# The Probability Flow in the Stock Market and Spontaneous Symmetry Breaking in Quantum Finance

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## Abstract

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## 1. Introduction

**the fundamental theorem of finance**[6,7]. In a previous paper, some of the authors formulated the spontaneous symmetry breaking in quantum finance [8]. Within this formalism, the martingale condition (state) appears as a vacuum condition which becomes degenerate under some circumstances [6,8]. In the most general sense, we have a multiplicity of vacuum states for the Black–Scholes (BS) and the Merton–Garman (MG) cases. This vacuum degeneracy is related to the symmetries under change of prices and the symmetry under changes in volatility for the MG case. Therefore, we cannot conclusively say that these symmetries are spontaneously broken [9,10] (their generators do not annihilate the vacuum state). The perfect vacuum condition is only recovered for some combination of parameters, for which the ground state (martingale state) is unique. For the regime analyzed in [8], it was possible to find a natural connection between the flow of information in the system and spontaneous symmetry breaking. Some suspects about this connection were mentioned in [11]. In [8], an extended version of the martingale state was also proposed, which includes not only prices but also the volatility as a variable. A degenerate vacuum condition again appears in these situations, with the corresponding symmetries spontaneously broken. Another interesting situation was analyzed in [8]. It corresponds to an ideal case where, for the MG and the BS case, additional non-derivative terms are included, such that the martingale condition is still satisfied. These potential terms are different to those analyzed before by some other authors [6,12]. The potential terms added in [8] are also different to the standard case studied in [13], where a double slit constraint was explored. Indeed, the potential terms analyzed in [8] correspond to collective decisions or collective behavior. Although the results obtained in [8] are correct, they were always focused on the analysis of situations, ignoring the kinetic contributions in the neighborhood of the martingale state. This corresponds to the strong-field regime to be defined in this paper. This regime, although valid, does not represent the whole scenario. For this reason, in this paper we also explore other regimes by extending the results obtained in [8]. We then explore situations where the kinetic terms behave as additional potential terms (weak-field regime and intermediate regimes). This certainly happens in reality when we consider the martingale condition in the BS equation as well as for the MG equation. Indeed, the kinetic terms cannot be ignored in general around the neighborhood of the martingale state when we consider its standard definition. Still, we can say that there are regimes where the results obtained in [8] are valid and regimes where the additional results obtained in this paper represent a more accurate picture of the reality. Finally, in this paper we fully connect the notions of spontaneous symmetry breaking and flow of information (probability) in the stock market. The paper is organized as follows: In Section 2, we explain the BS equation and we derive its Hamiltonian form. In Section 3, we explain the MG equation and again we derive its Hamiltonian. In Section 4, we illustrate the standard definition of martingale and then we justify why the martingale state (condition) can be perceived as a vacuum state in Quantum Finance. In Section 5, we evaluate the conditions under which additional terms in the potential can be included, such that the martingale condition is still preserved. In Section 6, we make some extensions of the results obtained in [8], in connection with the spontaneous symmetry breaking under changes of prices and volatility. In this paper we analyze regimes which were ignored in [8]. This means that in this paper we explore both weak and strong regimes for the quantum field, representing the series expansion of the martingale state. The martingale state then corresponds to the vacuum condition for this quantum field, denoted by ${\varphi}_{vac}$. Its explicit result depends on the regime under analysis, as well as the order of the series expansion when we express the Hamiltonian as a function of quantum fields. In Section 7, we analyze the details about the extended martingale condition, which depends not only on the prices of the options but also on the stochastic volatility. This is the section where the explicit results for the MG case are analyzed. Finally, in Section 8, we conclude.

## 2. The Black–Scholes Equation

- (1)
- The spot interest rate r is constant.
- (2)
- In order to create the hedged portfolio $\mathsf{\Pi}$, the stock is infinitely divisible, and in addition it is possible to short sell the stock.
- (3)
- The portfolio satisfies the no-arbitrage condition.
- (4)
- The portfolio $\mathsf{\Pi}$ can be re-balanced continuously.
- (5)
- There is no fee for transaction.
- (6)
- The stock price has a continuous evolution.

