Toward Black Scholes for Prediction Markets: A Unified Kernel and Market Maker’s Handbook

ArXiv ID: 2510.15205 “View on arXiv”

Authors: Shaw Dalen

Abstract

Prediction markets, such as Polymarket, aggregate dispersed information into tradable probabilities, but they still lack a unifying stochastic kernel comparable to the one options gained from Black-Scholes. As these markets scale with institutional participation, exchange integrations, and higher volumes around elections and macro prints, market makers face belief volatility, jump, and cross-event risks without standardized tools for quoting or hedging. We propose such a foundation: a logit jump-diffusion with risk-neutral drift that treats the traded probability p_t as a Q-martingale and exposes belief volatility, jump intensity, and dependence as quotable risk factors. On top, we build a calibration pipeline that filters microstructure noise, separates diffusion from jumps using expectation-maximization, enforces the risk-neutral drift, and yields a stable belief-volatility surface. We then define a coherent derivative layer (variance, correlation, corridor, and first-passage instruments) analogous to volatility and correlation products in option markets. In controlled experiments on synthetic risk-neutral paths and real event data, the model reduces short-horizon belief-variance forecast error relative to diffusion-only and probability-space baselines, supporting both causal calibration and economic interpretability. Conceptually, the logit jump-diffusion kernel supplies an implied-volatility analogue for prediction markets: a tractable, tradable language for quoting, hedging, and transferring belief risk across venues such as Polymarket.

Keywords: Logit jump-diffusion, Risk-neutral drift, Expectation-Maximization, Belief volatility surface, Derivative pricing, Prediction Markets

Complexity vs Empirical Score

• Math Complexity: 7.0/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail
• Why: The paper introduces advanced stochastic processes (logit jump-diffusion, expectation-maximization calibration) and derivative pricing PIDEs, indicating high mathematical complexity. It validates the model on synthetic data and real event data with quantitative forecast error metrics, supporting empirical rigor, though it lacks full implementation details like code or public datasets.

  flowchart TD
    A["Research Goal<br>Develop Black-Scholes equivalent<br>for prediction markets"] --> B["Methodology<br>Logit Jump-Diffusion Model"]
    B --> C["Data & Inputs<br>Synthetic Risk-Neutral Paths<br>Real Event Data"]
    C --> D["Computational Process<br>EM Calibration &<br>Belief Volatility Surface"]
    D --> E["Key Findings<br>Reduced Forecast Error &<br>Tradable Belief Risk Factors"]