Logit Jump-Diffusion Kernel

• The logit jump-diffusion kernel is a stochastic model that transforms probabilities via the logit function, combining diffusive and jump dynamics to enforce risk-neutrality.
• It explicitly separates key risk factors—belief volatility, jump intensity, and jump dependence—from drift, enabling precise calibration and derivative construction.
• Computational experiments validate its superior short-horizon variance forecasting and enhanced risk management in prediction markets.

A logit jump-diffusion kernel is a stochastic transition kernel defined by the evolution of a transformed process whose state variable is the logit of an underlying probability. This construction equips the logit variable with both diffusive and jump dynamics and explicitly enforces risk-neutrality under the risk-neutral measure. Originally proposed to bring a unifying, quotable stochastic kernel for prediction markets analogous to the Black-Scholes kernel in option markets, the logit jump-diffusion kernel separates tradable risk factors — belief volatility, jump intensity, and jump dependence — from drift, enabling standardized quoting, hedging, and transfer of belief risk (Dalen, 17 Oct 2025). The framework draws together advanced Itô-Lévy methods, mixture separation via EM/calibration pipelines, and derivative layers for volatility and correlation products. It is supported by theoretical, computational, and market calibration results.

1. Mathematical Construction and Risk-Neutral Measure

The kernel models the evolution of the Q-martingale probability $p_t$, treated in prediction markets as the risk-neutral belief that an event occurs. The logit transform,

$$x_t = \operatorname{logit}(p_t) = \log\left(\frac{p_t}{1-p_t}\right)$$

maps $p_t$ from $(0,1)$ onto $\mathbb{R}$, providing an unconstrained state-space for analysis. Its dynamics are given by a jump-diffusion SDE: $$dx_t = \mu(t, x_t)dt + \sigma_b(t, x_t)dW_t + \int_\mathbb{R} z \,\widetilde{N}(dt, dz)$$ where $\sigma_b$ is the belief volatility, and $\widetilde{N}$ is a compensated Poisson random measure with time-dependent Lévy measure $\nu_t(dz)$.

Risk-neutral dynamics are enforced by demanding that $p_t = S(x_t)$, with $S(x) = 1/(1 + e^{-x})$, is a Q-martingale. Applying Itô's formula with jumps to $S(x_t)$ yields the drift constraint,

$$0 = S'(x)\mu(t, x) + \frac{1}{2}S''(x)\sigma_b^2(t, x) + \int_\mathbb{R} [S(x+z) - S(x) - S'(x)\chi(z)] \nu_t(dz)$$

with truncation $\chi(z) = z\mathbf{1}_{|z|<1}$ and $S'(x)=p(1-p), S''(x)=p(1-p)(1-2p)$. Solving yields the explicit drift: $$\mu(t, x) = - \frac{ \frac{1}{2} S''(x)\sigma_b^2(t, x) + \int_\mathbb{R}[S(x+z) - S(x) - S'(x)\chi(z)]\nu_t(dz) }{ S'(x) }$$ This separation identifies belief volatility ($\sigma_b$), jump intensity and shape ($\nu_t$), and cross-event risk as distinct, quotable risk parameters.

2. Kernel Calibration: Filtering, Mixture Separation, and Surface Construction

Empirically, the state sequence $\{p_t\}$ is observed with microstructure or sampling noise. Filtering and calibration proceed via:

• State-space filtering: Observed logit-mids $$y_t = \operatorname{logit}(\widetilde{p}_t) = x_t + \eta_t$$ are denoised via Kalman/smoother techniques, with heteroskedastic variance $\sigma_\eta^2(t)$ modeled from bid-ask/spread data.
• Mixture separation with EM: On a grid, increments $\Delta x_t$ are modeled as a mixture:

$$\Delta x_t \sim \begin{cases} N(\mu_t\Delta,\, \sigma_b^2(t)\Delta), & \text{prob. }1-\lambda_t\Delta \ \text{jump from }f_J(\cdot;\theta_t), & \text{prob. }\lambda_t\Delta \end{cases}$$

The EM algorithm estimates posterior jump probabilities $\gamma_t$, then updates $\sigma_b$, $\lambda_t$, $\theta_t$ via moment matching.

