The goodness-of-fit of the fuel-switching price using the mean-reverting L´evy jump process Julien Chevallier and St´ ephane Goutte IPAG Business School (IPAG Lab) University Paris 8 & ESG Management School ISEFI 2014

Introduction



We propose in this paper to model the fuel-switching price by using continuous-time stochastic jump diffusions.

Introduction



We propose in this paper to model the fuel-switching price by using continuous-time stochastic jump diffusions.



We augment the model by C ¸ etin and Verschuere (2009, IJTAF) through the introduction of jumps in the underlying stochastic process of the fuel-switching behavior.

Introduction



We propose in this paper to model the fuel-switching price by using continuous-time stochastic jump diffusions.



We augment the model by C ¸ etin and Verschuere (2009, IJTAF) through the introduction of jumps in the underlying stochastic process of the fuel-switching behavior.



Aim: find the model that provides the best goodness-of-fit to the fuel switching price historical values.

Introduction



We propose in this paper to model the fuel-switching price by using continuous-time stochastic jump diffusions.



We augment the model by C ¸ etin and Verschuere (2009, IJTAF) through the introduction of jumps in the underlying stochastic process of the fuel-switching behavior.



Aim: find the model that provides the best goodness-of-fit to the fuel switching price historical values.



To do so, we conduct a ‘horse-race’ between 3 competing models:

Introduction



We propose in this paper to model the fuel-switching price by using continuous-time stochastic jump diffusions.



We augment the model by C ¸ etin and Verschuere (2009, IJTAF) through the introduction of jumps in the underlying stochastic process of the fuel-switching behavior.



Aim: find the model that provides the best goodness-of-fit to the fuel switching price historical values.



To do so, we conduct a ‘horse-race’ between 3 competing models:

1. Continuous process,

Introduction



We propose in this paper to model the fuel-switching price by using continuous-time stochastic jump diffusions.



We augment the model by C ¸ etin and Verschuere (2009, IJTAF) through the introduction of jumps in the underlying stochastic process of the fuel-switching behavior.



Aim: find the model that provides the best goodness-of-fit to the fuel switching price historical values.



To do so, we conduct a ‘horse-race’ between 3 competing models:

1. Continuous process, 2. Normal Inverse Gaussian (NIG) L´evy jump process,

Introduction



We propose in this paper to model the fuel-switching price by using continuous-time stochastic jump diffusions.



We augment the model by C ¸ etin and Verschuere (2009, IJTAF) through the introduction of jumps in the underlying stochastic process of the fuel-switching behavior.



Aim: find the model that provides the best goodness-of-fit to the fuel switching price historical values.



To do so, we conduct a ‘horse-race’ between 3 competing models:

1. Continuous process, 2. Normal Inverse Gaussian (NIG) L´evy jump process, 3. Variance Gamma (VG) L´evy jump process.

Outline

1. Fuel-switching in the power sector 2. Model 3. Empirical application

Fuel-switching in the power sector



The ability of power generators to switch between their fuel inputs

Fuel-switching in the power sector



The ability of power generators to switch between their fuel inputs (such as coal to gas)

Fuel-switching in the power sector





The ability of power generators to switch between their fuel inputs (such as coal to gas) is expected to be the primary source of CO2 emissions reduction in the power sector. Indeed, when the carbon price is above the switching point,

Fuel-switching in the power sector





The ability of power generators to switch between their fuel inputs (such as coal to gas) is expected to be the primary source of CO2 emissions reduction in the power sector. Indeed, when the carbon price is above the switching point, gas-fired power plants become more profitable than coal-fired ones.

Fuel-switching in the power sector (ctd.)



The introduction of carbon costs modifies the marginal cost for each plant by introducing the emissions factor,

Fuel-switching in the power sector (ctd.)



The introduction of carbon costs modifies the marginal cost for each plant by introducing the emissions factor, which depends on the fuel and the amount of fuel burnt:

Fuel-switching in the power sector (ctd.)



The introduction of carbon costs modifies the marginal cost for each plant by introducing the emissions factor, which depends on the fuel and the amount of fuel burnt: MC =

EF FC + EC η η

(1)

with M C the marginal cost, F C fuel costs, η the plant efficiency, EF the emissions factor, and EC emissions costs.

Fuel-switching in the power sector (ctd.)



The switching-point between a given coal plant and a given gas plant can be defined as the emissions cost that equalizes marginal costs, i.e. M Cgas = M Ccoal .

Fuel-switching in the power sector (ctd.)





