An Ising Model Approach to Financial Markets - Market Crashes as Phase Transitions (Warwick Mathematical Finance Magazine Submission)

Overview

This project investigates financial market dynamics through the framework of statistical mechanics, using the Ising model as a minimal agent-based system.

By modelling traders as interacting spins on a lattice, the project explores how local interactions can give rise to emergent macroscopic behaviour, including:

• Large-scale coordination (market trends)
• Abrupt regime shifts (crash-like behaviour)
• Heavy-tailed return distributions

The central aim is to assess whether key stylised facts of financial markets can be reproduced near criticality.

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Model Description

We consider a 2D Ising model on a square lattice with periodic boundary conditions:

• Spins: $s_i \in {-1, +1}$
• Hamiltonian:

$$H = -J \sum_{\langle i,j \rangle} s_i s_j - h \sum_i s_i$$

• Dynamics:
  • Metropolis-Hastings updates (standard Ising)
  • Heat-bath dynamics with Bornholdt-type feedback

Financial Interpretation

Physics Quantity	Financial Analogy
Spin $s_i$	Trader decision (buy/sell)
Magnetisation $m_t$	Aggregate market sentiment
Temperature $T$	Market uncertainty
Coupling $J$	Interaction strength between agents
Phase transition	Market instability / crashes

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Bornholdt Extension

To introduce feedback effects, we include a global term in the effective field:

$$h_i^{\text{eff}} = J \sum_{\text{nn}} s_j - \alpha s_i |m|$$

This term penalises alignment with the global state when the market is over-extended, generating:

• Intermittent switching
• Increased volatility
• More realistic return dynamics

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Synthetic Returns

We define a toy return series derived from the change in magnetisation:

$$r_t = m_{t+1} - m_t$$

This enables analysis of return distributions, volatility dynamics, and autocorrelation structures.

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Key Results

• Near the critical temperature ($T_c$):
  • Large fluctuations in magnetisation.
  • Fat-tailed return distributions (non-Gaussian).
• Bornholdt dynamics:
  • Increased intermittency.
  • Richer volatility structure.
• Limitations:
  • Limited long-memory effects.
  • Weak volatility clustering compared to real-world high-frequency data.

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Repository Structure

ising-market-model/
│
├── notebooks/          # Exploratory analysis and experiments
│
├── src/
│   ├── ising.py        # Core Ising model and dynamics
│   ├── analysis.py     # Statistical analysis of simulation output
│   ├── master.py       # Experiment orchestration (parameter sweeps)
│   └── plotting.py     # Visualisation tools
│
├── figures/            # Generated plots for analysis and paper
├── paper/              # Research article (PDF + LaTeX source)
│
├── requirements.txt
└── README.md

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Installation

Clone the repository:

git clone https://github.com/yourusername/ising-market-model.git
cd ising-market-model

Install dependencies:

pip install -r requirements.txt

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Usage

Running an Experiment

from src.master import IsingExperiment

exp = IsingExperiment(
    size=50,
    temperatures=[1.5, 2.0, 2.269, 2.5],
    alphas=[1.0, 2.0, 4.0],
    update_rule="bornholdt",
    seed=42,
)

results = exp.run_grid(
    n_equil=5000,
    n_steps=10000,
    sample_freq=10
)

Visualisation

from src.plotting import IsingPlotter

plotter = IsingPlotter(results)

plotter.plot_magnetisation_vs_T()
plotter.plot_susceptibility()

# Plot specific dynamics
plotter.plot_time_series(T=2.269, alpha=2.0)
plotter.plot_returns(T=2.269, alpha=2.0)
plotter.plot_acf(T=2.269, alpha=2.0)

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Analysis Pipeline

Step	Module	Description
1	ising.py	Generates magnetisation and energy time series
2	analysis.py	Computes returns, volatility, autocorrelation, and thermodynamic observables (Binder cumulant, heat capacity)
3	master.py	Runs parameter sweeps across temperature $T$ and feedback strength $\alpha$
4	plotting.py	Produces publication-quality figures

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Future Work

• Research Paper to be uploaded
• Time-varying temperature (market regimes)
• Heterogeneous agent models
• Calibration to real market data (e.g. S&P 500)
• Information-theoretic measures (entropy, KL divergence)

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License

This project is licensed under the MIT License.

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Acknowledgements

• Statistical mechanics and Ising model literature
• Econophysics research on financial markets
• Python scientific computing ecosystem