# Wind Put Barrier Options Pricing Based on the Nordix Index

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Barrier Options

#### 2.1. Barrier Options in Electricity Markets

#### 2.2. Barrier Options Based on Weather Indexes

#### 2.2.1. Wind Speed Modelling

#### 2.2.2. Nordix Index

#### 2.2.3. Put Barrier Option Pricing Structuring

## 3. Methodology

#### 3.1. Estimation of the ARFIMA Model for Wind Speeds

#### 3.2. Forecast of Daily Wind Speeds

#### 3.3. Structure of the Option Contract

^{3}, (ii) the area covered by the blades of the generator expressed in m

^{2}and (iii) the average of the cube of the wind speed. The area covered by the blades depends on the characteristics of the wind unit such as capacity, height of the tower, diameter, etc.

#### 3.4. Valuation of the European Wind Put Barrier Option

## 4. Case Study

#### 4.1. Estimation of the ARFIMA Model for Wind Speeds

#### 4.2. Forecast of Daily Wind Speeds

#### 4.3. Structure of the Option Contract

#### 4.4. Valuation of the European Wind Put Barrier Option

## 5. Results and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

- Alaton, P.; Djehiche, B.; Stillberger, D. On modelling and pricing weather derivatives. Appl. Math. Finance
**2002**, 9, 1–20. [Google Scholar] [CrossRef] - Alexandridis, A.K.; Zapranis, A. Weather Derivatives: Modeling and Pricing Weather-Related Risk; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Baillie, R.T. Long memory processes and fractional integration in econometrics. J. Econ.
**1996**, 73, 5–59. [Google Scholar] [CrossRef] - Benth, F.E.; Di Persio, L.; Lavagnini, S. Stochastic Modeling of Wind Derivatives in Energy Markets. Risks
**2018**, 6, 56. [Google Scholar] [CrossRef] [Green Version] - Benth, F.E. Pricing of Commodity and Energy Derivatives for Polynomial Processes. Mathmatics
**2021**, 9, 124. [Google Scholar] [CrossRef] - Benth, F.E.; Benth, J. Šaltytė Modeling and Pricing in Financial Markets for Weather Derivatives; World Scientific: Singapore, Singapore, 2012; Volume 17. [Google Scholar]
- Beran, J. Statistics for Long-Memory Processes; Chapman & Hall: Boca Raton, FL, USA, 1994. [Google Scholar]
- Berhane, T.; Shibabaw, A.; Awgichew, G.; Walelgn, A. Pricing of weather derivatives based on temperature by obtaining market risk factor from historical data. Model. Earth Syst. Environ.
**2020**, 1–14. [Google Scholar] [CrossRef] - Bludszuweit, H.; Dominguez-Navarro, J.A. A Probabilistic Method for Energy Storage Sizing Based on Wind Power Forecast Uncertainty. IEEE Trans. Power Syst.
**2011**, 26, 1651–1658. [Google Scholar] [CrossRef] - Botoş, H.M.; Ciumaş, C. The use of the Black-Scholes Model in the Field of Weather Derivatives. Procedia Econ. Financ.
**2012**, 3, 611–616. [Google Scholar] [CrossRef] [Green Version] - Boyle, C.F.; Haas, J.; Kern, J.D. Development of an irradiance-based weather derivative to hedge cloud risk for solar energy systems. Renew. Energy
**2020**, 164, 1230–1243. [Google Scholar] [CrossRef] - Brockett, P.L.; Wang, M.; Yang, C.; Zou, H. Portfolio Effects and Valuation of Weather Derivatives. Financ. Rev.
**2006**, 41, 55–76. [Google Scholar] [CrossRef] - Bui, V.-H.; Hussain, A.; Nguyen, T.-T.; Kim, H.-M. Multi-Objective Stochastic Optimization for Determining Set-Point of Wind Farm System. Sustainability
**2021**, 13, 624. [Google Scholar] [CrossRef] - Campbell, S.D.; Diebold, F.X. Weather Forecasting for Weather Derivatives. J. Am. Stat. Assoc.
**2005**, 100, 6–16. [Google Scholar] [CrossRef] [Green Version] - Cao, M.; Wei, J. Option market liquidity: Commonality and other characteristics. J. Financ. Mark.
**2010**, 13, 20–48. [Google Scholar] [CrossRef] - Caporin, M.; Preś, J. Modelling and forecasting wind speed intensity for weather risk management. Comput. Stat. Data Anal.
**2012**, 56, 3459–3476. [Google Scholar] [CrossRef] [Green Version] - Chiarella, C.; Kang, B.; Meyer, G.H. The evaluation of barrier option prices under stochastic volatility. Comput. Math. Appl.
**2012**, 64, 2034–2048. [Google Scholar] [CrossRef] [Green Version] - Cobos, N.G.; Arroyo, J.M.; Alguacil-Conde, N.; Street, A. Robust Energy and Reserve Scheduling Under Wind Uncertainty Considering Fast-Acting Generators. IEEE Trans. Sustain. Energy
**2018**, 10, 2142–2151. [Google Scholar] [CrossRef] - Contreras, J.; Rodríguez, Y.E. Incentives for wind power investment in Colombia. Renew. Energy
**2016**, 87, 279–288. [Google Scholar] [CrossRef] - Davis, M. Pricing weather derivatives by marginal value. Quant. Financ.
**2001**, 1, 305–308. [Google Scholar] [CrossRef] - Garcia, R.; Contreras, J.; Van Akkeren, M.; Garcia, J. A GARCH Forecasting Model to Predict Day-Ahead Electricity Prices. IEEE Trans. Power Syst.
**2005**, 20, 867–874. [Google Scholar] [CrossRef] - Gersema, G.; Wozabal, D. An equilibrium pricing model for wind power futures. Energy Econ.
**2017**, 65, 64–74. [Google Scholar] [CrossRef] - De la Guajira, G. Presentación de la Guajira. 2020. Available online: https://laguajira.gov.co/web/la-guajira/la-guajira.html (accessed on 11 February 2021).
- Göncü, A. Pricing temperature-based weather derivatives in China. J. Risk Financ.
