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

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.

• The paper introduces a logit jump-diffusion model that standardizes prediction markets, enabling quoting and hedging of belief risks similar to Black-Scholes.
• The model employs a logit transformation and separates diffusion from jump components using an EM algorithm while preserving the risk-neutral drift.
• Experimental results show lower belief-variance forecast error and robust causal calibration, highlighting its potential to enhance market liquidity.

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

The paper "Toward Black-Scholes for Prediction Markets: A Unified Kernel and Market Maker's Handbook" introduces a novel framework for standardizing prediction markets through a logit jump-diffusion model with risk-neutral (RN) drift, analogous to the Black-Scholes model in options markets. This framework aims to establish a stochastic kernel for prediction markets, providing standardized tools for quoting and hedging belief risks.

Introduction and Motivation

Prediction markets, using platforms like Polymarket, allow trading contracts based on future event outcomes. These markets lack a unified stochastic model akin to Black-Scholes, making it difficult to manage belief-volatility, jump, and cross-event risks effectively. The authors propose a logit jump-diffusion model where the traded probability $p_t$ functions as a $\mathbb{Q}$-martingale. This model exposes belief volatility, jump intensity, and cross-event dependence as tradable risk factors, facilitating risk management and hedging across various prediction venues.

Logit Jump-Diffusion Model

The proposed kernel uses a logit transformation, mapping traded probabilities into real-valued log-odds, allowing for the application of Itô-Lévy calculus while preserving the probabilistic boundaries. The model incorporates diffusive correlation and co-jumps, capturing how event probabilities evolve under both continuous market dynamics and discrete information shocks. The RN drift condition ensures that $p_t$ retains its martingale property under the risk-neutral measure, enabling the parametrization of belief volatility and jump features as economically interpretable and quotable risks.

Calibration Pipeline

A detailed calibration pipeline is proposed, leveraging filtering techniques to estimate the latent logit process, separate diffusion from jumps using an EM algorithm, and enforce RN drift for causal calibration. This process yields a stable belief-volatility surface, essential for quoting, hedging, and maintaining market liquidity. The pipeline adapts to microstructure noise and correlates jumps across events, resulting in a robust platform for implementing the RN-JD framework in real markets.

Derivative Layer

Inspired by the options markets, the paper suggests the development of derivative instruments specifically for belief dynamics in prediction markets, such as belief variance swaps, correlation swaps, and first-passage notes. These derivatives allow market participants to hedge against belief volatility and jump risks effectively, offering significant utility in dynamically complex and information-rich environments.

Experimental Results

In controlled experiments with synthetic RN-consistent paths and real event data, the RN-JD model demonstrates lower belief-variance forecast error compared to diffusion-only models and $p$-space baselines. This improvement underscores the model's superior causal calibration and economic interpretability, suggesting its potential to enhance market efficiency by standardizing belief risk across different market scenarios.

Conclusion

By introducing a logit jump-diffusion model for prediction markets, the authors provide a structured framework for standardized risk management, akin to the role Black-Scholes played in option markets. This model supports institutional market integration and offers tools for comprehensive risk assessment and hedging, paving the way for more liquid and efficient prediction markets.

In conclusion, the RN-JD kernel's establishment of a tractable, tradable language for belief risk in prediction markets is a significant step towards ensuring that these markets can function with the precision and reliability needed to facilitate large-scale institutional participation and robust market-making strategies.