Lattice $\phi^{4}$ field theory as a multi-agent system of financial markets

TL;DR

This paper models financial markets using a $^{4}$ lattice field theory with multi-agent dynamics, capturing key market features like fat tails and volatility clustering, and demonstrating its empirical relevance to FTSE 100 data.

Contribution

It introduces a novel $^{4}$ lattice field theory framework for financial markets, incorporating competing interactions and continuous agent states, advancing beyond Ising-based models.

Findings

Reproduces stylized facts of financial markets such as fat tails and volatility clustering.

Numerically verified alignment with FTSE 100 market data.

Shows continuous degrees of freedom enhance modeling capacity.

Abstract

We introduce a $\phi^{4}$ lattice field theory with frustrated dynamics as a multi-agent system to reproduce stylized facts of financial markets such as fat-tailed distributions of returns and clustered volatility. Each lattice site, represented by a continuous degree of freedom, corresponds to an agent experiencing a set of competing interactions which influence its decision to buy or sell a given stock. These interactions comprise a cooperative term, which signifies that the agent should imitate the behavior of its neighbors, and a fictitious field, which compels the agent instead to conform with the opinion of the majority or the minority. To introduce the competing dynamics we exploit the Markov field structure to pursue a constructive decomposition of the $\phi^{4}$ probability distribution which we recompose with a Ferrenberg-Swendsen acceptance or rejection sampling step. We then verify numerically that the multi-agent $\phi^{4}$ field theory produces behavior observed on empirical data from the FTSE 100 London Stock Exchange index. We conclude by discussing how the presence of continuous degrees of freedom within the $\phi^{4}$ lattice field theory enables a representational capacity beyond that possible with multi-agent systems derived from Ising models.