# Stochastic Modeling of Wind Derivatives in Energy Markets

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## Abstract

**:**

## 1. Introduction

## 2. Spot Price Model

#### Calibration

## 3. Models for Wind

#### Calibration

## 4. Income for a Wind Energy Company

#### 4.1. Normal Inverse Gaussian Approximation

#### 4.2. Income Formulas

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

## 5. Quanto Options

#### 5.1. Contract Structure

**Lemma**

**1.**

**Proof.**

#### 5.2. Futures Dynamics

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

#### 5.3. Option Price

**Theorem**

**1.**

**Proof.**

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of f_{Z}

## References

- Applebaum, David. 2009. Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press. [Google Scholar] [CrossRef]
- Asmussen, Søren, and Jan Rosiński. 2011. Approximations of Small Jumps of LéVy Processes with a View towards Simulation. Journal of Applied Probability 38: 432–93. [Google Scholar]
- Benth, Fred E., and Jūratė Š. Benth. 2004. The Normal Inverse Gaussian distribution and spot price modeling in energy markets. International Journal of Theoretical and Applied Finance 7: 177–92. [Google Scholar] [CrossRef]
- Benth, Fred E., and Jūratė Š. Benth. 2012. Modeling and Pricing in Financial Markets for Weather Derivatives. Singapore: World Scientific. [Google Scholar]
- Benth, Fred E., Jūratė Š. Benth, and Steen Koekebakker. 2008. Stochastic Modelling of Electricity and Related Markets. Singapore: World Scientific. [Google Scholar]
- Benth, Fred E., Nina Lange, and Tor Åge Myklebust. 2013. Pricing and hedging quanto options in energy markets. Journal of Energy Markets 8: 1–35. [Google Scholar] [CrossRef]
- Betz, Albert. 1966. Introduction to the Theory of Flow Machines. Oxford: Pergamon Press. [Google Scholar]
- Burton, Tony. 2011. Wind Energy: Handbook. Hoboken: John Wiley & Sons. [Google Scholar]
- Folland, Gerald B. 2007. Real Analysis: Modern Techniques and Their Applications, 2nd ed. Hoboken: Wiley. [Google Scholar]
- Jørgensen, Bent. 1982. Statistical Properties of the Generalised Inverse Gaussian Distribution. Lecture Notes in Statistics. New York: Springer, vol. 9. [Google Scholar]
- Schwartz, Eduardo S. 1997. The stochastic behavior of commodity prices: Implication for valuation and hedging. The Journal of Finance 52: 923–73. [Google Scholar] [CrossRef]
- Tankov, Peter, and Rama Cont. 2004. Financial Modeling with Jump Processes. CRC Financial Mathematics Series; Boca Raton: CRC Press. [Google Scholar]
- Villanueva, Daniel, and Feijoó Andrés. 2010. Wind power distributions: A review of their applications. Renewable and Sustainable Energy Reviews 14: 1490–95. [Google Scholar] [CrossRef]
- Wystup, Uwe. 2010. Quanto Options. In Encyclopedia of Quantitative Finance. Hoboken: Wiley. [Google Scholar] [CrossRef]

1 | We use the R function stl from package stats. |

2 | We use the R function arima from package stats. |

3 | We use the R function nigFit from package fBasics. |

4 | We use the R function boot from the package boot. |

5 | We use the R function lm from package stats. |

6 | We use the R function stl from package stats. |

7 | We use the R function arima from package stats. |

8 | We use the R function integrate from package stats, together with besselK from package base. |

**Figure 1.**(

**a**) QQ plot for the Gaussian distribution; (

**b**) QQ plot for the normal inverse Gaussian (NIG) distribution.

**Figure 4.**Power production time series (bold line) and power production obtained by the Betz law (dotted line), with respect to the first one hundred days.

**Figure 5.**The estimated price functions, ${f}_{P}$ (bold line) and ${f}_{W}$ (dotted line), with respect to the first one hundred days.