**2011**, 13, 32–44. [Google Scholar] [CrossRef] - Gourieroux, C.; Jasiak, J. Autoregressive gamma processes. J. Forecast.
**2006**, 25, 129–152. [Google Scholar] [CrossRef] - Groll, A.; Cabrera, B.L.; Meyer-Brandis, T. A consistent two-factor model for pricing temperature derivatives. Energy Econ.
**2016**, 55, 112–126. [Google Scholar] [CrossRef] [Green Version] - Hamisultane, H. Extracting Information from the Market to Price the Weather Derivatives. SSRN Electron. J.
**2006**. [Google Scholar] [CrossRef] [Green Version] - Hell, P.; Meyer-Brandis, T.; Rheinländer, T. Consistent factor models for temperature markets. Int. J. Theor. Appl. Financ.
**2012**, 15. [Google Scholar] [CrossRef] - Hess, M. A New Model for Pricing Wind Power Futures. SSRN Electron. J.
**2019**. [Google Scholar] [CrossRef] - Hosking, J.R.M. Fractional differencing. Biometrika
**1981**, 68, 165–176. [Google Scholar] [CrossRef] - Hull, J. Fundamentals of Futures and Options Markets; Prentice Hall: Upper Saddle River, NJ, USA, 2011. [Google Scholar]
- Hurst, H.E. Long-Term Storage Capacity of Reservoirs. Trans. Am. Soc. Civ. Eng.
**1951**, 116, 770–799. Available online: http://cedb.asce.org/CEDBsearch/record.jsp?dockey=0292165 (accessed on 23 November 2020). - Jewson, S.; Zervos, M. The Black-Scholes Equation for Weather Derivatives. SSRN Electron. J.
**2003**. [Google Scholar] [CrossRef] [Green Version] - Leobacher, G.; Ngare, P. On Modelling and Pricing Rainfall Derivatives with Seasonality. Appl. Math. Financ.
**2011**, 18, 71–91. [Google Scholar] [CrossRef] - Leroy, A. Design and Valuation of Wind Derivatives; White Papers; Université Libre de Bruxelles: Bruxelles, Belgium, 2004. [Google Scholar]
- Li, P. Pricing weather derivatives with partial differential equations of the Ornstein–Uhlenbeck process. Comput. Math. Appl.
**2018**, 75, 1044–1059. [Google Scholar] [CrossRef] - Li, P.; Lu, X.; Zhu, S.-P. Pricing weather derivatives with the market price of risk extracted from the utility indifference valuation. Comput. Math. Appl.
**2020**, 79, 3394–3409. [Google Scholar] [CrossRef] - Salgueiro, A.M.; Tarrazon-Rodon, M.-A. Approaching rainfall-based weather derivatives pricing and operational challenges. Rev. Deriv. Res.
**2019**, 23, 163–190. [Google Scholar] [CrossRef] - Matsumoto, T.; Yamada, Y. Simultaneous hedging strategy for price and volume risks in electricity businesses using energy and weather derivatives. Energy Econ.
**2021**, 95, 105101. [Google Scholar] [CrossRef] - Meissner, G.; Burke, J. Can we use the Black-Scholes-Merton model to value temperature options? Int. J. Financial Mark. Deriv.
**2011**, 2, 298. [Google Scholar] [CrossRef] - Palma, W.; Wiley InterScience (Online service). Long-Memory Time Series: Theory and Methods; Wiley-Interscience: Hoboken, NJ, USA, 2007. [Google Scholar]
- Peng, M.W.; Sun, S.L.; Pinkham, B.; Chen, H. The Institution-Based View as a Third Leg for a Strategy Tripod. Acad. Manag. Perspect.
**2009**, 23, 63–81. [Google Scholar] [CrossRef] [Green Version] - Pérez-González, F.; Yun, H. Risk Management and Firm Value: Evidence from Weather Derivatives. J. Financ.
**2013**, 68, 2143–2176. [Google Scholar] [CrossRef] - Roscoe, A.J.; Ault, G. Supporting high penetrations of renewable generation via implementation of real-time electricity pricing and demand response. IET Renew. Power Gener.
**2010**, 4, 369. [Google Scholar] [CrossRef] [Green Version] - Gao, S.; He, Y.; Chen, H. Wind speed forecast for wind farms based on ARMA-ARCH model. In Proceedings of the 2009 International Conference on Sustainable Power Generation and Supply, Nanjing, China, 6–7 April 2009; pp. 1–4. [Google Scholar]
- Sowell, F. Modeling long-run behavior with the fractional ARIMA model. J. Monet. Econ.
**1992**, 29, 277–302. [Google Scholar] [CrossRef] - Tang, W.; Chang, S. A Semi-Lagrangian method for the weather options of mean-reverting Brownian motion with jump–diffusion. Comput. Math. Appl.
**2016**, 71, 1045–1058. [Google Scholar] [CrossRef] - Tol, R.S.J. Autoregressive Conditional Heteroscedasticity in daily wind speed measurements. Theor. Appl. Clim.
**1997**, 56, 113–122. [Google Scholar] [CrossRef] - UPME. Integración de las energías renovables no convencionales en Colombia. 2015. Available online: http://bdigital.upme.gov.co/handle/001/1311 (accessed on 23 November 2020).
- Wieczorek-Kosmala, M. Weather Risk Management in Energy Sector: The Polish Case. Energies
**2020**, 13, 945. [Google Scholar] [CrossRef] [Green Version] - WWEA. Proceedings of the 12th World Wind Energy Conference & WWEC 2013 Trade Fair, Havana, Cuba, 3–5 June 2013.
- Zhang, X.; Wang, X.; Wang, X. Exotic options bundled with interruptible electricity contracts. In Proceedings of the 2005 International Power Engineering Conference, Singapore, 29 November–2 December 2005; pp. 1–115. [Google Scholar]
- Xiao, Y.; Wang, X.; Wang, X.; Wu, Z. Trading wind power with barrier option. Appl. Energy
**2016**, 182, 232–242. [Google Scholar] [CrossRef] - XM. Descripción del Sistema Eléctrico Colombiano. 2019. Available online: http://www.xm.com.co/Paginas/Mercado-de-energia/descripcion-del-sistema-electrico-colombiano.aspx (accessed on 23 November 2020).
- Yamada, Y. Valuation and hedging of weather derivatives on monthly average temperature. J. Risk
**2007**, 10, 101–125. [Google Scholar] [CrossRef] [Green Version] - Yamada, Y. Simultaneous optimization for wind derivatives based on prediction errors. In Proceedings of the 2008 American Control Conference, Seattle, WA, USA, 11–13 June 2008; pp. 350–355. [Google Scholar]

