# The Effect of Mean-Reverting Processes in the Pricing of Options in the Energy Market: An Arithmetic Approach

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## Abstract

**:**

## 1. Introduction

## 2. Setting

#### 2.1. Spot Price Dynamics

**Assumption**

**A1.**

#### 2.2. Swap-Price Value and Dynamics

**Theorem**

**1.**

**Theorem**

**2.**

## 3. Pricing a Call Option on the Swap

**Theorem**

**3.**

**Proof.**

#### Considerations on the Effect of the Jump Components’ Volatilities

## 4. The Effect of Mean-Reverting Jump Processes on the Call Option Price

**Lemma**

**1.**

**Theorem**

**4.**

**Proof.**

## 5. Numerical Illustration

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof of Theorem 1

**Proof.**

#### Appendix A.2. Proof of Theorem 2

**Proof.**

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**Figure 1.**A sample path of ${Y}_{j}$, $j=1,2$ (

**upper**), and of S (

**lower**). The jumps’ sizes of the ${Y}_{j}$s are normally distributed with zero mean and variance 2, while the jumps’ frequency is set ${\lambda}_{1}\equiv 2\equiv {\lambda}_{2}$. As for the plot of S, we have taken $\Lambda \equiv 0$, and, with regards to X, $\mu =0$ and $\sigma \equiv 0.6$.

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**MDPI and ACS Style**

Schmeck, M.D.; Schwerin, S.
The Effect of Mean-Reverting Processes in the Pricing of Options in the Energy Market: An Arithmetic Approach. *Risks* **2021**, *9*, 100.
https://doi.org/10.3390/risks9050100

**AMA Style**

Schmeck MD, Schwerin S.
The Effect of Mean-Reverting Processes in the Pricing of Options in the Energy Market: An Arithmetic Approach. *Risks*. 2021; 9(5):100.
https://doi.org/10.3390/risks9050100

**Chicago/Turabian Style**

Schmeck, Maren Diane, and Stefan Schwerin.
2021. "The Effect of Mean-Reverting Processes in the Pricing of Options in the Energy Market: An Arithmetic Approach" *Risks* 9, no. 5: 100.
https://doi.org/10.3390/risks9050100