# Improving the Efficiency of Hedge Trading Using Higher-Order Standardized Weather Derivatives for Wind Power

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Market Trading Model

#### 2.2. Minimum Variance Hedging Problem

#### 2.3. Non-Parametric Derivatives

#### 2.4. Standardized Derivatives

#### 2.5. Hedge Trading Strategy Using LASSO Regression

## 3. Results

- Wind power generation [MWh]: actual power generation in Eastern Denmark (DK2) (downloaded from https://www.nordpoolgroup.com/en/Market-data1/, accessed on 11 January 2021)
- Wind speed [m/s] and temperature [°C]: observed values at Copenhagen Airport (downloaded from http://rp5.ru/metar.php?metar=EKCH, accessed on 11 January 2021)

#### 3.1. Estimated Trend (Non-Parametric Derivatives)

#### 3.2. Measurement of Hedge Effects

#### 3.3. Trading Efficiency Using LASSO Regression

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Research on Wind Power Output Forecasting

## Appendix B. Verification of Non-Linearity by Ramsey’s RESET Test

Model | Diagnostic Variables | F-Statistic | p-Value |
---|---|---|---|

V ~ W | squares, cubes | 264.97 | $<2.2\times {10}^{16}$ |

V ~ T | squares | 305.21 | $<2.2\times {10}^{16}$ |

V ~ W + T | squares, cubes | 294.66 | $<2.2\times {10}^{16}$ |

## Appendix C. Hedge Trading Related Costs That Can Depend on Trading Volume

## Appendix D. The Case of Using Derivatives with No Average Value Correction for the Underlying Index

**Figure A1.**Relationship between contract volume and hedge effect by hedging model (cases in which no mean value correction was made for the underlying indexes). Note: the term “lasso” refers to the W3*T2 model estimated by LASSO regression. Hedge effect is measured in out-of-sample periods. Note: the blue dots represent the estimation results from OLS and the dashed curves represent the estimation results obtained by varying $\lambda $ in the LASSO regression.

**Figure A2.**Relationship between the regularization parameter $\lambda $ and the coefficients in LASSO regression (i.e., cases with no mean value correction for the underlying indexes). The graph on the right transforms the scale of the left graph using the hyperbolic tangent function ($tanh\left(100x\right)$) for ease of viewing.

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**Figure 1.**Overall picture of the transactions assumed in this study (from derivatives contracts to settlements).

**Figure 2.**Market trading model for wind derivatives. Note: “W” represents wind speed index and “T” represents temperature index.

**Figure 3.**Conceptual diagram of methods for determining standardized (non-parametric) derivatives’ trading volume (payoff functions). Note: this conceptual diagram is an example of using only wind speed derivatives.

**Figure 6.**Realized versus predicted values of wind power generation in the out-of-sample period (Upper: 2020, Lower: February 2020). Note: the upper graph plots daily averaged values for visibility. The realized data are the raw wind production data downloaded from https://www.nordpoolgroup.com/en/Market-data1/ (accessed on 11 January 2021).

**Figure 7.**Relationship between contract volume and hedge effect by hedging model. Note: the term “lasso” refers to the W3*T2 model estimated by LASSO regression. Hedge effect is measured in out-of-sample periods. Note: the blue dots represent the estimation results from OLS and the dashed curves represent the estimation results obtained by varying $\lambda $ in the LASSO regression.

**Figure 8.**Relationship between the regularization parameter $\lambda $ and the coefficients in LASSO regression. The graph on the right transforms the scale of the left graph using the hyperbolic tangent function ($tanh\left(100x\right)$) for ease of viewing.

Standardized Derivatives Model | Non-Parametric Derivatives Model | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

W1 | W2 | W3 | W3+T2 | W3+W*T2 | W3*T2 | Wd | Wd+Td | Wd*Td | ||

Standardized derivatives | $W$ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |||

${W}^{2}$ | ✓ | ✓ | ✓ | ✓ | ✓ | |||||

${W}^{3}$ | ✓ | ✓ | ✓ | ✓ | ||||||

$T+{T}^{2}$ | ✓ | ✓ | ✓ | |||||||

$WT+W{T}^{2}$ | ✓ | ✓ | ||||||||

${W}^{2}T+{W}^{2}{T}^{2}+{W}^{3}T+{W}^{3}{T}^{2}$ | ✓ | |||||||||

Non-parametric derivatives | ${f}_{W}\left(W\right)$ | ✓ | ✓ | |||||||

${f}_{T}\left(T\right)$ | ✓ | |||||||||

${f}_{te}\left(W,T\right)$ | ✓ |

Monthly Hedge Effect | All Period | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Nov | Dec | |||

