# The Role of the Volatility in the Option Market

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## Abstract

**:**

## 1. Introduction

## 2. The Option Market

## 3. The Black–Scholes Equation

#### 3.1. Black–Scholes Hamiltonian Formulation

#### 3.2. The Evolution of Probability in the Black–Scholes Equation

#### 3.3. Black–Scholes Pricing Kernel

#### 3.4. The Influence of the Non-Conservation of the Probability in the Prices of the Stocks under the Black–Scholes Formulation

## 4. The Merton–Garman Equation

#### 4.1. Derivation of the Merton–Garman Equation

#### 4.2. Hamiltonian Form of the Merton–Garman Equation

#### 4.3. The Merton–Garman Kernel

#### 4.4. The Flow of Probability in the MG Case

## 5. Spontaneous Symmetry Breaking from the Black–Scholes Hamiltonian

#### 5.1. The Martingale Condition as a Vacuum State

#### 5.2. Broken Symmetries in the Financial Equations

#### 5.3. Spontaneous Symmetry Breaking: Symmetries under Changes of Prices

## 6. Local Equivalence between the Black–Scholes and the Merton–Garman Equation

#### The Merton–Garman Equation Emerging from the Black–Scholes Equation

## 7. The Higgs Mechanism in Quantum Finance: The Dynamical Origin of the Volatility

#### The Dynamical Origin of the Volatility

## 8. Solutions for the Black-Scholes Equation

#### Implied Volatility and Making Decisions Based on the Volatility Estimation

## 9. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Example of the possible profits received by the holder of a European call option. The option price is 5 USD, and the strike price is 42 USD. This is an example of a zero-sum game. Figure taken from [2].

**Figure 2.**Example of the possible profits received by the writer of a European call option. The option price is 5 USD and the strike price is 42 USD. This is an example of a zero-sum game. Figure taken from [2].

**Figure 3.**Example of the possible profits received by the holder of a European put option. The option price is 3 USD and the strike price is 50 USD. This is an example of a zero-sum game. Figure taken from [2].

**Figure 4.**Example of the possible profits received by the writer of a European put option. The option price is 3 USD and the strike price is 50 USD. This is an example of a zero-sum game. Figure taken from [2].

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**MDPI and ACS Style**

Arraut, I.; Lei, K.-I.
The Role of the Volatility in the Option Market. *AppliedMath* **2023**, *3*, 882-908.
https://doi.org/10.3390/appliedmath3040047

**AMA Style**

Arraut I, Lei K-I.
The Role of the Volatility in the Option Market. *AppliedMath*. 2023; 3(4):882-908.
https://doi.org/10.3390/appliedmath3040047

**Chicago/Turabian Style**

Arraut, Ivan, and Ka-I Lei.
2023. "The Role of the Volatility in the Option Market" *AppliedMath* 3, no. 4: 882-908.
https://doi.org/10.3390/appliedmath3040047