Toward Black–Scholes for Prediction Markets

Options markets did not standardize because Black–Scholes was true; they standardized because it gave every desk the same small set of state variables — implied volatility, Greeks — to quote and hedge against. This paper (Dalen, Daedalus Research, arXiv:2510.15205, Oct 2025, v2 Apr 2026) argues prediction markets are at the same pre-BS stage and proposes the missing kernel: model the log-odds of the traded probability as a jump–diffusion, let the martingale property pin the drift, and trade what remains. Two thirds of the paper is an unusually concrete engineering manual: a calibration pipeline from raw ticks to a "belief-volatility surface" and an operator-level market-making recipe.

The problem: making markets in beliefs without a model #

An event contract pays $1 if the event occurs, $0 otherwise, so its price $p_t \in (0,1)$ reads as a risk-neutral probability. Execution is fragmented across LMSR scoring rules, constant-product AMMs, and central limit order books — each sets prices, none says how probabilities should evolve across time, news shocks, or related events. Market makers eat the consequences: adverse selection from informed flow, inventory risk that concentrates near resolution and cannot be hedged with the underlying until settlement, and for LMSR an expected deficit that grows in proportion to the liquidity offered. There is no standard instrument to lay off "belief risk" — the risk that the market-implied probability moves.

The "why now" is institutional scale. The paper cites ICE (NYSE's parent) announcing up to $2B for roughly a 20% stake in Polymarket at about $8B pre-money in 2025, over $200M of prior Polymarket funding across 2024–2025, September 2025 monthly volumes around $1.4B for Polymarket and $1–3B for Kalshi depending on week and product mix (sports alone above $2.4B), and the CFTC's 2024 proposed rulemaking on event contracts.

The kernel: logit jump–diffusion with risk-neutral drift #

The state variable is the log-odds $x_t = \log\frac{p_t}{1-p_t}$, with $p_t = S(x_t) = (1+e^{-x_t})^{-1}$. Beliefs follow an Itô–Lévy SDE on the real line:

$$dx_t = \mu(t,x_t)\,dt + \sigma_b(t,x_t)\,dW_t + \int_{\mathbb{R}} z\,\tilde N(dt,dz)$$

The one structural constraint is that $p_t = S(x_t)$, being a discounted price of a $1 payoff, must be a martingale under the risk-neutral measure. Applying Itô's formula for jump processes and setting the drift of $S(x_t)$ to zero pins down $\mu$ completely:

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

The drift is not a free parameter — it exists only to offset sigmoid convexity and jump compensation. Everything left over is tradable: $\sigma_b$, the jump intensity and moments in the Lévy measure $\nu_t$, and, for multiple events, a diffusive correlation $\rho_{ij}$ plus a co-jump measure capturing simultaneous news shocks. A short-maturity expansion gives the instantaneous probability covariance as $S_i' S_j' \sigma_b^i \sigma_b^j \rho_{ij}\,dt$ plus a co-jump integral — two separately hedgeable terms.

The proposed derivative layer #

On this kernel the paper defines the event-market analogues of the equity volatility stack:

General payoffs price via a backward partial integro-differential equation (PIDE) — finite differences with fast convolution for the jump integral, Fourier methods when coefficients are constant, Monte Carlo otherwise. None of these instruments currently trade anywhere; this is a specification, not a market description.

The market-maker handbook #

The practical core is Section 4, written in operator terms. Greeks are taken in logit units: for the vanilla contract $\Delta_x = p(1-p)$ and $\Gamma_x = p(1-p)(1-2p)$, so delta vanishes at the boundaries and curvature peaks in the swing zone near $p \approx 0.5$. Risk decomposes into four buckets: directional ($\Delta_x$), curvature/news ($\Gamma_x$), information intensity (belief-vega plus jump moments), and cross-event ($\rho$-vega plus co-jump covariance).

Quoting is Avellaneda–Stoikov transplanted into $x$-space. With inventory $q_t$, risk aversion $\gamma$, horizon $T$, and fill-intensity decay $k$:

$$r_x(t) = x_t - q_t\,\gamma\,\sigma_b^2\,(T-t), \qquad 2\delta_x(t) \approx \gamma\,\sigma_b^2\,(T-t) + \frac{2}{\gamma}\log\Big(1+\frac{\gamma}{k}\Big)$$

Post $x$-space bid/ask around the reservation quote, then map through the sigmoid for display. The Jacobian gives $\delta_p \approx p(1-p)\,\delta_x$, so displayed spreads auto-compress near $p \approx 0$ or $1$ — which the paper counters with a tick-size floor on the display spread and an inventory cap scaling like $1/\max\{S'(x), \varepsilon\}$. The refresh loop runs at 100–500 ms: update filtered state and toxicity flags, recompute quotes, widen or pull on VPIN-style imbalance spikes or unscheduled jump alarms, pre-widen before scheduled news, then rebalance cross-event and calendar hedges.

Cross-event hedging uses the ratio $\beta_{i \leftarrow j} \approx (S_i'/S_j')\,\rho_{ij}$, shrunk by a factor $\alpha \in [0.5, 1)$ and clamped when either $S'$ approaches zero to avoid explosive hedges near the boundaries, with a separate additive correction when co-jump covariance is material. The suggested day-one menu is deliberately minimal: vanilla contracts, $x$-variance strips around scheduled news, a few correlation strikes between the most coupled events, and corridor variance centered on $p \in [0.35, 0.65]$.

