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

TL;DR

The paper tackles the lack of a canonical stochastic kernel for prediction markets and proposes a minimal yet actionable framework: a logit jump–diffusion on log‑odds $x= ext{logit}(p)$ with risk‑neutral drift, ensuring $p_t=S(x_t)$ is a $\,\mathbb{Q}$‑martingale. It couples this kernel with a data‑driven calibration pipeline (state filtering, EM separation of diffusion and jumps, and drift enforcement) and a derivative layer (belief variance/volatility swaps, correlation/covariance swaps, corridor variance, and first‑passage notes) to enable quoting and hedging belief risk. In end‑to‑end experiments on synthetic RN‑consistent paths and real‑world data, the RN–JD framework reduces short‑horizon belief‑variance forecast error relative to diffusion‑only and probability‑space baselines, supporting causal interpretation and economic plausibility. The work provides a practical, interpretable language for belief risk, analogous to implied volatility in options, and outlines a path toward institutional, cross‑venue liquidity for prediction markets through standardized hedges and risk factoring.

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.