Estimate | Confidence interval (95%) | Estimate | Confidence interval (95%) | ||
---|---|---|---|---|---|

${\widehat{\theta}}_{S}$ | $1.23$ | $[1.08,1.42]$ | ${\widehat{\sigma}}_{\epsilon}$ | $9.87\times {10}^{-4}$ | $[9.13,10.5]\times {10}^{-4}$ |

$\widehat{\alpha}$ | $4.41$ | $[3.49,5.33]$ | $\widehat{\mu}$ | $2.61\times {10}^{-2}$ | $[1.24,3.99]\times {10}^{-2}$ |

$\widehat{\beta}$ | $-7.80\times {10}^{-1}$ | $[-12.8,-2.77]\times {10}^{-1}$ | $\widehat{\delta}$ | $1.53\times {10}^{-1}$ | $[1.33,1.73]\times {10}^{-1}$ |

$m=0$ |

$M=+\infty $ |

$\widehat{h}=8.42\pm 0.38\times {10}^{6}$ |

**Table 3.**Estimated parameters from the calibration of the wind speed and wind power production time series.

Estimate | Confidence Interval (95%) | Estimate | Confidence Interval (95%) | ||
---|---|---|---|---|---|

${\widehat{\theta}}_{W}$ | $7.97\times {10}^{-1}$ | $[7.00,9.04]\times {10}^{-1}$ | ${\widehat{\theta}}_{P}$ | $5.90\times {10}^{-1}$ | $[5.16,6.71]\times {10}^{-1}$ |

${\widehat{\sigma}}_{W}^{2}$ | $1.15\times {10}^{-1}$ | $[1.07,1.24]\times {10}^{-1}$ | ${\widehat{\sigma}}_{P}^{2}$ | $4.93\times {10}^{-1}$ | $[4.59,5.31]\times {10}^{-1}$ |

${t}_{0}$: | 31/12/2014 |

${\tau}_{1}$: | 01/01/2015 |

${\tau}_{2}$: | 31/12/2015 |

$\mathit{\rho}$ | Three-Year Calibration | Two-Year Calibration | One-Year Calibration | |||
---|---|---|---|---|---|---|

$-0.9$ | $15.25$ | $(-96.73\%)$ | $270.76$ | $(-95.94\%)$ | $419.21$ | $(-95.56\%)$ |

$-0.5$ | $163.15$ | $(-65.02\%)$ | $2459.24$ | $(-63.11\%)$ | $3559.76$ | $(-62.28\%)$ |

0 | $466.43$ | $(0\%)$ | $6666.03$ | $(0\%)$ | $9437.49$ | $(0\%)$ |

$0.5$ | $882.27$ | $(89.15\%)$ | $12,253.93$ | $(83.83\%)$ | $17,143.00$ | $(81.65\%)$ |

$0.9$ | $1309.76$ | $(180.80\%)$ | $17,920.39$ | $(168.83\%)$ | $24,911.99$ | $(163.967\%)$ |

Three-Year Calibration | Two-Year Calibration | One-Year Calibration |
---|---|---|

$\widehat{\rho}=4.53\times {10}^{-1}$ | $\widehat{\rho}=8.58\times {10}^{-1}$ | $\widehat{\rho}=8.32\times {10}^{-1}$ |

© 2018 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**

Benth, F.E.; Di Persio, L.; Lavagnini, S.
Stochastic Modeling of Wind Derivatives in Energy Markets. *Risks* **2018**, *6*, 56.
https://doi.org/10.3390/risks6020056

**AMA Style**

Benth FE, Di Persio L, Lavagnini S.
Stochastic Modeling of Wind Derivatives in Energy Markets. *Risks*. 2018; 6(2):56.
https://doi.org/10.3390/risks6020056

**Chicago/Turabian Style**

Benth, Fred Espen, Luca Di Persio, and Silvia Lavagnini.
2018. "Stochastic Modeling of Wind Derivatives in Energy Markets" *Risks* 6, no. 2: 56.
https://doi.org/10.3390/risks6020056