# A New Approach to Measure Volatility in Energy Markets

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## Abstract

**:**

## 1. Introduction

## 2. Permutation and Topological Entropies

- (c1)
- ${x}_{r+{i}_{1}}\le {x}_{r+{i}_{2}}\le \dots {x}_{r+{i}_{m}}.$
- (c2)
- ${i}_{s-1}<{i}_{s}$ if ${x}_{r+{i}_{s-1}}={x}_{r+{i}_{s}}.$

## 3. Simulated data

**Figure 1.**The (

**a**) PE, (

**b**) MPE and (

**c**) TE of the two simulated time series ${({x}_{n})}_{n=1}^{T}$ and ${({y}_{n})}_{n=1}^{T}$, using $m=5$, $w=600$. Single lines correspond to the logistic map ${({y}_{n})}_{n}$ and the dotted lines correspond to the random series ${({x}_{n})}_{n}$.

**Figure 2.**The standard deviation of (

**a**) the two simulated series, (

**b**) their returns series and (

**c**) their logarithmic return series. Single lines correspond to the logistic map ${({y}_{n})}_{n}$ and the dotted lines correspond to the random series ${({x}_{n})}_{n}$.

**Figure 3.**The PE (single line) and MPE (dotted line) of the simulated time series ${({z}_{n})}_{n}$.

**Figure 4.**(

**a**) The simulated time series ${({z}_{n})}_{n}$ resulted in merging two Gaussian processes with the same standard deviation ($\sigma =1$) and different means ($\mu =5$ and 10, respectively). (

**b**) The PE (single line) and MPE (dotted line) of ${({z}_{n})}_{n}$ using $m=5$ and $w=600$.

**Figure 5.**(

**a**) The simulated time series resulted in merging the Gaussian processes and the logistic map given in Figure 1. (

**b**) The PE (single line) and MPE (dotted line) of ${({z}_{n})}_{n}$ using $m=5$ and $w=600$.

## 4. Applications to Some Energy Markets

Countries | Date of entry |

Norway | 1/1/1993 |

Sweeden | 1/1/1996 |

Finland | 29/12/1997 |

West Denmark | 1/7/1999 |

East Denmark | 1/10/2000 |

KONTEK zone (Germany) | 5/10/2005 |

**Figure 7.**(

**a**) The Nord Pool spot prices from May 1992 to December 1998, in NOK/MWh. (

**b**) The PE (dotted line) and its mean for each season (single line). (

**c**) The MPE (dotted line) and its mean for each season (single line).

**Figure 8.**(

**a**) The Nord Pool spot prices from January of 1999 to December 2009, in EUR/MWh. (

**b**) The PE (dotted line) and its mean for each season (single line). (

**c**) The MPE (dotted line) and its mean for each season (single line).

**Table 2.**The mean of the MPE and PE in the periods determined by the incorporation of a new country.

Period | MPE mean | PE mean |
---|---|---|

From May 1992 to January 1993 | 1.47 | 1.45 |

From January 1993 to January 1996 | 1.70 | 1.68 |

From January 1996 to January 1997 | 3.53 | 3.43 |

From January 1997 to July 1999 | 3.19 | 3.15 |

From July 1999 to October 2000 | 3.29 | 3.23 |

From October 2000 to October 2005 | 3.30 | 3.20 |

From October 2005 to December 2009 | 3.35 | 3.13 |

**Figure 9.**(

**a**) The Ontario price series. (

**b**) The PE (dotted line) and its mean for each season (single line). (

**c**) The MPE (dotted line) and its mean for each season (single line).

**Figure 10.**(

**a**) The OMEL price series. (

**b**) The PE (dotted line) and its mean for each season (single line). (

**c**) The MPE (dotted line) and its mean for each season (single line).

**Figure 12.**(

**a**) The standard deviation of the four price series. (

**b**) The standard deviation of their return series.

Market | PE mean | MPE mean |

Ontario | 4.14 | 4.95 |

VIC | 3.89 | 3.95 |

SA | 3.88 | 3.92 |

NSW | 3.87 | 3.93 |

QLD | 3.87 | 3.93 |

OMEL | 3.41 | 4.13 |

Nord Pool | 3.16 | 3.34 |

## 5. Conclusions

## Acknowledgments

## References

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© 2012 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 license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Ruiz, M.d.C.; Guillamón, A.; Gabaldón, A. A New Approach to Measure Volatility in Energy Markets. *Entropy* **2012**, *14*, 74-91.
https://doi.org/10.3390/e14010074

**AMA Style**

Ruiz MdC, Guillamón A, Gabaldón A. A New Approach to Measure Volatility in Energy Markets. *Entropy*. 2012; 14(1):74-91.
https://doi.org/10.3390/e14010074

**Chicago/Turabian Style**

Ruiz, María del Carmen, Antonio Guillamón, and Antonio Gabaldón. 2012. "A New Approach to Measure Volatility in Energy Markets" *Entropy* 14, no. 1: 74-91.
https://doi.org/10.3390/e14010074