholar
- Karatzas I, Shreve SE (1997) Brownian motion and stochastic calculus, 2nd edn. Springer, New YorkGoogle Scholar
- Kassberg S, Kiesel R, Liebmann T (2008) Fair valuation of insurance contracts under Lévy process specifications. Insurance Math Econ 42:419–433CrossRefGoogle Scholar
- Kling C, Rubin J (1997) Bankable permits for the control of environmental pollution. J Public Econ 64:101–115CrossRefGoogle Scholar
- Knill O (2009) Probability and stochastic processes with applications. Overseas India Press, New DelhiGoogle Scholar
- Kolb RW, Overdahl JA (2006) Understanding futures markets. Blackwell Publishing, New YorkGoogle Scholar
- Kong X-B (2012) Is a pure jump process fitting the high frequency data better than a jump-diffusion process? J Stat Plan Inference, (forthcoming)Google Scholar
- Kroese DP, Taimre T, Botev ZI (2011) Handbook of Monte Carlo methods. Wiley Series in Probabilities and Statistics, ChichesterCrossRefGoogle Scholar
- Leiby P, Rubin J (2001) Intertemporal permit trading for the control of greenhouse gas emissions. Environ Resour Econ 19:229–256CrossRefGoogle Scholar
- Lin YN, Lin AY (2007) Pricing the cost of carbon dioxide emission allowance futures. Rev Futur Mark 16:1–16Google Scholar
- Liu J, Longstaff F, Pan J (2003) Dynamic asset allocation with event risk. J Financ 58:231–259CrossRefGoogle Scholar
- Maeda A (2004) Impact of banking and forward contracts on tradable permit markets. Environ Econ Policy Stud 6:81–102Google Scholar
- Martzoukos SH, Trigeorgis L (2002) Real (investment) options with multiple sources of rare events. Eur J Oper Res 136:696–706CrossRefGoogle Scholar
- Montgomery DW (1972) Markets in licenses and efficient pollution control programs. J Econ Theory 5: 395–418Google Scholar
- Pakkanen MS (2010) Microfoundations for diffusion price processes. Math Financ Econ 3:89–114CrossRefGoogle Scholar
- Paolella MS, Taschini L (2008) An econometric analysis of emission allowance prices. J Banking Financ 32:2022–2032CrossRefGoogle Scholar
- Rolski T, Schmidli H, Schmidt V, Teugels J (1999) Stochastic processes for insurance and finance. Wiley Series in Probabilities and Statistics, ChichesterCrossRefGoogle Scholar
- Rubin J (1996) A model of intertemporal emission trading, banking, and borrowing. J Environ Econ Manag 31:269–286CrossRefGoogle Scholar
- Rydberg TH (1997) The normal inverse Gaussian Lévy process: simulation and approximation. Commun Stat Stoch Model 13:887–910CrossRefGoogle Scholar
- Schennach SM (2000) The economics of pollution permit banking in the context of title IV of the 1990 clean air act amendments. J Environ Econ Manag 40:189–210CrossRefGoogle Scholar
- Schoutens W (2003) Lévy processes in finance. Wiley Series in Probability and Statistics, New YorkCrossRefGoogle Scholar
- Schwartz ES (1997) The stochastic behavior of commodity prices: implications for valuation and hedging. J Financ 52:923–973Google Scholar
- Schwartz ES, Smith JE (2000) Short-term variations and long-term dynamics in commodity prices. Manag Sci 46:893–911CrossRefGoogle Scholar
- Seifert J, Uhrig-Homburg M, Wagner M (2008) Dynamic behavior of \(\text{ CO }_2\) prices. J Environ Econ Manag 56:180–194CrossRefGoogle Scholar
- Speyer JL, Chung WH (2008) Stochastic processes, estimation, and control. SIAM Press, Society for Industrial and Applied Mathematics, PhiladelphiaCrossRefGoogle Scholar
- Stevens B, Rose A (2002) A dynamic analysis of the marketable permits approach to global warming policy: a comparison of spatial and temporal flexibility. J Environ Econ Manag 44:45–69CrossRefGoogle Scholar
- Stirzaker D (2005) Stochastic processes and models. Oxford University Press, OxfordGoogle Scholar
- Todorov V, Tauchen G (2010) Activity signature functions for high-frequency data analysis. J Econom 154:125–138CrossRefGoogle Scholar
- Todorov V, Tauchen G (2011) Volatility jumps. J Bus Econ Stat 29:356–371CrossRefGoogle Scholar
- Zhang L, Mykland PA, Aït-Sahalia Y (2005) A tale of two time scales: determining integrated volatility with noisy high frequency data. J Am Stat Assoc 100:1394–1411Google Scholar
- Zhu Z, Graham P, Reedman L, Lo T (2009) A scenario-based integrated approach for modeling carbon price risk. Decis Econ Financ 32:35–48CrossRefGoogle Scholar