---
title: Settlement-Discount Term Structure
url: https://www.emergentmind.com/topics/settlement-discount-term-structure
type: topic
---

# Settlement-Discount Term Structure

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 [2605.31431].

## 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 [2605.31431].

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 [2605.31431].

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 $T$ [1606.03899; 1404.0340]. 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 [2605.31431].

## 2. Formal representation and annualized wedges

For (near-)certain \$1 claims with maturity $T$, the reduced-form parameterization writes
$$
P_{i,t}=\mathbb{E}_t[X_i]\,D(\tau_{i,t}), \qquad
D(\tau)=\exp\!\bigl(-r_{\mathrm{PM}}(\tau)\,\tau\bigr),
$$
where $r_{\mathrm{PM}}(\tau)$ is the reduced-form required return for capital locked in unresolved positions. The term structure is therefore the horizon-dependent curve $\tau \mapsto D(\tau)$, recovered empirically from near-certainty price frontiers [2605.31431].

The paper operationalizes this through a high-quantile frontier $P_q(\tau)$ of near-certain prices at horizon $\tau$, using minimum, $0.1$-percentile, and $0.5$-percentile envelopes. The implied lock-up rate and annualized settlement wedge are
$$
r_q(\tau) = -\frac{1}{\tau}\log P_q(\tau), \qquad
\mathrm{ASW}_q(\tau)=\exp\!\bigl(365\,r_q(\tau)\bigr)-1,
$$
with corresponding discount factor
$$
D_q(\tau)=\exp\!\bigl(-r_q(\tau)\,\tau\bigr).
$$
ASW is therefore an annualized representation of the discount embedded in a near-certain claim [2605.31431].

The decomposition of the price gap separates residual uncertainty from settlement discounting:
$$
P_{i,t}=(1-\delta_{i,t})D(\tau_{i,t}), \qquad \delta_{i,t}:=1-\mathbb{E}_t[X_i],
$$
so that
$$
1-P_{i,t}
=
\underbrace{1-\mathbb{E}_t[X_i]}_{\text{residual uncertainty}}
+
\underbrace{\mathbb{E}_t[X_i]\bigl(1-D(\tau_{i,t})\bigr)}_{\text{settlement discount}}.
$$
This is the key formal reason why price discounts at long horizons need not be interpreted as pure miscalibration [2605.31431].

The recovered wedge is explicitly reduced-form. If residual failure probability persists even on the near-certainty frontier, then the implied rate is upward biased:
$$
\hat{r}_{q}(\tau)=-\frac{1}{\tau}\log P_{q}(\tau)
=
r_{\mathrm{PM}}(\tau)-\frac{1}{\tau}\log(1-\delta_q(\tau))
\ge r_{\mathrm{PM}}(\tau).
$$
Accordingly, ASW measures a required return for locked capital rather than a pure latent-belief curve [2605.31431].

## 3. Identification, data, and estimation workflow

Identification uses “persistent near-certain contracts,” defined as markets where one side, YES or NO, has a midpoint price at least $0.90$ for seven consecutive daily snapshots. Markets with subsequent large reversals are excluded. Realized settlement time is proxied by realized `closeDate`, interpreted as the first time winning claims can be redeemed:
$$
\tau^{\mathrm{obs}}_{i,t}=T_i^{\mathrm{settle}}-t=\tau^e_{i,t}+u_{i,t}.
$$
This is a noisy but informative proxy for the ex ante priced horizon [2605.31431].

The main empirical setting is Polymarket, using hourly quotes via the Polymarket Data API, on-chain trades via Polygon, and UMA oracle timestamps. The data fields `endDate`, `closeDate`, and `umaEndDate` distinguish scheduled market end, settlement proxy, and realized oracle resolution time. External benchmarks include AAVE supply rates from Dune and the 2-year U.S. Treasury yield from FRED [2605.31431].

Sample construction is large-scale. The full universe contains **141,848 events** and **323,342 markets**. The eligible CLOB sample contains **10,608 events**, **62,291 markets**, and **4,647,145 hourly price observations**. The near-certainty sample contains **4,483 events**, **36,349 markets**, and **3,712,775 hourly observations**; one daily snapshot per market is used for calibration, and the baseline frontier excludes negRisk markets [2605.31431].

