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Estimation of the Number of Factors in a Multi-Factorial Heath-Jarrow-Morton Model in Power Markets

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Quantitative Energy Finance

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Abstract

We study the calibration of specific multi-factorial Heath-Jarrow-Morton models to power market prices, with a focus on the estimation of the optimal number of Gaussian factors. We describe a common statistical procedure based on likelihood maximisation and Akaike/Bayesian information criteria, in the case of a joint calibration on both spot and futures prices. We perform a detailed analysis on three national markets within Europe: Belgium, France, and Germany. The results show a lot of similarities among all the markets we consider, especially on the optimal number of factors and on the behaviour of the different factors.

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Notes

  1. 1.

    In practice the number of observed prices may vary, even after removing the redundant products (e.g. a quarterly contract when all the corresponding monthly contracts are observed), see Sect. 4.1.

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Authors and Affiliations

  1. EDF R&D and Fime Lab, Palaiseau, France

    Olivier Féron & Pierre Gruet

Authors
  1. Olivier Féron
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  2. Pierre Gruet
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Corresponding author

Correspondence to Pierre Gruet .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Oslo, Oslo, Norway

    Fred Espen Benth

  2. Department of Mathematics, Imperial College London, London, UK

    Almut E. D. Veraart

Appendix: Estimation Results

Appendix: Estimation Results

See Tables 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, and 28.

1.1 Belgian Power Market Data

N

Estimated correlation matrix

7

\( \begin {bmatrix} 1.00 & 0.02 & -0.02 & 0.04 & -0.06 & 0.09 & -0.11 \\ 0.02 & 1.00 & -1.00 & 0.95 & -0.91 & 0.82 & -0.73 \\ -0.02 & -1.00 & 1.00 & -0.98 & 0.94 & -0.86 & 0.76 \\ 0.04 & 0.95 & -0.98 & 1.00 & -0.99 & 0.93 & -0.84 \\ -0.06 & -0.91 & 0.94 & -0.99 & 1.00 & -0.97 & 0.89 \\ 0.09 & 0.82 & -0.86 & 0.93 & -0.97 & 1.00 & -0.98 \\ -0.11 & -0.73 & 0.76 & -0.84 & 0.89 & -0.98 & 1.00 \end {bmatrix}\)

8

\( \begin {bmatrix} 1.00 & 0.03 & -0.04 & 0.07 & -0.10 & 0.12 & -0.11 & 0.09 \\ 0.03 & 1.00 & -0.99 & 0.95 & -0.84 & 0.65 & -0.43 & 0.29 \\ -0.04 & -0.99 & 1.00 & -0.98 & 0.89 & -0.70 & 0.48 & -0.34 \\ 0.07 & 0.95 & -0.98 & 1.00 & -0.96 & 0.82 & -0.62 & 0.47 \\ -0.10 & -0.84 & 0.89 & -0.96 & 1.00 & -0.94 & 0.79 & -0.66 \\ 0.12 & 0.65 & -0.70 & 0.82 & -0.94 & 1.00 & -0.95 & 0.86 \\ -0.11 & -0.43 & 0.48 & -0.62 & 0.79 & -0.95 & 1.00 & -0.97 \\ 0.09 & 0.29 & -0.34 & 0.47 & -0.66 & 0.86 & -0.97 & 1.00 \end {bmatrix}\)

Table 9 Estimated correlation matrices in the French Power Market, period 2020. N is the number of factors
Full size table

N

Estimated correlation matrix

6

\( \begin {bmatrix} 1.00 & 0.15 & -0.14 & 0.14 & -0.14 & 0.14 \\ 0.15 & 1.00 & -0.69 & 0.63 & -0.50 & 0.43 \\ -0.14 & -0.69 & 1.00 & -0.99 & 0.95 & -0.90 \\ 0.14 & 0.63 & -0.99 & 1.00 & -0.98 & 0.95 \\ -0.14 & -0.50 & 0.95 & -0.98 & 1.00 & -0.99 \\ 0.14 & 0.43 & -0.90 & 0.95 & -0.99 & 1.00 \end {bmatrix}\)

7

\( \begin {bmatrix} 1.00 & 0.26 & -0.10 & 0.09 & -0.06 & 0.03 & -0.00 \\ 0.26 & 1.00 & -0.61 & 0.58 & -0.50 & 0.39 & -0.31 \\ -0.10 & -0.61 & 1.00 & -1.00 & 0.97 & -0.89 & 0.81 \\ 0.09 & 0.58 & -1.00 & 1.00 & -0.98 & 0.92 & -0.85 \\ -0.06 & -0.50 & 0.97 & -0.98 & 1.00 & -0.98 & 0.93 \\ 0.03 & 0.39 & -0.89 & 0.92 & -0.98 & 1.00 & -0.98 \\ -0.00 & -0.31 & 0.81 & -0.85 & 0.93 & -0.98 & 1.00 \end {bmatrix}\)

