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On the multiplicity of the martingale condition: Spontaneous symmetry breaking in Quantum Finance

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  • Ivan Arraut
  • Alan Au
  • Alan Ching-biu Tse

Abstract

We demonstrate that the martingale condition in the stock market can be interpreted as a vacuum condition when we express the financial equations in the Hamiltonian form. We then show that the symmetry under the changes of the prices is spontaneously broken in general and the symmetry under changes in the volatility, for the case of the Merton-Garman (MG) equation, is also spontaneously broken. This reproduces a vacuum degeneracy for the system. In this way, we find the conditions under which, the martingale condition can be considered to be a non-degenerate vacuum. This gives us a surprising connection between spontaneous symmetry breaking and the flow of information through the boundaries for the financial systems. Subsequently, we find an extended martingale condition for the MG equation, depending not only prices but also on the volatility and finally, we show what happens if we include additional non-derivative terms on the Black Scholes and on the MG equations, breaking then some other symmetries of the system spontaneously.

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  • Ivan Arraut & Alan Au & Alan Ching-biu Tse, 2020. "On the multiplicity of the martingale condition: Spontaneous symmetry breaking in Quantum Finance," Papers 2004.11270, arXiv.org.
    • Handle: RePEc:arx:papers:2004.11270
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    References listed on IDEAS

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    1. Arraut, Ivan & Au, Alan & Tse, Alan Ching-biu & Segovia, Carlos, 2019. "The connection between multiple prices of an Option at a given time with single prices defined at different times: The concept of weak-value in quantum finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 526(C).
    2. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
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    7. Baaquie, Belal E. & Corianò, Claudio & Srikant, Marakani, 2004. "Hamiltonian and potentials in derivative pricing models: exact results and lattice simulations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 334(3), pages 531-557.
    8. Jones, E. Philip, 1984. "Option arbitrage and strategy with large price changes," Journal of Financial Economics, Elsevier, vol. 13(1), pages 91-113, March.
    9. Ivan Arraut & Alan Au & Alan Ching-biu Tse & Carlos Segovia, 2019. "The connection between multiple prices of an Option at a given time with single prices defined at different times: The concept of weak-value in quantum finance," Papers 1905.05813, arXiv.org.
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