RN–JD Kernel: Prediction Market Calibration

• RN–JD Kernel is a unified stochastic framework that models prediction market probabilities as risk-neutral martingales incorporating both diffusive and jump processes.
• It employs a robust calibration pipeline including noise filtering, EM-based separation of diffusion and jump components, and risk-neutral drift enforcement.
• The model yields explicit risk factors and closed-form derivative pricing formulas, improving forecast accuracy over diffusion-only methods.

The RN–JD Kernel, or Risk-Neutral Logit Jump-Diffusion Kernel, is a unified stochastic process and calibration framework for prediction market probabilities, designed to play an analogous role to the Black–Scholes kernel in option pricing. It rigorously models the dynamics of traded probabilities as martingales under a risk-neutral measure, capturing both diffusive and jump-driven belief revisions, and yielding well-defined quotable risk factors to standardize quoting, hedging, and the design of belief risk derivatives across prediction venues.

1. Mathematical Specification

The RN–JD Kernel posits that the log-odds $x_t = \logit(p_t)$, with $p_t$ the traded probability, evolves under a risk-neutral ($\Q$) measure via an Itô–Lévy stochastic differential equation: $$dx_t = \mu(t, x_t)\,dt + \sigma_b(t, x_t)\,dW_t + \int_\mathbb{R} z\, \widetilde{N}(dt,dz),$$ where $W_t$ is $\Q$-Brownian motion, $\widetilde{N}$ is the compensated jump measure for a Lévy process with intensity $\nu_t$, and $\sigma_b(t,x)$ is the belief-volatility.

The traded probability process is recovered through the logit link, $p_t = S(x_t) = (1 + e^{-x_t})^{-1}$, embedding the process on $(0,1)$. Applying the Itô–Lévy formula yields the SDE for $p_t$, where the drift term is set to zero to satisfy the $\Q$-martingale condition: $$\mu(t, x) = - \frac{\frac{1}{2}S''(x)\sigma_b^2(t, x) + \int_\mathbb{R}[S(x+z) - S(x) - S'(x)\chi(z)]\,\nu_t(dz)}{S'(x)},$$ ensuring arbitrage-free evolution of probability prices.

2. Calibration Pipeline

Calibration proceeds in three structured steps on filtered mid/bid–ask/trade data:

• Microstructure Noise Filtering: Observed logits $y_t = \logit(\tilde{p}_t) = x_t + \eta_t$ are denoised via a Gaussian Kalman filter to estimate latent log-odds $\hat{x}_t$.
• Diffusion-vs-Jump Separation via EM: Discrete increments $\Delta x_t$ are modeled as a mixture:

$$\Delta x_t \sim (1 - \lambda_t\Delta)\,N(\mu\Delta, \sigma_b^2\Delta) + \lambda_t\Delta\,f_J(\cdot;\theta),$$

with EM steps estimating diffusive variance ($\sigma_b^2$), jump intensity ($\lambda$), and jump size variance ($s_J^2$), with soft partitioning by posterior jump probabilities.

• Risk-Neutral Drift Enforcement: Given updated $\sigma_b, \nu$, the martingale drift $\mu(t,x)$ is recomputed, and the Kalman smoother is re-applied to refine the path $\hat{x}_t$.

Regularization is applied when fitting surfaces in time-to-resolution ($\tau$) and logit-moneyness ($m$), by solving

$$\min_f \sum_g w_g[\hat f(g) - f(\tau_g, m_g)]^2 + \alpha \|\nabla^2 f\|_2^2,$$

subject to nonnegativity and mild boundary constraints.

3. Quotable Risk Factors and Kernel Surfaces

The calibrated RN–JD Kernel exposes explicit, quotable risk factors:

• Belief-Volatility Surface: $\sigma_b(\tau, m)$ serves as an analogue to the implied volatility surface, quantifying locally expected diffusive variance for each point in time–moneyness space.
• Jump-Intensity Surface: $\lambda(\tau, m)$, measuring event risk from abrupt belief jumps.
• Jump Second Moment Surface: $s_J^2(\tau, m)$, capturing the mean square jump size.

Swap and corridor derivative strikes are synthesized via frozen-coefficient expansions such as: $$K^{x\text{-var}}_{t, t+\Delta} \approx \int_t^{t+\Delta} \sigma_b^2(u)\,du + \int_t^{t+\Delta} \lambda(u)\,\mathbb{E}[Z^2]\,du.$$ These surfaces standardize the quoting and transfer of belief risk, supporting consistent hedging and valuation across prediction venues.

4. Derivative Layer: Instruments and Pricing

The RN–JD Kernel induces a menu of coherent derivatives in direct analogy to variance and correlation products in option markets:

• Variance Swaps: Payout equals the realized quadratic variation in $x$ or $p$.
• Correlation/Covariance Swaps: Integrated covariation between two event probabilities, incorporating both diffusive and co-jump components.
• Corridor Variance Products: Variance accruals are restricted to intervals in $p$, focusing hedging on uncertainty zones.
• First-Passage Notes: Payments are contingent on hitting a probability threshold before expiry.

Pricing for many payoffs—especially under piecewise-constant coefficients—admits closed-form or short-maturity approximations. General claims are priced by solving the associated partial integro-differential equation (PIDE): $$\partial_t V + \mu(t, x)\partial_xV + \frac{1}{2}\sigma_b^2(t,x)\partial_{xx}V + \int \big[V(t, x+z) - V(t, x) - \partial_xV\,\chi(z)\big]\nu_t(dz) = 0,$$ with terminal condition $V(T, x) = g(x)$, using methods such as finite differences, Fourier inversion, or Monte Carlo with jump thinning.

5. Empirical Performance

On synthetic risk-neutral data, the RN–JD Kernel delivers lower short-horizon uncertainty forecast error than diffusion-only or probability-space baselines. For instance, in 60s-forward prediction:

Model	MSE$_\text{all}$	QLIKE$_\text{all}$
RN–JD	70.28	1.46
Diffusion-only (best)	77–105	2.66–4.73

Competing baselines included RW–logit (const. σ), pure logit diffusion, Wright–Fisher in $p$, and AR–GARCH in $p$. Enforcing the risk-neutral drift removed bias in the filtered path, ensuring the probability process remains a true $\Q$-martingale.

6. Economic and Structural Interpretation

The RN–JD framework provides an explicit, market-consistent unification of diffusive and abrupt belief revisions. Empirically:

• EM separation adapts to shifting dominance of volatility and jump risk.
• Jump intensities spike during macroeconomic news releases and unexpected events.
• Surfaces reveal market toxicity and anticipated information content, concentrating risk management where prediction markets are most sensitive.

Practitioners leverage $\sigma_b(\tau, m)$ and $\lambda(\tau, m)$ for quoting belief-vega and jump-vega, employ derivative strips for event-interval and calendar hedges, and use corridor/first-passage notes to target liquidity and manage gap risk.

7. Applications and Standardization

The RN–JD Kernel establishes a tractable process for traded probabilities with:

• Q-martingale structure and exposed calendar and jump risk factors,
• Scalable calibration pipeline: noise filtering → diffusion/jump EM separation → drift enforcement → risk factor surface smoothing,
• Closed-form pricing and replication of variance, correlation, corridor, and first-passage derivatives.

This enables consistent quoting and hedging of belief risk and underpins the emergence of standard risk metrics and derivative products for prediction markets, comparable in rigor and universality to the role of implied volatility in options markets (Dalen, 17 Oct 2025).