8

\( \begin {bmatrix} 1.00 & 0.23 & -0.06 & 0.06 & -0.05 & 0.07 & -0.13 & 0.16 \\ 0.23 & 1.00 & -0.58 & 0.54 & -0.44 & 0.32 & -0.24 & 0.21 \\ -0.06 & -0.58 & 1.00 & -1.00 & 0.95 & -0.83 & 0.62 & -0.49 \\ 0.06 & 0.54 & -1.00 & 1.00 & -0.98 & 0.87 & -0.67 & 0.54 \\ -0.05 & -0.44 & 0.95 & -0.98 & 1.00 & -0.95 & 0.78 & -0.65 \\ 0.07 & 0.32 & -0.83 & 0.87 & -0.95 & 1.00 & -0.93 & 0.83 \\ -0.13 & -0.24 & 0.62 & -0.67 & 0.78 & -0.93 & 1.00 & -0.98 \\ 0.16 & 0.21 & -0.49 & 0.54 & -0.65 & 0.83 & -0.98 & 1.00 \end {bmatrix}\)

Table 10 Estimated correlation matrices in the Belgian Power Market, period 2021–2022. N is the number of factors
Full size table

N

Estimated correlation matrix

6

\( \begin {bmatrix} 1.00 & -1.00 & 0.09 & -0.09 & 0.08 & -0.08 \\ -1.00 & 1.00 & -0.10 & 0.10 & -0.09 & 0.08 \\ 0.09 & -0.10 & 1.00 & -1.00 & 0.98 & -0.86 \\ -0.09 & 0.10 & -1.00 & 1.00 & -0.99 & 0.89 \\ 0.08 & -0.09 & 0.98 & -0.99 & 1.00 & -0.92 \\ -0.08 & 0.08 & -0.86 & 0.89 & -0.92 & 1.00 \end {bmatrix}\)

7

\( \begin {bmatrix} 1.00 & -0.99 & 0.03 & -0.02 & 0.00 & 0.29 & -0.33 \\ -0.99 & 1.00 & -0.05 & 0.04 & -0.02 & -0.27 & 0.30 \\ 0.03 & -0.05 & 1.00 & -1.00 & 0.99 & -0.33 & 0.19 \\ -0.02 & 0.04 & -1.00 & 1.00 & -1.00 & 0.34 & -0.20 \\ 0.00 & -0.02 & 0.99 & -1.00 & 1.00 & -0.36 & 0.22 \\ 0.29 & -0.27 & -0.33 & 0.34 & -0.36 & 1.00 & -0.99 \\ -0.33 & 0.30 & 0.19 & -0.20 & 0.22 & -0.99 & 1.00 \end {bmatrix}\)

8

\( \begin {bmatrix} 1.00 & -0.99 & 0.08 & -0.07 & 0.06 & -0.01 & 0.01 & -0.01 \\ -0.99 & 1.00 & -0.10 & 0.09 & -0.08 & 0.01 & -0.01 & 0.00 \\ 0.08 & -0.10 & 1.00 & -1.00 & 0.99 & -0.66 & 0.56 & -0.45 \\ -0.07 & 0.09 & -1.00 & 1.00 & -1.00 & 0.69 & -0.59 & 0.48 \\ 0.06 & -0.08 & 0.99 & -1.00 & 1.00 & -0.72 & 0.62 & -0.50 \\ -0.01 & 0.01 & -0.66 & 0.69 & -0.72 & 1.00 & -0.96 & 0.87 \\ 0.01 & -0.01 & 0.56 & -0.59 & 0.62 & -0.96 & 1.00 & -0.97 \\ -0.01 & 0.00 & -0.45 & 0.48 & -0.50 & 0.87 & -0.97 & 1.00 \end {bmatrix}\)

