  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.
