# Modelling of Fuel- and Energy-Switching Prices by Mean-Reverting Processes and Their Applications to Alberta Energy Markets

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Overview

#### 1.2. Literature Review

## 2. Economic Background and the Data

#### 2.1. Economic Content

#### 2.2. The Fuel-Switching Data

## 3. Parameter Estimation

#### 3.1. Inhomogeneous Geometric Brownian Motion

#### 3.2. OU Process and Lévy-Driven OU Process

#### 3.2.1. Parameter Estimation

#### 3.2.2. Parameters Initialization

#### 3.3. Regime-Switching Models

#### 3.3.1. Parameter Estimation of RSOU Models Using EM Algorithm

#### 3.3.2. Parameter Estimation of the NIG Part of RSLDOU Model Using MLE Algorithm

## 4. Empirical Results

#### 4.1. Estimation Results for Fuel-Switching Data

#### 4.1.1. Estimation of IGBM

#### 4.1.2. Estimation of OU and LDOU Processes

#### 4.1.3. Estimation of RSOU and RSLDOU Processes

#### 4.2. Estimation Results for Energy-Switching Data

#### 4.2.1. Estimation of IGBM

#### 4.2.2. Estimation of OU and LDOU Processes

#### 4.2.3. Estimation of RSOU and RSLDOU Processes

## 5. Models Comparison

#### 5.1. AIC and BIC

#### 5.2. Cross Validation (CV) for Time Series Data: The h-Block Cross Validation

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

IGBM | Inhomogeneous Geometric Brownian Motion |

OU | Ornstein–Uhlenbeck |

LDOU | Levy-driven Ornstein–Uhlenbeck |

RSOU | Regime-switching Ornstein–Uhlenbeck |

RSLDOU | Regime-swithching Levy-driven Ornstein–Uhlenbeck |

NIG | Normal Inverse Gaussian |

AIC | Akaike Information Criteria |

BIC | Beysian Information Criteria |

GHG | Green House Gas |

EU-ETS | EU Emissions Trading System |

BCA | Border Carbon Adijustment |

EITE | Emissions-Intensive and Trade-Exposed |

UNFCC | United Nations Framework Convention108on Climate Change |

## Appendix A. Mean-Reverting Stochastic Models

#### Appendix A.1. Inhomogeneous Geometric Brownian Motion (IGBM) and Change of Time Method

**Definition**

**A1.**

**Theorem**

**A1.**

#### Appendix A.2. Lévy Process and Normal Inverse Gaussian Distribution

**Definition**

**A2.**

- ${L}_{0}=0$;
- for any three times $0\le s<t<u$, the increments of this process ${L}_{t}-{L}_{s}$ and ${L}_{u}-{L}_{t}$ are independent;
- for any two times $0\le s<t$, ${L}_{t}-{L}_{s}$ have equal distribution to ${L}_{t-s}$.
- The sample path of ${L}_{t}$ should have right continuous and admit left limits, i.e., a CadLag path.

**Definition**

**A3.**

#### Appendix A.3. Ornstein–Uhlenbeck (OU) Process and Lévy-Driven Ornstein–Uhlenbeck (LDOU) Process

**Definition**

**A3.**

**Definition**

**A4.**

#### Appendix A.4. Regime-Switching Ornstein–Uhlenbeck (RSOU) Process and Regime-Switching Lévy-Driven Ornstein–Uhlenbeck (RSLDOU) Process