Calibration: from ticks to a belief-vol surface #

flowchart LR
  A[mid / bid-ask / trades] --> B[conditioning:<br/>trade-weighted mid,<br/>clamp, resample]
  B --> C[heteroskedastic KF<br/>filters latent x]
  C --> D[EM mixture:<br/>diffusion vs jumps]
  D --> E[enforce RN drift,<br/>re-smooth x]
  E -->|1-2 outer loops| C
  E --> F[penalized spline<br/>over tau and m]
  F --> G[belief-vol surface<br/>+ jump + dependence layers]

The pipeline is venue-agnostic and each stage is specified concretely. Conditioning builds a trade-weighted mid, clamps $p$ to $[10^{-5}, 1-10^{-5}]$ to keep logits finite, and resamples to a uniform 100 ms–1 s grid. The observed logit is modeled as the latent $x_t$ plus heteroskedastic microstructure noise whose variance is regressed on observable frictions — squared spread, inverse depth, trade rate, squared aggressor imbalance — which the paper reports markedly improves filtering in illiquid stretches. A Kalman filter (unscented or particle variants when $p$ is pinned or jumps are frequent) recovers $\hat x_t$.

Jump separation is a per-increment two-component mixture fitted by EM: each $\Delta \hat x_t$ is Gaussian diffusion with probability $1-\lambda_t\Delta$ or a draw from a centered jump law with probability $\lambda_t\Delta$; posterior "jump responsibilities" $\gamma_t$ above a threshold (e.g. 0.7) mark jump-dominant intervals. The M-step re-estimates $\sigma_b^2$, $\lambda$, and jump second moments by responsibility-weighted moments — see probability distributions for the mixture machinery. Then the risk-neutral drift is recomputed analytically from the freshly estimated parameters and the smoother re-run with it — the model's economics feed back into the filter. Point estimates are finally smoothed across time-to-resolution and moneyness by penalized splines with shape constraints (nonnegativity, edge stability, term smoothness relaxed at known news dates). Cross-event dependence is estimated twice on purpose: de-jumped diffusive correlation from jump-free intervals, and co-jumps separately — because they hedge different risks.

The experiment and its numbers #

The evaluation is a single causal forecasting task mirroring desk operation: at each second, forecast the next 60 s of realized logit variance (diffusion sum plus a jump term weighted by a scalar $c_J$ tuned on validation by QLIKE over a grid $\{0.3,\dots,1.0\}$). Setup: 6000 steps at 1 Hz; 20 high-volume Polymarket event series corrupted with regime-varying heteroskedastic noise; all models share the same Kalman prefilter and a train/validation/test thirds split; rolling EM window of 400 s; 600 Monte Carlo draws per step for the drift's jump-compensation integral; scheduled-news windows encoded as Gaussian kernels of width 90 s boosting jump intensity ex-ante.

What an engineer can actually use #

Four things transfer even if you never trade a belief-variance swap. First, work in log-odds: it removes boundary pathologies, makes Gaussian tooling legitimate, and the $p(1-p)$ Jacobian gives boundary-aware spreads and inventory caps for free. Second, the martingale drift is a free regularizer: any model of prediction-market dynamics whose implied $p$ carries drift under the market measure is fitting something that shouldn't exist; enforcing the analytical drift removed systematic bias in their filter. Third, the noise-aware filter + EM jump separation is a complete, implementable recipe for turning dirty bid/ask streams into a variance estimate — the heteroskedastic measurement model driven by spread/depth/trade-rate is the load-bearing detail. Fourth, the quoting loop (A–S in logit units, toxicity widening, scheduled-news pre-widening, kill-switches on feed gaps and repeated pick-offs) is a sane starting architecture for any event-market maker.

How much to trust it #

Treat the framework as solid and the empirical claims as weak. The kernel math is standard Itô–Lévy machinery correctly applied, and the handbook is mostly careful transplantation of proven results (Avellaneda–Stoikov, variance-swap theory, bi-power variation, co-jump tests) — the value is the assembly, not new theory. Against that: this is a single-author preprint from "Daedalus Research Team," whose stated affiliation is an X account; it is not peer-reviewed; the evaluation is one table, one 60 s horizon, semi-synthetic data, with the internal inconsistencies flagged above; there are no live-trading results, and the entire derivative layer is hypothetical. The authors themselves list single-event focus, a parsimonious symmetric light-tailed jump law, and stylized microstructure as limitations. Read it as a well-organized design document with a thin proof-of-concept attached.

Visual audit #

The 25-page PDF contains no mechanism diagram or empirical plot. Its only information-bearing visual is Table 1 on physical page 19. The table was inspected at full resolution and transcribed above because its values are needed to audit the paper's claims; it was not promoted into the durable topic as performance evidence. The equation-heavy pages 6–19 were also inspected against the extracted text, with no clipped or substituted mathematical glyphs observed. The green and red rectangles visible in Poppler renders are the PDF's native link annotations, not content or rendering errors.

The reusable, independently verified account now lives in Prediction Markets: Pricing, Mechanisms & Market Making. This note remains the source-specific record for the preprint's proposed logit jump–diffusion, calibration pipeline, derivative layer, and internally inconsistent evaluation.