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

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.

💡 Research Summary

The paper proposes a novel interpretation of the martingale condition in financial markets as a vacuum condition by recasting the Black‑Scholes (BS) and Merton‑Garman (MG) pricing equations into Hamiltonian form. Starting from the standard stochastic differential equations for the underlying asset price and, in the MG case, its stochastic volatility, the authors construct risk‑neutral portfolios that eliminate the stochastic terms, leading to the familiar BS partial differential equation and the more complex MG equation. By applying logarithmic transformations (S = eˣ, V = eʸ) the time evolution of the option price can be written as a Schrödinger‑type equation ∂ψ/∂t = Ĥψ, where Ĥ is a non‑Hermitian operator for BS and a two‑dimensional operator containing mixed derivatives for MG.

The central claim is that the martingale condition—i.e., the requirement that the discounted option price be a martingale under the risk‑neutral measure—is mathematically equivalent to demanding that the system be in its “vacuum” state. In quantum‑mechanical language, a vacuum is a state annihilated by the Hamiltonian’s symmetry generators. The authors identify two continuous symmetries: (i) translation in the log‑price variable (price shift symmetry) and (ii) translation in the log‑volatility variable (volatility‑scale symmetry). They argue that, in general, these symmetries are spontaneously broken because the vacuum does not remain invariant under the corresponding generators. Consequently, the vacuum is degenerate (multiple martingale states exist), which they interpret as a multiplicity of admissible risk‑neutral dynamics.

A key insight is that only for special parameter combinations—such as r = σ²/2 in the BS case, or λ − β = 0 together with specific values of the correlation ρ in the MG case—does the vacuum become non‑degenerate. In these “fine‑tuned” regimes the symmetries are restored, and the authors associate this restoration with the absence of information flow across market boundaries (e.g., no arbitrage opportunities, no cross‑market price leakage). Thus, they link spontaneous symmetry breaking to the flow of information in financial systems.

The paper further extends the martingale condition to include volatility as an explicit argument of the option price, yielding an “extended martingale condition” for the MG equation. This extension requires the elimination of stochastic volatility terms, which is achieved only when the market price of volatility risk (parameter β) vanishes and when there is no external white‑noise contribution from the volatility process.

Finally, the authors introduce non‑derivative (potential) terms into the Hamiltonians. By adding two free parameters they construct Hermitian potentials that preserve the martingale condition while breaking additional symmetries. In the limit where kinetic terms become negligible, the BS and MG dynamics converge, suggesting that the potential dominates the market’s behavior in certain regimes (e.g., during sudden shocks or policy interventions). The analysis shows that these potentials can re‑induce vacuum degeneracy, leading to further spontaneous symmetry breaking.

Overall, the work offers a conceptual bridge between quantum field theory concepts (vacuum, spontaneous symmetry breaking, degeneracy) and financial mathematics. While the analogy is intellectually appealing, the paper lacks rigorous quantitative validation: there is no empirical calibration, no numerical simulation of the proposed extended martingale dynamics, and the economic interpretation of the added potentials remains vague. The reliance on formal analogies rather than testable predictions limits the immediate applicability of the results. Nonetheless, the study opens an intriguing line of inquiry into how symmetry principles might inform the structure of risk‑neutral pricing and the role of information flow in market dynamics, suggesting several avenues for future research, including empirical testing of the degenerate versus non‑degenerate regimes, and a deeper exploration of the economic meaning of the introduced potential terms.