The switching-point between a given coal plant and a given gas plant can be defined as the emissions cost that equalizes marginal costs, i.e. M Cgas = M Ccoal . It represents the allowance cost that leads to switch between two plants in the merit order.

Fuel-switching in the power sector (ctd.)







The switching-point between a given coal plant and a given gas plant can be defined as the emissions cost that equalizes marginal costs, i.e. M Cgas = M Ccoal . It represents the allowance cost that leads to switch between two plants in the merit order. This price depends on each plant’s fuel costs, efficiency and emissions factor: ECswitch =

ηcoal F Cgas − ηgas F Ccoal ηgas EFcoal − ηcoal EFgas

(2)

Fuel-switching in the power sector (ctd.)







The switching-point between a given coal plant and a given gas plant can be defined as the emissions cost that equalizes marginal costs, i.e. M Cgas = M Ccoal . It represents the allowance cost that leads to switch between two plants in the merit order. This price depends on each plant’s fuel costs, efficiency and emissions factor: ECswitch =



ηcoal F Cgas − ηgas F Ccoal ηgas EFcoal − ηcoal EFgas

If the EUA price is lower than this cost, generating electricity from coal is more profitable than from gas.

(2)

Fuel-switching in the power sector (ctd.) 40 COAL NGAS

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Figure: EEX Coal and Natural Gas Prices (in EUR/MWh) from January

Fuel-switching in the power sector (ctd.) 80 SWITCH 70

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Figure: Switch Price (in EUR/MWh) from January 01, 2007 to December

Fuel-switching in the power sector (ctd.)



In theory, the switching point could occur between all the technologies available for power generation.

Fuel-switching in the power sector (ctd.)



In theory, the switching point could occur between all the technologies available for power generation.



However, in practice, the main abatement opportunities are expected to come from the switching from coal to gas.

Fuel-switching in the power sector (ctd.)



In theory, the switching point could occur between all the technologies available for power generation.



However, in practice, the main abatement opportunities are expected to come from the switching from coal to gas.



Switching from coal to oil (or from oil to gas) is possible, but it appears very limited in the European electricity generation mix, and it is usually more expensive.

Fuel-switching in the power sector (ctd.)



In theory, the switching point could occur between all the technologies available for power generation.



However, in practice, the main abatement opportunities are expected to come from the switching from coal to gas.



Switching from coal to oil (or from oil to gas) is possible, but it appears very limited in the European electricity generation mix, and it is usually more expensive.



Switching from coal to nuclear (or gas to nuclear) also appears unlikely, since nuclear energy is not flexible and has to operate at high-load to be profitable.

Fuel-switching in the power sector (ctd.)



Several factors may impact fuel-switching opportunities:

Fuel-switching in the power sector (ctd.)



Several factors may impact fuel-switching opportunities:

1. Fuel prices: the switching point may be seen as the EUA price at which unused available gas-fired capacity is substituted for coal-fired generation.

Fuel-switching in the power sector (ctd.)



Several factors may impact fuel-switching opportunities:

1. Fuel prices: the switching point may be seen as the EUA price at which unused available gas-fired capacity is substituted for coal-fired generation. 2. The load: the fuel-switching potential varies throughout the year, depending on the season (winter or summer), the time of the week (day-of-week or week-end), and the period of the day (day or night).

Fuel-switching in the power sector (ctd.)



Several factors may impact fuel-switching opportunities:

1. Fuel prices: the switching point may be seen as the EUA price at which unused available gas-fired capacity is substituted for coal-fired generation. 2. The load: the fuel-switching potential varies throughout the year, depending on the season (winter or summer), the time of the week (day-of-week or week-end), and the period of the day (day or night). 3. High EUA prices are more likely to fall within the switching band.

Outline

1. Fuel-switching in the power sector 2. Model 3. Empirical application

Model 

The fuel-switching price exhibits salient features departing from Gaussianity such as fat tails (or semi-heavy tails), excess skewness, and jumps.

Model 

The fuel-switching price exhibits salient features departing from Gaussianity such as fat tails (or semi-heavy tails), excess skewness, and jumps.



We propose to model the fuel-switching price using the mean-reverting L´evy jump model, by evaluating its performance against other competitors:

Model 

The fuel-switching price exhibits salient features departing from Gaussianity such as fat tails (or semi-heavy tails), excess skewness, and jumps.