Table 11 Estimated \(\alpha _i\)’s (yr\({ }^{-1}\)) in the French Power Market, period 2018–2019. N is the number of factors
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Table 12 Estimated \(\alpha _i\)’s (yr\({ }^{-1}\)) in the French Power Market, period 2020. N is the number of factors
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Table 13 Estimated \(\alpha _i\)’s (yr\({ }^{-1}\)) in the French Power Market, period 2021–2022. N is the number of factors
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Table 14 Estimated \(\sigma _i\)’s (yr\({ }^{-1/2}\)) in the French Power Market, period 2018–2019. N is the number of factors
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Table 15 Estimated \(\sigma _i\)’s (yr\({ }^{-1/2}\)) in the French Power Market, period 2020. N is the number of factors
Full size table
Table 16 Estimated \(\sigma _i\)’s (yr\({ }^{-1/2}\)) in the French Power Market, period 2021–2022. N is the number of factors
Full size table
Table 17 Estimated correlation matrices in the French Power Market, period 2018–2019. N is the number of factors
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Table 18 Estimated correlation matrices in the French Power Market, period 2020. N is the number of factors
Full size table
Table 19 Estimated correlation matrices in the French Power Market, period 2021–2022. N is the number of factors
Full size table
Table 20 Estimated \(\alpha _i\)’s (yr\({ }^{-1}\)) in the German Power Market, period 2018-2019. N is the number of factors
Full size table
Table 21 Estimated \(\alpha _i\)’s (yr\({ }^{-1}\)) in the German Power Market, period 2020. N is the number of factors
Full size table
Table 22 Estimated \(\alpha _i\)’s (yr\({ }^{-1}\)) in the German Power Market, period 2021–2022. N is the number of factors
Full size table
Table 23 Estimated \(\sigma _i\)’s (yr\({ }^{-1/2}\)) in the German Power Market, period 2018-2019. N is the number of factors
Full size table
Table 24 Estimated \(\sigma _i\)’s (yr\({ }^{-1/2}\)) in the German Power Market, period 2020. N is the number of factors
Full size table
Table 25 Estimated \(\sigma _i\)’s (yr\({ }^{-1/2}\)) in the German Power Market, period 2021–2022. N is the number of factors
Full size table
Table 26 Estimated correlation matrices in the German Power Market, period 2018–2019. N is the number of factors
Full size table

N

Estimated correlation matrix

7

\( \begin {bmatrix} 1.00 & -0.04 & 0.03 & 0.01 & -0.05 & 0.08 & -0.09 \\ -0.04 & 1.00 & -1.00 & 0.98 & -0.94 & 0.84 & -0.74 \\ 0.03 & -1.00 & 1.00 & -0.98 & 0.95 & -0.85 & 0.75 \\ 0.01 & 0.98 & -0.98 & 1.00 & -0.99 & 0.91 & -0.81 \\ -0.05 & -0.94 & 0.95 & -0.99 & 1.00 & -0.97 & 0.89 \\ 0.08 & 0.84 & -0.85 & 0.91 & -0.97 & 1.00 & -0.98 \\ -0.09 & -0.74 & 0.75 & -0.81 & 0.89 & -0.98 & 1.00 \end {bmatrix}\)

8

\( \begin {bmatrix} 1.00 & 0.01 & -0.01 & 0.01 & 0.00 & -0.02 & 0.02 & -0.00 \\ 0.01 & 1.00 & -1.00 & 0.95 & -0.84 & 0.67 & -0.46 & 0.17 \\ -0.01 & -1.00 & 1.00 & -0.96 & 0.87 & -0.71 & 0.50 & -0.20 \\ 0.01 & 0.95 & -0.96 & 1.00 & -0.97 & 0.85 & -0.67 & 0.36 \\ 0.00 & -0.84 & 0.87 & -0.97 & 1.00 & -0.96 & 0.82 & -0.55 \\ -0.02 & 0.67 & -0.71 & 0.85 & -0.96 & 1.00 & -0.95 & 0.75 \\ 0.02 & -0.46 & 0.50 & -0.67 & 0.82 & -0.95 & 1.00 & -0.92 \\ -0.00 & 0.17 & -0.20 & 0.36 & -0.55 & 0.75 & -0.92 & 1.00 \end {bmatrix}\)

Table 27 Estimated correlation matrices in the German Power Market, period 2020. N is the number of factors
Full size table

1.2 French Power Market Data

1.3 German Power Market Data

N

Estimated correlation matrix

7

\( \begin {bmatrix} 1.00 & 0.07 & -0.07 & 0.07 & -0.08 & 0.06 & -0.04 \\ 0.07 & 1.00 & -1.00 & 0.96 & -0.88 & 0.72 & -0.57 \\ -0.07 & -1.00 & 1.00 & -0.97 & 0.90 & -0.74 & 0.59 \\ 0.07 & 0.96 & -0.97 & 1.00 & -0.97 & 0.85 & -0.70 \\ -0.08 & -0.88 & 0.90 & -0.97 & 1.00 & -0.94 & 0.84 \\ 0.06 & 0.72 & -0.74 & 0.85 & -0.94 & 1.00 & -0.97 \\ -0.04 & -0.57 & 0.59 & -0.70 & 0.84 & -0.97 & 1.00 \end {bmatrix}\)