**Definition**

**A6.**

## References

- Kaufmann, B. “You Probably Won’t Notice it,” Economist Says of 50% Hike in Alberta Carbon Tax. Interview Given to the Calgary Herarld and Bill Kaufmann, 28 December 2017. [Google Scholar]
- Ambasta, A.; Buonocore, J. Carbon pricing: A win-win environmental and public health policy. Can. J. Public Health
**2018**, 109, 779–781. [Google Scholar] [CrossRef] - Sijm, J.P.M.; Bakker, S.J.A.; Chen, Y.; Harmsen, H.W.; Lise, W. CO
_{2}price dynamics: The implications of EU emissions trading for electricity prices & operations. In Proceedings of the 2006 IEEE Power Engineering Society General Meeting, Montreal, QC, Canada, 18–22 June 2006; p. 4. [Google Scholar] [CrossRef] - Seifert, J.; Uhrig-Homburg, M.; Wagner, M. Dynamic behavior of CO
_{2}prices. J. Environ. Econ. Manag.**2008**, 56, 180194. [Google Scholar] [CrossRef] - Carmona, R.; Fehr, M.; Hinz, J. Optimal stochastic control and carbon price formation. SIAM J. Control. Optim.
**2009**, 48, 21682190. [Google Scholar] [CrossRef] - Arora, S.; Taylor, J.W. Orecasting electricity smart meter data using conditional kernel density estimation. Omega
**2016**, 59, 47–59. [Google Scholar] [CrossRef] [Green Version] - Erik, D.; William, D. Greenhouse Gas Emission Reduction By Means Of Fuel Switching In Electricity Generation: Addressing The Potentials. Energy Convers. Manag.
**2008**, 49, 843–853. [Google Scholar] - Arrigoni, A.; Lu, W.; Swishchuk, A.V.; Goutte, S. Energy-Switching Using Lévy Processes—An Application to Canadian and North American Data (Working Paper). 2019. Available online: https://ssrn.com/abstract=3408174 (accessed on 24 October 2019).
- Lu, W. Applications of Mean Reverting Process in Alberta Market. Master’s Thesis, University of Calgary, Calgary, AB, Canada, 2020. [Google Scholar]
- Swishchuk, A. Explicit Option Pricing Formula for a Mean-reverting Asset in Energy Market. J. Numer. Appl. Math.
**2008**, 1, 216–233. [Google Scholar] - Yang, J.W.; Tsai, S.Y.; Shyu, S.D.; Chang, C.C. Pairs trading: The performance of a stochastic spread model with regime switching-evidence from the S&P 500. Int. Rev. Econ. Financ.
**2016**, 43, 139–150. [Google Scholar] - Bai, Y.; Wu, L. Analytic value function for optimal regime-switching pairs trading rules. Quant. Financ.
**2018**, 18, 637–654. [Google Scholar] [CrossRef] - Stübinger, J.; Endres, S. Pairs trading with a mean-reverting jump-diffusion model on high-frequency data. Quant. Financ.
**2018**, 18, 1735–1751. [Google Scholar] [CrossRef] [Green Version] - Swishchuk, A.; Roldan-Contreras, A.; Soufiani, E.; Martinez, G.; Seifi, M.; Agrawal, N.; Yao, Y. Practical option valuation of futures contracts with negative underlying prices. arXiv
**2020**, arXiv:2009.12350v1. [Google Scholar] - Chevalier, J.; Goutte, S. Estimation of Lévy-driven Ornstein–Uhlenbeck processes: Application to modeling of CO
_{2}and fuel-switching. Ann. Oper. Res.**2015**, 255, 169–197. [Google Scholar] [CrossRef] - Arrigoni, A.; Lu, W. Energy-Switching Using Lévy Processes—An Application to Canadian and North American Data. In Proceedings of the 42nd International Association for Energy Economics (IAEE) Annual Conference in Montréal, Montreal, QC, Canada, 29 May–1 June 2019. [Google Scholar]
- Fragros, P.; Fragkiadakis, K.; Paroussos, L. Reducing the decarbonisation cost burden for EU energy-intensive industries. Energies
**2021**, 14, 236. [Google Scholar] [CrossRef] - Montenegro, R.C.; Fragkos, P.; Dobbins, A.H.; Schmid, D.; Pye, S.; Fah, U. Beyond the Energy System: Modeling FrameworksDepicting Distributional Impacts for Interdisciplinary PolicyAnalysis. Energy Technol.
**2021**, 9, 2000668. [Google Scholar] [CrossRef] - Georgescu-Roegen, N. The Entropy Law and the Economic Process; Harvard University Press: Cambridge, MA, USA, 1971. [Google Scholar]
- Ellerman, A.D.; Buchner, B.K. The European Union emissions trading scheme: Origins, allocation, and early results. Rev. Environ. Econ. Policy
**2007**, 1, 66–87. [Google Scholar] [CrossRef] - Barndorff-Nielsen, O.E.; Halgreen, C. Infinite divisibility of the hyperbolisch and generalized inverse Gaussian distributions. Z. Wahrscheinlichkeitstheorie Verwandte Geb.
**1977**, 38, 309–312. [Google Scholar] [CrossRef] - Jordan, S. Inverse Gaussian Distribution and the Moment Problem. J. Appl. Statist. Sci.
**1999**, 9, 61–71. [Google Scholar] - Chevallier, J.; Goutte, S. On the estimation of regime-switching Lévy models. Stud. Nonlinear Dyn. E
**2017**, 21, 3–29. [Google Scholar] [CrossRef] [Green Version] - Lachenbruch, P.A.; Mickey, M.R. Estimation of Error Rates in Discriminant Analysis. Technometrics
**1968**, 10, 1–11. [Google Scholar] [CrossRef] - Burman, P.; Chow, E.; Nolan, D. A Cross-validatory method for dependent data. Biometrika
**1994**, 81, 351–358. [Google Scholar] [CrossRef] - Zachmann, G. A stochastic fuel switching model for electricity prices. Energy Econ.
**2013**, 35, 5–13. [Google Scholar] [CrossRef] - Ikeda, N.; Watanabe, S. Stochastic Dierential Equations and Diusion Processes; Kodansha Ltd.: Tokyo, Japan, 1981. [Google Scholar]
- Elliott, R. Stochastic Calculus and Applications; Springer: New York, NY, USA, 1982. [Google Scholar]
- Barndorff-Nielsen, O.E.; Shephard, N. Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in fnancial economics. J. R. Stat. Soc. Ser. B
**2001**, 63, 167–241. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) The fuel cost and (

