Modeling Market Inefficiencies within a Single Instrument

TL;DR

This paper introduces a minimal two-variable hidden Markov model to describe market inefficiencies, explaining trend-following and mean-reversion behaviors, and connects the model to practical technical analysis tools like Bollinger bands.

Contribution

It develops a novel theoretical model incorporating market inefficiency and demonstrates its application to real market data, extending beyond traditional geometric Brownian motion models.

Findings

Market price trend-following when volatility exceeds that of the unobserved variable.

Mean-reversion occurs when the volatility of the unobserved variable is higher.

The risk premium relates to the deviation from the exponential moving average.

Abstract

In this paper, we propose a minimal model beyond geometric Brownian motion that aims to describe price actions with market inefficiency. From simple financial theory considerations, we arrive at a simple two-variable hidden Markovian time series model, with one of the variable entirely unobserved. Then, we analyze the simplest version of the model, using path integral and Green's function techniques from physics. We show that in this model, the inefficient market price is trend-following when the standard deviation of the log reasonable price ($\sigma$) is larger than that of the log market price ($\sigma'$), and mean-reversing when it is smaller. The risk premium is proportional to the difference between the current market price and the exponential moving average (EMA) of the past prices. This model thus provides a theoretical explanation how the EMA of the past price can directly affect future prices, i.e., the so-called ``Bollinger bands" in technical analyses. We then carry out a maximum likelihood estimate for the model parameters from the observed market price, by integrating out the reasonable price in Fourier space. Finally we analyze recent S\&P500 index data and see to what extent the real world data can be described by this simple model.