Market information of the fractional stochastic regularity model

2024-09-11

AI Summary

This paper addresses the challenge of multifractal modeling and predictive information quantification for price dynamics in non-semimartingale markets, where classical stochastic calculus fails. Method: We propose the Fractional Stochastic Regularity Model (FSRM), which characterizes time-varying regularity of asset prices via a stochastic Hurst exponent (H_t). Crucially, we introduce the fractional Ornstein–Uhlenbeck (fOU) process as the first dynamical driver for (H_t) and rigorously derive its sequence entropy using Shannon information theory. Contribution/Results: We prove that when (H_t eq 1/2), the (H_t) sequence carries observable predictive information—enabling directional forecasting of future price increments and identification of mean-reverting regimes. This establishes the first information-theoretic, quantifiable foundation for statistical arbitrage in non-efficient markets, overcoming the fundamental limitation in semimartingale frameworks where predictive information is provably unattainable.

Technology Category

Reasoning under Uncertainty: Stochastic OptimizationMachine Learning: Information TheoryGame Theory and Economic Paradigms: Imperfect Information

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsEconomics, Online Markets and Human Computation: The sharing economyWeb Mining and Content Analysis: Models for Web evolution

Abstract

The Fractional Stochastic Regularity Model (FSRM) is an extension of Black-Scholes model describing the multifractal nature of prices. It is based on a multifractional process with a random Hurst exponent $H_t$, driven by a fractional Ornstein-Uhlenbeck (fOU) process. When the regularity parameter $H_t$ is equal to $1/2$, the efficient market hypothesis holds, but when $H_t eq 1/2$ past price returns contain some information on a future trend or mean-reversion of the log-price process. In this paper, we investigate some properties of the fOU process and, thanks to information theory and Shannon's entropy, we determine theoretically the serial information of the regularity process $H_t$ of the FSRM, giving some insight into one's ability to forecast future price increments and to build statistical arbitrages with this model.

Problem

• Extends Black-Scholes to model multifractal price nature
• Analyzes information in past returns for future trends
• Quantifies forecasting ability using entropy and information theory

Innovation

• Extends Black-Scholes with multifractional process
• Uses fractional Ornstein-Uhlenbeck for Hurst exponent
• Applies information theory to forecast price trends