8

\( \begin {bmatrix} 1.00 & 0.11 & -0.11 & 0.10 & -0.09 & 0.05 & -0.01 & 0.01 \\ 0.11 & 1.00 & -1.00 & 0.97 & -0.90 & 0.77 & -0.61 & 0.53 \\ -0.11 & -1.00 & 1.00 & -0.98 & 0.92 & -0.79 & 0.62 & -0.54 \\ 0.10 & 0.97 & -0.98 & 1.00 & -0.97 & 0.86 & -0.70 & 0.60 \\ -0.09 & -0.90 & 0.92 & -0.97 & 1.00 & -0.95 & 0.82 & -0.72 \\ 0.05 & 0.77 & -0.79 & 0.86 & -0.95 & 1.00 & -0.95 & 0.88 \\ -0.01 & -0.61 & 0.62 & -0.70 & 0.82 & -0.95 & 1.00 & -0.98 \\ 0.01 & 0.53 & -0.54 & 0.60 & -0.72 & 0.88 & -0.98 & 1.00 \end {bmatrix}\)

Table 28 Estimated correlation matrices in the German Power Market, period 2021–2022. N is the number of factors
Full size table

N

Estimated correlation matrix

6

\( \begin {bmatrix} 1.00 & -0.90 & 0.03 & -0.02 & 0.02 & -0.03 \\ -0.90 & 1.00 & -0.09 & 0.09 & -0.08 & 0.08 \\ 0.03 & -0.09 & 1.00 & -1.00 & 0.98 & -0.88 \\ -0.02 & 0.09 & -1.00 & 1.00 & -0.99 & 0.91 \\ 0.02 & -0.08 & 0.98 & -0.99 & 1.00 & -0.95 \\ -0.03 & 0.08 & -0.88 & 0.91 & -0.95 & 1.00 \end {bmatrix}\)

7

\( \begin {bmatrix} 1.00 & -0.99 & 0.06 & -0.05 & 0.04 & -0.01 & 0.00 \\ -0.99 & 1.00 & -0.07 & 0.07 & -0.06 & 0.01 & -0.01 \\ 0.06 & -0.07 & 1.00 & -1.00 & 0.99 & -0.80 & 0.77 \\ -0.05 & 0.07 & -1.00 & 1.00 & -1.00 & 0.82 & -0.80 \\ 0.04 & -0.06 & 0.99 & -1.00 & 1.00 & -0.84 & 0.83 \\ -0.01 & 0.01 & -0.80 & 0.82 & -0.84 & 1.00 & -1.00 \\ 0.00 & -0.01 & 0.77 & -0.80 & 0.83 & -1.00 & 1.00 \end {bmatrix}\)

8

\( \begin {bmatrix} 1.00 & -0.99 & 0.07 & -0.06 & 0.06 & -0.02 & 0.02 & -0.02 \\ -0.99 & 1.00 & -0.09 & 0.08 & -0.07 & 0.03 & -0.02 & 0.02 \\ 0.07 & -0.09 & 1.00 & -1.00 & 0.99 & -0.75 & 0.72 & -0.68 \\ -0.06 & 0.08 & -1.00 & 1.00 & -1.00 & 0.77 & -0.74 & 0.70 \\ 0.06 & -0.07 & 0.99 & -1.00 & 1.00 & -0.79 & 0.76 & -0.72 \\ -0.02 & 0.03 & -0.75 & 0.77 & -0.79 & 1.00 & -0.99 & 0.94 \\ 0.02 & -0.02 & 0.72 & -0.74 & 0.76 & -0.99 & 1.00 & -0.98 \\ -0.02 & 0.02 & -0.68 & 0.70 & -0.72 & 0.94 & -0.98 & 1.00 \end {bmatrix}\)

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Féron, O., Gruet, P. (2024). Estimation of the Number of Factors in a Multi-Factorial Heath-Jarrow-Morton Model in Power Markets. In: Benth, F.E., Veraart, A.E.D. (eds) Quantitative Energy Finance. Springer, Cham. https://doi.org/10.1007/978-3-031-50597-3_1

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