A Quantum Oscillator Model of Stock Markets

Abstract

Classical oscillators have long been used to model financial time series, but suffer from the drawback that the underlying system does not behave like a mechanical spring: there are no regular oscillations, and price is not a well-defined mechanical quantity but always has a degree of uncertainty. In recent years a number of authors have attempted to address these problems by basing their models on quantum harmonic oscillators, which always have a non-zero volatility, however the role of other features that are characteristic of quantum systems, such as discrete energy levels, has not been clear. This paper develops a quantum model in which the energy level corresponds to an integer number of transactions. The model is derived by quantizing entropic forces which represent the intentions of buyers and sellers to transact as a function of price. It is shown that the model captures the non-Gaussian nature of financial statistics, and correctly predicts empirical phenomena including the square-root law of price impact, along with its associated variance.