Settlement-Discount Term Structure

• Settlement-discount term structure is a maturity-dependent schedule that quantifies the discount on near-certain claims due to delayed settlement in prediction markets.
• The annualized settlement wedge (ASW) is derived from high-quantile frontiers and represents the required return on capital locked during the settlement period.
• Market design elements like yield-bearing collateral and negRisk conversion significantly influence the discount curve, affecting price calibration and capital efficiency.

A settlement-discount term structure is the maturity-dependent schedule of discounts embedded in contingent claims whose economic uncertainty can disappear before winning claims become redeemable. In collateralized, oracle-settled prediction markets, a winning payoff that is effectively certain may still trade below par because collateral remains locked until oracle settlement; in that setting, a near-certain dollar is a delayed dollar. The resulting term structure is recovered from persistent near-certain contracts and summarized by the annualized settlement wedge (ASW), a reduced-form measure of the required return on locked capital. Empirically, the recovered wedges are positive, maturity-dependent, and time-varying, and they change with market architecture, capital recycling, and collateral productivity (Gebele et al., 29 May 2026).

1. Conceptual basis

The settlement discount captures the pricing wedge induced by delayed redeemability of winning claims in collateralized prediction markets. The core reduced-form pricing relation is

$P_{i,t} = \mathbb{E}_t[X_i]\,D(\tau_{i,t}),$

with $D(\tau)\in(0,1]$, where $X_i$ is the terminal payoff and $\tau_{i,t}$ is time remaining to settlement. Even when an event is economically resolved, winners realize cash only when the oracle finalizes and settlement opens. The delay imposes opportunity costs, liquidity costs, and residual platform/oracle risk on locked capital, so prices of near-certain contracts can remain below $1$ and vary with maturity (Gebele et al., 29 May 2026).

This mechanism creates a near-certainty horizon gradient. When beliefs are already close to certainty, raw prices reflect both the residual failure probability and the settlement discount. With longer $\tau$, the discount factor is smaller, so the observed price lies farther below par even when $\mathbb{E}_t[X_i]\approx 1$. Raw underpricing at long horizons therefore need not be evidence of forecast error alone (Gebele et al., 29 May 2026).

In standard term-structure language, a discount curve is a mapping from maturity to present-value discount factors, with associated forward rates and yields defined by

$$D(T)=\exp\Big\{-\int_0^T r(u)\,du\Big\}, \qquad f(T)=-\frac{d}{dT}\log D(T),$$

and admissible discount curves satisfy $0<D(0,T)\le 1$ and are nonincreasing in $T$ [(Filipović et al., 2016); (Cousin et al., 2014)]. The settlement-discount term structure in prediction markets is analogous in form but not in economic content: it is not a risk-free curve, but a platform-specific discount schedule induced by lock-up and settlement mechanics (Gebele et al., 29 May 2026).

2. Formal representation and annualized wedges

For (near-)certain \$D(\tau)\in(0,1]$0T$D(\tau)\in(0,1]$1$ P_{i,t}=\mathbb{E}t[X_i]\,D(\tau{i,t}), \qquad D(\tau)=\exp!\bigl(-r_{\mathrm{PM}}(\tau)\,\tau\bigr),

$D(\tau)\in(0,1]$2

r_q(\tau) = -\frac{1}{\tau}\log P_q(\tau), \qquad \mathrm{ASW}_q(\tau)=\exp!\bigl(365\,r_q(\tau)\bigr)-1,

$D(\tau)\in(0,1]$3

D_q(\tau)=\exp!\bigl(-r_q(\tau)\,\tau\bigr).

$D(\tau)\in(0,1]$4

P_{i,t}=(1-\delta_{i,t})D(\tau_{i,t}), \qquad \delta_{i,t}:=1-\mathbb{E}_t[X_i],

$D(\tau)\in(0,1]$5

1-P_{i,t}

\underbrace{1-\mathbb{E}t[X_i]}{\text{residual uncertainty}} + \underbrace{\mathbb{E}t[X_i]\bigl(1-D(\tau{i,t})\bigr)}_{\text{settlement discount}}.

$D(\tau)\in(0,1]$6

\hat{r}{q}(\tau)=-\frac{1}{\tau}\log P{q}(\tau)

r_{\mathrm{PM}}(\tau)-\frac{1}{\tau}\log(1-\delta_q(\tau)) \ge r_{\mathrm{PM}}(\tau).