• Drift enforcement: Following risk-factor estimation, one recomputes $\mu(t, x)$ to enforce the Q-martingale constraint.
• Surface construction: Aggregated/smoothed $\sigma_b(t, x)$ and jump parameters are rendered as a belief-volatility surface parameterized in $(\tau, x)$ or $(\tau, p)$ for quoting and hedging across time and moneyness.

3. Belief Risk Factors and Derivatives

The kernel exposes risk factors for quoting and hedging:

Factor	Mathematical Basis	Economic Interpretation
Belief Volatility	$\sigma_b$	Sensitivity of beliefs to new information
Jump Intensity	$\nu_t$, $\lambda_t$	Frequency of abrupt belief updates
Leap Magnitude	$f_J(\cdot;\theta_t)$	Distribution of belief jumps (severity)
Correlation	Cross-event covariance	Co-movement of event beliefs

The derivative layer includes:

• Variance swaps: Realized quadratic variation

$$\text{QV}_{t,T}^{x} = \int_t^T \sigma_b^2(u, x_u)\,du + \sum_{t < u \leq T} (\Delta x_u)^2$$

with swap strike $$K_{t,T}^{x,\text{var}} \approx \int_t^T \sigma_b^2(u) du + \int_t^T \lambda(u)\mathbb{E}[z^2(u)] du$$.

• Corridor swaps: Variance accrued only within specific regions (barriers).
• First-passage notes: Payout linked to whether $p_t$ crosses a level.

4. Quoting and Hedging: Avellaneda–Stoikov Mechanics in Logit Space

Optimized quoting adapts the Avellaneda–Stoikov protocol to the logit kernel:

• Reservation price in logit space:

$r_x(t) = x_t - q_t\gamma\overline{\sigma_b^2}(T-t)$

with spread:

$$2\delta_x(t) \approx \gamma\overline{\sigma_b^2}(T-t) + \frac{2}{k}\log\left(1+\frac{\gamma}{k}\right)$$

Quoting is then performed in logit coordinates and mapped back to probability space via $S(x)$, which automatically compresses spreads near $p=0,1$ owing to vanishing $S'(x)$. Inventory management and spread floor/capping logic are established for boundary effects.

5. Computational Experiments and Short-Horizon Variance Forecasts

Controlled tests on synthetic risk-neutral paths and real data confirm:

• The logit jump–diffusion kernel yields lower forecast error in short-horizon quadratic variability compared to diffusion-only and probability-space (direct $p_t$) models.
• The model more accurately isolates and anticipates volatility bursts, particularly those induced by scheduled macro/news events.
• Separating jump and diffusion contributions further improves forecast accuracy, with “jump boosting” near major information events markedly decreasing short-run error metrics.

6. Theoretical Generalizations, Related Models, and Extensions

The kernel is a direct specialization of path-integral, Fourier, and convolutional pricing techniques for jump–diffusion models (Liang et al., 2010), with relationship to Kolmogorov–Feller representation (Kim et al., 2013); it imports mixture-separation logic and large deviation principles for jump-drift/jump-diffusion processes (Monthus, 2021); adapts time-change and subordination results from comparison frameworks (Liu et al., 2021); and is theoretically receptive to polynomial extensions (Filipović et al., 2017), moment-based expansions, and score-based sampling (Baule, 9 Mar 2025).

The explicit separation of risk factors, drift enforcement for Q-martingale property, and construction of volatility and correlation derivatives establish a scalable lattice for quoting, hedging, and risk transfer at institutional scale.

7. Economic and Practical Implications

By mapping probability to logit space and modeling with a jump-diffusion kernel subject to risk-neutral constraints, prediction markets gain a coherent stochastic machinery for:

• Standardized belief risk quotation analogous to the implied-volatility surface of options
• Inventory and adverse selection management grounded in economic interpretable volatility and jump statistics
• A deep derivative layer (variance, corridor, correlation, barrier, first-passage products) enabling hedging and liquidity transfer
• Algorithmic calibration pipelines that filter microstructure noise, effect mixture separation, and surface construction for transparent market making (Dalen, 17 Oct 2025)

The logit jump-diffusion kernel thus equips prediction markets with tools to address belief volatility, jump and correlation risk, providing an efficient and theoretically justified foundation for derivative construction, quoting, hedging, and risk transfer in bounded probability domains.