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.

翻译：预测市场（如Polymarket）将分散信息聚合为可交易概率，但至今仍缺乏类似于期权市场从布莱克-舒尔斯模型获得的统一随机核函数。随着这些市场在机构参与、交易所整合以及选举和宏观数据发布期间交易量的增长，做市商面临信念波动率、跳跃和跨事件风险，却缺乏标准化的报价或对冲工具。我们提出这样一个基础框架：采用逻辑跳跃扩散模型，引入风险中性漂移，将交易概率p_t视为Q鞅，并将信念波动率、跳跃强度及相关性作为可报价风险因子。在此基础上，我们构建了一套校准流程，可过滤微观结构噪声，利用期望最大化算法分离扩散与跳跃，强制满足风险中性漂移条件，并生成稳定的信念波动率曲面。随后，我们定义了连贯的衍生品层（方差、相关性、通道和首达时间工具），类比于期权市场中的波动率和相关性产品。在基于合成风险中性路径和真实事件数据的受控实验中，与纯扩散模型和概率空间基线相比，该模型降低了短期信念方差预测误差，同时支持因果校准和经济可解释性。从概念上讲，逻辑跳跃扩散核函数为预测市场提供了隐含波动率的对等物：一种易于处理、可交易的语言，用于在Polymarket等平台间实现信念风险的报价、对冲与转移。