Log-ergodicity: A New Concept for Modeling Financial Markets

TL;DR

This paper introduces log-ergodicity, a new concept for modeling financial markets by transforming stochastic processes into mean-ergodic processes, enabling better pricing of contingent claims through empirical validation.

Contribution

It proposes a parametric operator that induces ergodic behavior in financial models, offering a novel method for pricing and analyzing market dynamics.

Findings

The new approach effectively captures ergodic behavior in financial data.

Empirical examples show improved pricing accuracy over traditional models.

Comparison indicates the method's robustness and practical applicability.

Abstract

Although financial models violate ergodicity in general, observing the ergodic behavior in the markets is not rare. Policymakers and market participants control the market behavior in critical and emergency states, which leads to some degree of ergodicity as their actions are intentional. In this paper, we define a parametric operator that acts on the space of positive stochastic processes, transforming a class of positive stochastic processes into mean-ergodic processes. With this mechanism, we extract the data regarding the ergodic behavior hidden in the financial model, apply it to mathematical finance, and establish a novel method for pricing contingent claims. We provide some empirical examples and compare the results with existing ones to demonstrate the efficacy of this new approach.