The Probability Flow in the Stock Market and Spontaneous Symmetry Breaking in Quantum Finance
2. The Black–Scholes Equation
- The spot interest rate r is constant.
- In order to create the hedged portfolio , the stock is infinitely divisible, and in addition it is possible to short sell the stock.
- The portfolio satisfies the no-arbitrage condition.
- The portfolio can be re-balanced continuously.
- There is no fee for transaction.
- 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 : Weak and Strong Field Regimes
- Weak-field regime:For this regime, from Equation (45), we obtainNote that this result is trivial for 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 (zero volatility). Zero volatility means zero fluctuations and then zero possibility of changing the prices in the market.
- Strong Field regime:In this regime, Equation (45) gives the following conditionWe have two solutions; one of them is the trivial and the second solution isHere again, this result is trivial if , 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 , is in the definition of the volatility , 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 and ; for Any Value of n, m
7.1.5. Weak-Field Condition and ; for Any Value of n, m
7.1.6. Weak and Strong-Field Approximation: and
7.1.7. Weak and Strong-Field Approximation: and
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Informed Consent Statement
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
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/math9212777Chicago/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