  This paper presents a tractable model of non-linear dynamics of market
returns using a Langevin approach. Due to non-linearity of an interaction
potential, the model admits regimes of both small and large return
fluctuations. Langevin dynamics are mapped onto an equivalent quantum
mechanical (QM) system. Borrowing ideas from supersymmetric quantum mechanics
(SUSY QM), a parameterized ground state wave function (WF) of this QM system is
used as a direct input to the model, which also fixes a non-linear Langevin
potential. Using a two-component Gaussian mixture as a ground state WF with an
asymmetric double well potential produces a tractable low-parametric model with
interpretable parameters, referred to as the NES (Non-Equilibrium Skew) model.
Supersymmetry (SUSY) is then used to find time-dependent solutions of the model
in an analytically tractable way. Additional approximations give rise to a
final practical version of the NES model, where real-measure and risk-neutral
return distributions are given by three component Gaussian mixtures. This
produces a closed-form approximation for option pricing in the NES model by a
mixture of three Black-Scholes prices, providing accurate calibration to option
prices for either benign or distressed market environments, while using only a
single volatility parameter. These results stand in stark contrast to the most
of other option pricing models such as local, stochastic, or rough volatility
models that need more complex specifications of noise to fit the market data.