We propose to model the fuel-switching price using the mean-reverting L´evy jump model, by evaluating its performance against other competitors:

1. Continuous process

Model 

The fuel-switching price exhibits salient features departing from Gaussianity such as fat tails (or semi-heavy tails), excess skewness, and jumps.



We propose to model the fuel-switching price using the mean-reverting L´evy jump model, by evaluating its performance against other competitors:

1. Continuous process dXt = κ (θ − Xt ) dt + σdWt 

with parameters κ, θ in R and σ in R+ and where Wt is a Brownian motion.

(3)

Model 

The fuel-switching price exhibits salient features departing from Gaussianity such as fat tails (or semi-heavy tails), excess skewness, and jumps.



We propose to model the fuel-switching price using the mean-reverting L´evy jump model, by evaluating its performance against other competitors:

1. Continuous process dXt = κ (θ − Xt ) dt + σdWt 



with parameters κ, θ in R and σ in R+ and where Wt is a Brownian motion. κ denotes the mean-reverting rate.

(3)

Model 

The fuel-switching price exhibits salient features departing from Gaussianity such as fat tails (or semi-heavy tails), excess skewness, and jumps.



We propose to model the fuel-switching price using the mean-reverting L´evy jump model, by evaluating its performance against other competitors:

1. Continuous process dXt = κ (θ − Xt ) dt + σdWt



with parameters κ, θ in R and σ in R+ and where Wt is a Brownian motion. κ denotes the mean-reverting rate.



θ denotes the long-run mean.



(3)

Model 

The fuel-switching price exhibits salient features departing from Gaussianity such as fat tails (or semi-heavy tails), excess skewness, and jumps.



We propose to model the fuel-switching price using the mean-reverting L´evy jump model, by evaluating its performance against other competitors:

1. Continuous process dXt = κ (θ − Xt ) dt + σdWt 

  

with parameters κ, θ in R and σ in R+ and where Wt is a Brownian motion. κ denotes the mean-reverting rate. θ denotes the long-run mean. σ. denotes the volatility of X.

(3)

Model (ctd.)

2. Normal Inverse Gaussian L´ evy jump process

Model (ctd.)

2. Normal Inverse Gaussian L´ evy jump process dXt = κ (θ − Xt ) dt + σdLt 

with parameters κ,θ in R and σ in R+ and where Lt follows a Normal Inverse Gaussian (NIG) process.

(4)

Model (ctd.)

2. Normal Inverse Gaussian L´ evy jump process dXt = κ (θ − Xt ) dt + σdLt 

with parameters κ,θ in R and σ in R+ and where Lt follows a Normal Inverse Gaussian (NIG) process.

3. Variance Gamma L´ evy jump process

(4)

Model (ctd.)

2. Normal Inverse Gaussian L´ evy jump process dXt = κ (θ − Xt ) dt + σdLt 

(4)

with parameters κ,θ in R and σ in R+ and where Lt follows a Normal Inverse Gaussian (NIG) process.

3. Variance Gamma L´ evy jump process dXt = κ (θ − Xt ) dt + σdLt 

with parameters κ,θ in R and σ in R+ and where Lt follows a Variance Gamma (VG) process.

(5)

Model (ctd.) Parameters estimation: 

We aim at estimating the set of parameters Θ that contains the mean-reverting diffusion parameters m and a, the volatility of the diffusion s, as well as the set of parameters of the distribution laws of Z.

Model (ctd.) Parameters estimation: 

We aim at estimating the set of parameters Θ that contains the mean-reverting diffusion parameters m and a, the volatility of the diffusion s, as well as the set of parameters of the distribution laws of Z.



The estimation procedure unfolds in two steps.

Model (ctd.) Parameters estimation: 

We aim at estimating the set of parameters Θ that contains the mean-reverting diffusion parameters m and a, the volatility of the diffusion s, as well as the set of parameters of the distribution laws of Z.



The estimation procedure unfolds in two steps.

1. we estimate the subset of parameter {m, a, s} using a least squares method.

Model (ctd.) Parameters estimation: 

We aim at estimating the set of parameters Θ that contains the mean-reverting diffusion parameters m and a, the volatility of the diffusion s, as well as the set of parameters of the distribution laws of Z.



The estimation procedure unfolds in two steps.

1. we estimate the subset of parameter {m, a, s} using a least squares method. 2. we estimate the second subset of parameters corresponding to the law distribution of Z (i.e. {α, β, δ, μ} in the NIG case) using a maximum likelihood method.

Model (ctd.) Parameters estimation: 

We aim at estimating the set of parameters Θ that contains the mean-reverting diffusion parameters m and a, the volatility of the diffusion s, as well as the set of parameters of the distribution laws of Z.