**Figure 4.**Tick size, depending on the monetary quantity in the case of the Almirante Padilla station.

**Figure 5.**Comparison of electricity prices with the El Niño phenomenon, cost of generation of wind and wind put barrier option price (1 January 2016–31 March 2016).

**Figure 6.**Comparison of electricity prices without the El Niño phenomenon, cost of generation of wind and wind put barrier option price (1 April 2016–30 June 2016).

Statistic | Almirante Padilla Station |
---|---|

Minimum | 1.10 |

Maximum | 8.32 |

Daily average | 3.72 |

Standard deviation | 1.21 |

Variation coefficient | 3.08 |

Parameters | Almirante Padilla Station | |
---|---|---|

Coeffs. | p-Values | |

d | 0.3405484 | 0.000 |

ϕ_{1} | 0.3068596 | 0.000 |

ϕ_{3} | −0.0772107 | 0.079 |

Month in Summer Season | Monthly Error |
---|---|

First | 7.036% |

Second | 9.372% |

Third | 4.672% |

Characteristic | Parameters |
---|---|

Diameter (m) | 62 |

Covered area (m^{2}) | 3019 |

Blades (#) | 3 |

Height of tower (m) | 50–68 |

Characteristics | Parameters |
---|---|

Tick size for a capacity of 1300 MW | $0.00048 USD/kWh per Nordix |

Strike index | 215.797431 |

Tick value | 73.652589 |

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**MDPI and ACS Style**

Rodríguez, Y.E.; Pérez-Uribe, M.A.; Contreras, J.
Wind Put Barrier Options Pricing Based on the Nordix Index. *Energies* **2021**, *14*, 1177.
https://doi.org/10.3390/en14041177

**AMA Style**

Rodríguez YE, Pérez-Uribe MA, Contreras J.
Wind Put Barrier Options Pricing Based on the Nordix Index. *Energies*. 2021; 14(4):1177.
https://doi.org/10.3390/en14041177

**Chicago/Turabian Style**

Rodríguez, Yeny E., Miguel A. Pérez-Uribe, and Javier Contreras.
2021. "Wind Put Barrier Options Pricing Based on the Nordix Index" *Energies* 14, no. 4: 1177.
https://doi.org/10.3390/en14041177