Hedge effect (1-VRR) | W1 | 68.6% | 76.1% | 65.4% | 59.1% | 58.7% | 40.9% | 66.3% | 15.4% | 63.5% | 62.8% | 72.8% | 60.9% | 65.9% |

W3 | 67.5% | 79.4% | 72.3% | 59.9% | 59.5% | 43.9% | 69.1% | 25.3% | 68.8% | 62.5% | 75.9% | 60.3% | 68.1% | |

W3*T2 | 68.6% | 80.7% | 71.5% | 59.2% | 61.1% | 46.9% | 74.0% | 33.0% | 68.8% | 63.8% | 76.0% | 59.8% | 69.3% | |

Improvement rate (to “W1”) | W3 | −1.6% | 4.4% | 10.6% | 1.4% | 1.4% | 7.2% | 4.3% | 64.2% | 8.3% | −0.4% | 4.2% | −1.1% | 3.4% |

W3*T2 | 0.0% | 6.1% | 9.3% | 0.2% | 4.0% | 14.7% | 11.6% | 114.6% | 8.4% | 1.7% | 4.4% | −1.8% | 5.2% |

**Table 3.**Derivatives contract volume by hedging model, including LASSO regression-based trading strategies.

W1 | W2 | W3 | W3+T2 | W3+W*T2 | W3*T2 | $\mathbf{lasso}.{\mathit{\lambda}}_{\mathit{m}\mathit{i}\mathit{n}}$ | $\mathbf{lasso}.{\mathit{\lambda}}_{1\mathit{s}\mathit{e}}$ | Stdev of Payoffs | ||
---|---|---|---|---|---|---|---|---|---|---|

Contract volumes of each derivative (absolute values of coefficients) | ||||||||||

W | 0.98465 | 0.97895 | 1.13905 | 1.11814 | 1.16114 | 1.18486 | 1.18426 | 1.05063 | 2.3 | |

I(W^2) | - | 0.00378 | 0.06930 | 0.06541 | 0.06286 | 0.05867 | 0.05820 | 0.03367 | 8.3 | |

I(W^3) | - | - | 0.01406 | 0.01369 | 0.01376 | 0.01460 | 0.01455 | 0.00728 | 62.3 | |

T | - | - | - | 0.03324 | 0.03283 | 0.04237 | 0.04244 | 0.03022 | 6.4 | |

I(T^2) | - | - | - | 0.00569 | 0.00613 | 0.00694 | 0.00697 | 0.00579 | 46.3 | |

I(W * T) | - | - | - | - | 0.00355 | 0.00246 | 0.00229 | 0.00080 | 13.8 | |

I(W * T^2) | - | - | - | - | 0.00113 | 0.00180 | 0.00177 | 0.00006 | 128.2 | |

I(W^2 * T) | - | - | - | - | - | 0.00258 | 0.00262 | 0.00051 | 56.3 | |

I(W^2 * T^2) | - | - | - | - | - | 0.00016 | 0.00017 | 0.00011 | 428.9 | |

I(W^3 * T) | - | - | - | - | - | 0.00015 | 0.00017 | . | 361.8 | |

I(W^3 * T^2) | - | - | - | - | - | 0.00004 | 0.00004 | 0.00003 | 2651.4 | |

Contract volumes of derivatives portfolio | ||||||||||

Simple sum | 0.98465 | 0.98273 | 1.22241 | 1.23617 | 1.28140 | 1.31463 | 1.31348 | 1.12909 | ||

Weighted sum | 2.22805 | 2.24632 | 4.02585 | 4.40015 | 4.69229 | 5.31285 | 5.30554 | 3.74113 | ||

Hedge effects of derivatives portfolio | ||||||||||

1-VRR | 0.65892 | 0.65929 | 0.68146 | 0.69343 | 0.69797 | 0.70006 | 0.70009 | 0.69348 |

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## Share and Cite

**MDPI and ACS Style**

Matsumoto, T.; Yamada, Y.
Improving the Efficiency of Hedge Trading Using Higher-Order Standardized Weather Derivatives for Wind Power. *Energies* **2023**, *16*, 3112.
https://doi.org/10.3390/en16073112

**AMA Style**

Matsumoto T, Yamada Y.
Improving the Efficiency of Hedge Trading Using Higher-Order Standardized Weather Derivatives for Wind Power. *Energies*. 2023; 16(7):3112.
https://doi.org/10.3390/en16073112

**Chicago/Turabian Style**

Matsumoto, Takuji, and Yuji Yamada.
2023. "Improving the Efficiency of Hedge Trading Using Higher-Order Standardized Weather Derivatives for Wind Power" *Energies* 16, no. 7: 3112.
https://doi.org/10.3390/en16073112