The estimation procedure unfolds in two steps.

1. we estimate the subset of parameter {m, a, s} using a least squares method. 2. we estimate the second subset of parameters corresponding to the law distribution of Z (i.e. {α, β, δ, μ} in the NIG case) using a maximum likelihood method. 

This two-step approach greatly simplifies the task of the econometrician, as it reduces the optimisation problem.

Outline

1. Fuel-switching in the power sector 2. Model 3. Empirical application

Empirical application

Table: Descriptive Statistics

Statistics Mean Median Minimum Maximum Std Skewness Kurtosis

Data 21.3375 20.1400 -12.9800 77.4800 19.5542 0.4336 2.2116

Table: Estimated parameters of the continuous time process

Process

κ 2.9787

θ 21.8044

σ 67.9655

Empirical application (ctd.)

Table: Estimated parameters of the NIG case

NIG

α 0.1290

β 0.0101

δ 1.2792

μ 0.1527

Table: Estimated parameters of the VG case

VG

λ 0.6226

α 0.3971

β -0.0007

μ 0.2960

Each parameter can be interpreted as having a different effects on the shape of the distribution:  α - tail heaviness of steepness.  

β - symmetry. δ - scale.



μ - location.

Empirical application (ctd.)

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Figure: Residuals

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Empirical application (ctd.)

Probability Density Function 0.35

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Figure: Histogram of historical residuals of our fuel switching prices and the corresponding NIG (in black), VG (in green) and Normal (in red) distributed pdf.

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Empirical application (ctd.) QQ−Plot versus NIG 30

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QQ−Plot versus Normal

Quantiles of Input Sample

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Figure: QQ-plots for the residuals of our L´evy model against the NIG (top), VG (middle) and the Normal (bottom) distributed quantiles

Empirical application (ctd.)

Table: Results of the Kolmogorov-Smirnov tests with respect to different levels α

α 0.5 0.3 0.2 0.1 0.05 0.025 0.01 0.005 0.001

NIG 1 1 1 0 0 0 0 0 0

VG 1 1 1 1 1 1 0 0 0

Normal 1 1 1 1 1 1 1 1 1

Empirical application (ctd.)

Table: Values of the Cramer-von Mises test statistic for both NIG, VG and Normal distributions

Statistic Stats H

NIG 0.1957(0.7239) 0

VG 0.5112 (0.9624) 1

Note: In parenthesis the corresponding p-values.

Normal 6.2542 (1) 1

Conclusional remarks 

We show that a L´evy-type jump model offers satisfactory results to model the fuel-switching price, taking into account carbon emissions.

Conclusional remarks 



We show that a L´evy-type jump model offers satisfactory results to model the fuel-switching price, taking into account carbon emissions. The assumption that the model is driven by a L´evy jump process and not a Continuous Gaussian process is clearly demonstrated in this paper.

Conclusional remarks 





We show that a L´evy-type jump model offers satisfactory results to model the fuel-switching price, taking into account carbon emissions. The assumption that the model is driven by a L´evy jump process and not a Continuous Gaussian process is clearly demonstrated in this paper. We confirm that the NIG distribution provides overall a better fit to the fuel-switching price than the Gaussian distribution.

Conclusional remarks 



We show that a L´evy-type jump model offers satisfactory results to model the fuel-switching price, taking into account carbon emissions. The assumption that the model is driven by a L´evy jump process and not a Continuous Gaussian process is clearly demonstrated in this paper.



We confirm that the NIG distribution provides overall a better fit to the fuel-switching price than the Gaussian distribution.



Moreover, the NIG distribution gives better results than other Hyperbolic generalized distribution, and especially the Variance Gamma one.

Conclusional remarks 



We show that a L´evy-type jump model offers satisfactory results to model the fuel-switching price, taking into account carbon emissions. The assumption that the model is driven by a L´evy jump process and not a Continuous Gaussian process is clearly demonstrated in this paper.



We confirm that the NIG distribution provides overall a better fit to the fuel-switching price than the Gaussian distribution.



Moreover, the NIG distribution gives better results than other Hyperbolic generalized distribution, and especially the Variance Gamma one. Evidence of Heavy Tails in the fuel-switching price, hence superiority of the methodologies used in our paper.



Thanks for your attention!

Contact:

julien [dot] chevallier04 [at] univ-paris8 [dot] fr

sites.google.com/site/jpchevallier/