# A Compound Up-and-In Call like Option for Wind Projects Pricing

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Up-and-In Option

#### 2.2. Compound Up-and-in-Like Option for Wind Projects Valuation

#### 2.3. Valuation of Vega

## 3. Case Study

#### Discussion of Results: Findings and Implications

## 4. Conclusions and Limitations

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Notes

1 | In this work, we assume that the project value can change over the time. This means that we consider changing revenues that are affected by various variables, such as unpredictable market demand or other reasons. |

2 | Notice that the risk-neutral approach is often required to be applied by the Solvency II Directive to valuate the best estimate of many financial products. This approach implies that all assets considered in the financial projection have a risk-free return independently from their credit characteristics and are consequently discounted with risk-free rates. Other valuations, differently from the risk-neutral world, may lead to arbitrage opportunities. Moreover, in some contexts, such as the insurance liabilities, it is proved that other approaches (e.g., the real world deflator) lead to similar results both at inception and for subsequent measurement. For more details, a reader can refer to Ouelega (2013), Jouini et al. (2005). |

3 | The power generation has been chosen considering likely wind speed circumstances. |

4 | The proportions of costs has been chosen considering the study of Bufalo et al. (2022). |

5 | A FiT is a measure to encourage the projects in renewable energy sources. This policy is often realized by providing producers with an above-market price for the renewable energy produced. It is just an assumption. It is not excluded that there could be other mechanisms according to the various governments of the world. |

6 | The amount of FiT has been chosen considering the study of Loncar et al. (2017). |

7 | The discount rate equal to 8% has been extrapolated by the study of Loncar et al. (2017). |

8 | |

9 | The value of ${B}_{2}$ satisfies the up-option condition ${B}_{2}>{V}_{0}$ (Gaudenzi and Lepellere 2006). |

10 | |

11 | The value of ${s}_{{t}_{0}}$ satisfies the up-option condition ${B}_{1}>{s}_{{t}_{0}}$. |

12 | This result is obtained by considering our assumptions and and it is not extended to all wind farm projects. Results could drastically change by varying data or by changing the assumptions. |

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**Figure 5.**Vega ${\mathcal{V}}_{{n}_{p}}\left(\sigma \right)$ computed for ${K}_{2}=0.4\times {10}^{6},T=14\phantom{\rule{0.166667em}{0ex}}Y,\phantom{\rule{0.166667em}{0ex}}\Delta =1/360,\phantom{\rule{0.166667em}{0ex}}\sigma =0.5,\phantom{\rule{0.166667em}{0ex}}{r}_{f}=0.04$ (see Section 3), when ${V}_{{n}_{p}}$ ranges in $[1\times {10}^{6},\phantom{\rule{0.166667em}{0ex}}28\times {10}^{6}]$.

**Figure 7.**Binomial tree of simple call up-and-in-like option where the black line represents ${B}_{2}$ and the red line represents ${B}_{1}$.

**Figure 8.**Binomial tree of compound call up-and-in-like option where the black line represents ${B}_{2}$ and the red line represents ${B}_{1}$.

Years | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Revenues | 3.14 | 3.14 | 3.14 | 3.14 | 3.14 | 3.14 | 3.14 | 3.14 | 3.14 | |||||

Costs | −2.4 | −17.2 | 0.4 |

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## Share and Cite

**MDPI and ACS Style**

Bufalo, M.; Di Bari, A.; Villani, G.
A Compound Up-and-In Call like Option for Wind Projects Pricing. *Risks* **2023**, *11*, 90.
https://doi.org/10.3390/risks11050090

**AMA Style**

Bufalo M, Di Bari A, Villani G.
A Compound Up-and-In Call like Option for Wind Projects Pricing. *Risks*. 2023; 11(5):90.
https://doi.org/10.3390/risks11050090

**Chicago/Turabian Style**

Bufalo, Michele, Antonio Di Bari, and Giovanni Villani.
2023. "A Compound Up-and-In Call like Option for Wind Projects Pricing" *Risks* 11, no. 5: 90.
https://doi.org/10.3390/risks11050090