#### Black–Scholes Hamiltonian Formulation

## 3. The Merton–Garman Equation: Preliminaries and Derivation

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

#### 3.2. Hamiltonian form of the Merton–Garman Equation

## 4. The Martingale Condition in Finance

#### The Martingale Condition as a Vacuum Condition for a Hamiltonian

## 5. Non-Derivative Terms Introduced in the Financial Hamiltonians

## 6. Deeper Analysis for the Black–Scholes and the Merton–Garman Equation

#### 6.1. Standard Definition of Spontaneous Symmetry Breaking

#### 6.2. Reinterpretation of the Martingale Condition

#### Vacuum Conditions for $n\ne 0,1$: Weak and Strong Field Regimes

- (1)
**Weak-field regime:**${\varphi}^{n}<<\varphi $For this regime, from Equation (45), we obtain$$-\frac{{\sigma}^{2}}{2}(n-1)+\left(\frac{1}{2}{\sigma}^{2}-r\right)\varphi \approx 0,$$$${\varphi}_{vac}\approx \frac{{\sigma}^{2}}{{\sigma}^{2}-2r}(n-1).$$Note that this result is trivial for $n=1$ for any combination of parameters. This means that in these situations, again random fluctuations cannot change the prices in the market. The case (51) is also trivial when ${\sigma}^{2}=0$ (zero volatility). Zero volatility means zero fluctuations and then zero possibility of changing the prices in the market.- (2)
**Strong Field regime:**${\varphi}^{n}>>\varphi $In this regime, Equation (45) gives the following condition$$r{\varphi}^{2}+\left(\frac{1}{2}{\sigma}^{2}-r\right)n\varphi \approx 0.$$We have two solutions; one of them is the trivial ${\varphi}_{vac}\approx 0$ and the second solution is$${\varphi}_{vac}=\left(1-\frac{1}{2}\frac{{\sigma}^{2}}{r}\right)n.$$Here again, this result is trivial if ${\sigma}^{2}=2r$, which is again the condition for the Hamiltonian to be Hermitian and then preserve information. Therefore, it seems that this is the combination of parameters which avoids changes of information due to random fluctuations. Before going to the next section, we have to remark that the results obtained for the BS case will be valid for the MG case because the standard definition of martingale is independent of the stochastic volatility. The only change to be done for the solutions inside the MG equation, when we consider ${\varphi}_{vac}$, is in the definition of the volatility ${\sigma}^{2}={e}^{y}$, with y representing the variable connected to the stochastic volatility.

## 7. A More General Condition for the Symmetry Breaking in the Merton Garman Equation

#### 7.1. The Extended Martingale Condition and the Flow of Information

#### 7.1.1. Extended Martingale with n = 0 and m = 1

#### 7.1.2. Extended Martingale with n = 1 and m = 0

#### 7.1.3. Extended martingale with n = 1 and m = 1

#### 7.1.4. Strong Field Condition ${\varphi}_{x}^{n}>>{\varphi}_{x}$ and ${\varphi}_{y}^{m}>>{\varphi}_{y}$; for Any Value of n, m

#### 7.1.5. Weak-Field Condition ${\varphi}_{x}^{n}<<{\varphi}_{x}$ and ${\varphi}_{y}^{m}<<{\varphi}_{y}$; for Any Value of n, m

#### 7.1.6. Weak and Strong-Field Approximation: ${\varphi}_{x}^{n}>>{\varphi}_{x}$ and ${\varphi}_{y}^{m}<<{\varphi}_{y}$

#### 7.1.7. Weak and Strong-Field Approximation: ${\varphi}_{x}^{n}<<{\varphi}_{x}$ and ${\varphi}_{y}^{m}>>{\varphi}_{y}$

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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Arraut, I.; Lobo Marques, J.A.; Gomes, S. The Probability Flow in the Stock Market and Spontaneous Symmetry Breaking in Quantum Finance. *Mathematics* **2021**, *9*, 2777.
https://doi.org/10.3390/math9212777

**AMA Style**

Arraut I, Lobo Marques JA, Gomes S. The Probability Flow in the Stock Market and Spontaneous Symmetry Breaking in Quantum Finance. *Mathematics*. 2021; 9(21):2777.
https://doi.org/10.3390/math9212777

**Chicago/Turabian Style**

Arraut, Ivan, João Alexandre Lobo Marques, and Sergio Gomes. 2021. "The Probability Flow in the Stock Market and Spontaneous Symmetry Breaking in Quantum Finance" *Mathematics* 9, no. 21: 2777.
https://doi.org/10.3390/math9212777