**b**) fuel-switching data in Arrigoni et al. (2019), expressed in CAD/MWh.

Mean | Standard Deviation | Kurtosis | Skewness | Min | Max | |
---|---|---|---|---|---|---|

fuel-switching | 2.553058 | 3.976492 | −0.5539386 | 0.04961635 | −7.15667 | 14.88465 |

energy-switching | 61.08916 | 15.98072 | 0.6200243 | −0.7082977 | 15.60934 | 95.58245 |

a | L | $\mathit{\sigma}$ |
---|---|---|

$\times {2.514925}^{-12}$ | −2.212315 | 0.02235206 |

$\mathit{\kappa}$ | $\mathit{\theta}$ | $\mathit{\sigma}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\delta}$ | $\mathit{\mu}$ |
---|---|---|---|---|---|---|

0.008600791 | 1.10466 | 0.4005829 | 0.2418017 | −0.05377233 | 0.2637261 | 0.06009332 |

${\mathit{\kappa}}_{1}$ | ${\mathit{\kappa}}_{2}$ | ${\mathit{\theta}}_{1}$ | ${\mathit{\theta}}_{2}$ | ${\mathit{\sigma}}_{1}^{2}$ | ${\mathit{\sigma}}_{2}^{2}$ | ${\mathit{\pi}}_{11}$ | ${\mathit{\pi}}_{22}$ |
---|---|---|---|---|---|---|---|

0.00310 | 0.03306 | 4.9722 | −1.7579 | 0.01146 | 0.4887 | 0.9370 | 0.8528 |

${\mathit{\alpha}}_{1}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{\delta}}_{1}$ | ${\mathit{\mu}}_{1}$ |
---|---|---|---|

0.5565004 | −0.07445405 | 0.1021734 | 0.009274105 |

${\mathit{\alpha}}_{\mathbf{2}}$ | ${\mathit{\beta}}_{\mathbf{2}}$ | ${\mathit{\delta}}_{\mathbf{2}}$ | ${\mathit{\mu}}_{\mathbf{2}}$ |

1.026705 | −0.4044685 | 0.1459743 | 0.1332535 |

a | L | $\mathit{\sigma}$ |
---|---|---|

0.001173194 | 100.2216 | 0.1184857 |

$\mathit{\kappa}$ | $\mathit{\theta}$ | $\mathit{\sigma}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\delta}$ | $\mathit{\mu}$ |
---|---|---|---|---|---|---|

