Marketron Through the Looking Glass: From Equity Dynamics to Option Pricing in Incomplete Markets

ArXiv ID: 2508.09863 “View on arXiv”

Authors: Igor Halperin, Andrey Itkin

Abstract

The Marketron model, introduced by [“Halperin, Itkin, 2025”], describes price formation in inelastic markets as the nonlinear diffusion of a quasiparticle (the marketron) in a multidimensional space comprising the log-price $x$, a memory variable $y$ encoding past money flows, and unobservable return predictors $z$. While the original work calibrated the model to S&P 500 time series data, this paper extends the framework to option markets - a fundamentally distinct challenge due to market incompleteness stemming from non-tradable state variables. We develop a utility-based pricing approach that constructs a risk-adjusted measure via the dual solution of an optimal investment problem. The resulting Hamilton-Jacobi-Bellman (HJB) equation, though computationally formidable, is solved using a novel methodology enabling efficient calibration even on standard laptop hardware. Having done that, we look at the additional question to answer: whether the Marketron model, calibrated to market option prices, can simultaneously reproduce the statistical properties of the underlying asset’s log-returns. We discuss our results in view of the long-standing challenge in quantitative finance of developing an unified framework capable of jointly capturing equity returns, option smile dynamics, and potentially volatility index behavior.

Keywords: Marketron model, Hamilton-Jacobi-Bellman equation, Option pricing, Non-tradable state variables, Risk-adjusted measure, Equity Options

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail
• Why: The paper employs advanced mathematics, including HJB equations, nonlinear PDEs, integral equations, and active particle physics, reflecting high mathematical complexity. It demonstrates empirical rigor through calibration to market data (S&P 500, SPY options), backtesting, and implementation details, though it leans more toward theoretical methodology than fully practical backtest-ready code.

  flowchart TD
    A["Research Goal<br>Extend Marketron Model<br>to Option Pricing"] --> B["Data & Inputs<br>SPX Option Prices<br>Underlying Log-Returns"]
    B --> C["Methodology<br>Utility-Based Pricing<br>via Dual Solution"]
    C --> D["Computational Process<br>Efficient HJB Solver<br>for Calibration"]
    D --> E{"Key Findings / Outcomes"}
    E --> F["Unified Framework<br>Jointly Captures:<br>Equity Returns & Option Smile"]
    E --> G["Calibration Feasibility<br>Solvable on Standard Hardware"]
    E --> H["Market Incompleteness<br>Handled via Risk-Adjustment"]