$D(\tau)\in(0,1]$7

\tau^{\mathrm{obs}}{i,t}=T_i^{\mathrm{settle}}-t=\tau^e{i,t}+u_{i,t}.

$D(\tau)\in(0,1]$8

\tilde{P}{i,t}=\min!\left{1,\frac{P{i,t}}{\hat{D}(\tau_{i,t})}\right}, \qquad \hat{D}(\tau)=\exp!\bigl(-\hat{r}_q(\tau)\,\tau\bigr).

$D(\tau)\in(0,1]$9

X_i-P_{i,t}=\alpha+\beta \tau_{i,t}+\varepsilon_{i,t},

$X_i$0

X_i-\tilde{P}^{(q)}_{i,t}=\alpha_q+\beta_q \tau_{i,t}+\varepsilon_{i,t}.

$X_i$1

\hat{\beta}=0.00016780,\qquad p=3.75\times 10^{-7}.

$X_i$2

\sum_{k\in S}N_k \equiv (m-1)\cdot \mathbf{1}+\sum_{j\notin S}Y_j.

$X_i$3

V_S(t)\approx (m-1)+D(\tau),

$X_i$4

\bar{P}_N(\tau;m)\approx 1-\frac{1-D(\tau)}{m}.

$X_i$5

\bar{P}_N(\tau;n-1)\approx 1-\frac{1-D(\tau)}{n-1}.

$X_i$6

r_{\mathrm{PM}}(\tau)\approx r_{\text{opportunity}}(\tau)-r_c+\varphi(\tau),

$X_i$7

D_{\text{total}}(T)=D_{rf}(T)\times D_{\text{settlement}}(T). $X_i$8 is a reduced-form required return in the platform environment rather than a Treasury or OIS rate (Gebele et al., 29 May 2026).

This places the topic adjacent to several established term-structure literatures. In exact smooth term-structure estimation, the discount curve is constructed as the unique exact-fit, maximally smooth curve in a Hilbert space, with discount factors, yields, and forwards related through standard fixed-income identities (Filipović et al., 2016). In admissible-curve analysis, discount factors are required to be nonincreasing, market-fitting, and smooth, and the literature emphasizes that curve construction itself can involve substantial uncertainty when market anchors are sparse (Cousin et al., 2014). In constrained kriging approaches, market-fit conditions are linear equalities and no-arbitrage shape restrictions become inequality constraints, yielding both a “most likely curve” and confidence bands (Cousin et al., 2016). A plausible implication is that prediction-market settlement curves could be studied with the same nonparametric concern for smoothness, admissibility, and uncertainty quantification, even though their economic interpretation differs.

The analogy with collateralized derivatives is also instructive. In derivatives with imperfect collateral, discounting can depart from OIS because the relevant rate becomes a derivative financing rate reflecting collateral type, haircuts, rehypothecation, and repo conditions (Lou, 2017). The prediction-market setting is not the same institutional environment, but the parallel is direct at the level of mechanism: discounting depends on how collateral actually finances or immobilizes positions, not solely on abstract probabilistic beliefs (Gebele et al., 29 May 2026, Lou, 2017).

Several limitations qualify interpretation. Residual uncertainty along the frontier biases ASWs upward; midpoints can be stale or wide in thin long-horizon markets; realized settlement is only a proxy for expected horizon; and cross-platform comparisons such as Kalshi versus Polymarket are interpretive comparative statics because platforms differ in users, fees, regulation, liquidity, and resolution (Gebele et al., 29 May 2026). These points constrain any belief-purification exercise based on de-wedged prices.

The principal practical implication is that prices should be adjusted for the settlement-discount curve before being read as pure probabilities, especially at longer horizons. The design implications run in the same direction: conversion mechanisms such as negRisk increase capital efficiency; yield-bearing collateral reduces the opportunity cost of lock-up; and tick-size and fee choices near par matter because persistent short-end wedges can annualize into large carry (Gebele et al., 29 May 2026). More generally, pricing quality in prediction markets is endogenous to settlement mechanics, collateral productivity, and capital-recycling design, so information aggregation occurs through a financial infrastructure whose funding conditions are measurable and economically important (Gebele et al., 29 May 2026).