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

TL;DR

This paper explores the martingale condition in financial markets through the lens of spontaneous symmetry breaking, revealing deep connections between symmetry properties, vacuum states, and information flow in quantum finance models.

Contribution

It introduces a Hamiltonian framework for the martingale condition, demonstrating spontaneous symmetry breaking in the Merton-Garman equation and extending the martingale condition to include volatility.

Findings

Martingale condition as a vacuum state in Hamiltonian form

Spontaneous symmetry breaking occurs in price and volatility changes

Extended martingale condition includes volatility dependence

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.