0.08757514 | 64.3041 | 6.655671 | 1.494409 | −0.005857812 | 1.502579 | 0.005655803 |

${\mathit{\kappa}}_{1}$ | ${\mathit{\kappa}}_{2}$ | ${\mathit{\theta}}_{1}$ | ${\mathit{\theta}}_{2}$ | ${\mathit{\sigma}}_{1}^{2}$ | ${\mathit{\sigma}}_{2}^{2}$ | ${\mathit{\pi}}_{11}$ | ${\mathit{\pi}}_{22}$ |
---|---|---|---|---|---|---|---|

0.075023 | 1.0327 | 62.3686 | 77.8781 | 25.8930 | 109.0701 | 0.9888 | 0.9209 |

${\mathit{\alpha}}_{1}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{\delta}}_{1}$ | ${\mathit{\mu}}_{1}$ |
---|---|---|---|

0.1550563 | 0.002312699 | 7.641092 | 4.604966 |

${\mathit{\alpha}}_{\mathbf{2}}$ | ${\mathit{\beta}}_{\mathbf{2}}$ | ${\mathit{\delta}}_{\mathbf{2}}$ | ${\mathit{\mu}}_{\mathbf{2}}$ |

0.07798173 | −0.006995654 | 9.699337 | 40.40929 |

**Table 10.**Model comparison for fuel-switching data using AIC and BIC, where $5\times {10}^{-324}$ represents the smallest positive double in R.

Likelihood-Value | AIC | BIC | |
---|---|---|---|

OU process | <$5\times {10}^{-324}$ | 3613.942 | 3629.387 |

Levy-driven OU process | <$5\times {10}^{-324}$ | 2414.07 | 2450.108 |

regime switching OU process | $5.36326\times {10}^{139}$ | −623.4778 | −571.9943 |

regime switching Levy driven OU process | $2.500369\times {10}^{18}$ | −48.72594 | 43.94428 |

**Table 11.**Model comparison for energy-switching data using AIC and BIC, where $5\times {10}^{-324}$ represents the smallest positive double in R.

Likelihood-Value | AIC | BIC | |
---|---|---|---|

Inhomogeneous Geometric Brownian Motion | <$5\times {10}^{-324}$ | 1598.655 | 1608.877 |

OU process | $9.50999\times {10}^{-138}$ | 637.0088 | 647.2303 |

Levy-driven OU process | $5.098482\times {10}^{-138}$ | 646.2556 | 670.1058 |

regime switching OU process | $5.287361\times {10}^{-314}$ | 1462.693 | 1496.765 |

regime switching Levy driven OU process | $5.41323\times {10}^{-319}$ | 1501.672 | 1563.001 |

Fuel-Switching Data | Energy-Switching Data | |
---|---|---|

Inhomogeneous Geometric Brownian Motion | 96.95243 | |

OU process | 0.1973981 | 88.33951 |

Levy-driven OU process | 0.1896627 | 125.8895 |

regime switching OU process | 0.06602948 | 106.7323 |

regime switching Levy driven OU process | 0.2099495 | 95.20393 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lu, W.; Arrigoni, A.; Swishchuk, A.; Goutte, S.
Modelling of Fuel- and Energy-Switching Prices by Mean-Reverting Processes and Their Applications to Alberta Energy Markets. *Mathematics* **2021**, *9*, 709.
https://doi.org/10.3390/math9070709

**AMA Style**

Lu W, Arrigoni A, Swishchuk A, Goutte S.
Modelling of Fuel- and Energy-Switching Prices by Mean-Reverting Processes and Their Applications to Alberta Energy Markets. *Mathematics*. 2021; 9(7):709.
https://doi.org/10.3390/math9070709

**Chicago/Turabian Style**

Lu, Weiliang, Alexis Arrigoni, Anatoliy Swishchuk, and Stéphane Goutte.
2021. "Modelling of Fuel- and Energy-Switching Prices by Mean-Reverting Processes and Their Applications to Alberta Energy Markets" *Mathematics* 9, no. 7: 709.
https://doi.org/10.3390/math9070709