Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Dimitrios Bachtis

Abstract

We introduce a $φ^{4}$ 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 $φ^{4}$ probability distribution which we recompose with a Ferrenberg-Swendsen acceptance or rejection sampling step. We then verify numerically that the multi-agent $φ^{4}$ 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 $φ^{4}$ lattice field theory enables a representational capacity beyond that possible with multi-agent systems derived from Ising models.

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

Abstract

We introduce a 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 probability distribution which we recompose with a Ferrenberg-Swendsen acceptance or rejection sampling step. We then verify numerically that the multi-agent 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 lattice field theory enables a representational capacity beyond that possible with multi-agent systems derived from Ising models.

Paper Structure

This paper contains 7 sections, 12 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Constructive decomposition of the $\phi^{4}$ theory
  3. Multi-agent $\phi^{4}$ theory as a model of financial markets
  4. Stylized facts of $\phi^{4}$ financial markets
  5. Conclusions
  6. Data Availability Statement
  7. Acknowledgements

Figures (5)

  • Figure 1: Configurations of the multi-agent $\phi^{4}$ theory for $20500$, $28311$, and $30000$ Monte Carlo sweeps from left to right, respectively. The configurations are obtained for lattice size $L=64$ and for couplings $m^{2}=-3.0$, $\lambda=0.7$, and $a=5.0$. White color corresponds to the largest positive value of the fields, black color to the largest negative value, and shades of grey represent the intermediate values. We observe transitions from a metastable state of $20500$ sweeps, to a phase of chaotic dynamics in $28311$ sweeps, and a subsequent metastable state of $30000$ sweeps. Chartists dominate in the state of chaotic dynamics, and metastable states indicate the emergence of an expectation bubble.
  • Figure 2: Returns $r_{\phi^{4}}(t)$ as obtained from the multi-agent $\phi^{4}$ theory (top) and log-returns $r(t)$ from the FTSE 100 index (bottom) versus time $t$. We observe on both cases the presence of intermittent phases with large fluctuations.
  • Figure 3: Histograms of returns $r_{\phi^{4}}(t)$ as obtained from the multi-agent $\phi^{4}$ theory (top) and histograms of log-returns $r(t)$ as obtained from the FTSE 100 index (bottom) versus the value of returns. We observe the emergence of fat-tailed probability distributions. The y axis is logarithmic.
  • Figure 4: Complementary cumulative distribution histograms of the absolute value $|r_{\phi^{4}}(t)|$ on logarithmic scales. Both axes are logarithmic.
  • Figure 5: Autocorrelation function for the absolute returns $|r_{\phi^{4}}|(t)$ as calculated on the multi-agent $\phi^{4}$ theory. Both axes are logarithmic.