The practical construction of the curve proceeds in seven steps. Markets are screened for event duration of at least $14$ days, outcome-token volume of at least $100$, usable CLOB histories, and non-stale daily midpoints. Near-certainty markets are then selected using the seven-day upper-tail screen, with reversals excluded and negRisk omitted for the baseline curve. For each daily snapshot, the horizon proxy is computed as $\tau_{i,t}=\texttt{closeDate}-t$. Frontier prices $P_q(\tau)$ are constructed within days-to-settlement bins, converted to $r_q(\tau)$ and ASW, smoothed over $\tau$, and then used to de-wedge prices through
$$
\tilde{P}_{i,t}=\min\!\left\{1,\frac{P_{i,t}}{\hat{D}(\tau_{i,t})}\right\},
\qquad
\hat{D}(\tau)=\exp\!\bigl(-\hat{r}_q(\tau)\,\tau\bigr).
$$
Manually audited economically resolved but unsettled windows, where $P\approx D(\tau)$, provide a direct validation of the discount component [2605.31431].

## 4. Empirical properties of the curve

The recovered settlement-discount term structure on Polymarket is positive across horizons, maturity-dependent, and time-varying. Its shape is elevated at the short end, falls by about $20$ days, and then exhibits a long-end hump around $230$-$260$ days. The short-end elevation is consistent with the fact that tiny near-par discounts annualize strongly at short holding periods; the paper also notes possible shape features from tick size near par and a long-end hump in empirical curves [2605.31431].

Time-series behavior is also nontrivial. Lower-tail ASWs are high early, when capital is scarce, and later compress with thicker support post-late 2024. Composition shifts toward shorter maturities later in the sample. Daily minimum and $0.1$-percentile ASWs move together, which rules out single-observation artifacts. Polymarket ASWs co-move positively with AAVE during periods with longer frontier maturities, but the correlations are regime-dependent and do not imply a stable parity with Treasuries [2605.31431].

A central empirical test regresses raw and de-wedged pricing errors on settlement horizon:
$$
X_i-P_{i,t}=\alpha+\beta \tau_{i,t}+\varepsilon_{i,t},
$$
and
$$
X_i-\tilde{P}^{(q)}_{i,t}=\alpha_q+\beta_q \tau_{i,t}+\varepsilon_{i,t}.
$$
In the near-certainty sample with market-clustered standard errors, the raw slope is
$$
\hat{\beta}=0.00016780,\qquad p=3.75\times 10^{-7}.
$$
After adjustment using date-varying frontier-implied curves, the slope falls materially [2605.31431].

| Frontier | Mean ASW | Adjusted slope |
|---|---:|---:|
| Minimum-ASW frontier | 3.06% | 0.00008757 |
| 0.1-percentile frontier | 4.36% | 0.00006686 |
| 0.5-percentile frontier | 6.89% | 0.00002076 |

These adjusted slopes correspond to reductions of **47.8%**, **60.2%**, and **87.6%**, respectively; the $0.5$-percentile case yields $p=0.5285$, statistically indistinguishable from zero [2605.31431].

The same attenuation survives out of sample. Cross-fitted, held-out-event estimates reduce the slope by **56.4–92.9%**, preserving the effect beyond the calibration set. Reliability diagnostics show that discount adjustment materially moves long-horizon near-certain reliability toward the diagonal. Brier improvements are modest, which is consistent with the continued presence of residual uncertainty, but calibration-gap MSE improvements are strong and increase with horizon, reaching up to about **20%** in the greater-than-180-day bucket [2605.31431].

These findings undercut a common misconception that long-horizon near-certainty underpricing is primarily a forecasting problem. The measured pattern is instead consistent with priced settlement frictions, with residual uncertainty remaining but no longer carrying the entire maturity effect [2605.31431].

## 5. Architecture, synthetic collateral, and collateral productivity

Market design changes the settlement-discount term structure. In negRisk markets, baskets of NO positions can be economically converted into deterministic par units plus residual YES exposure. For a NO basket over a subset $S$ of $m$ outcomes in an $n$-outcome event, the payoff identity is
$$
\sum_{k\in S}N_k \equiv (m-1)\cdot \mathbf{1}+\sum_{j\notin S}Y_j.
$$
In the tail case $p_S(t)\approx 0$, reduced-form discounting gives
$$
V_S(t)\approx (m-1)+D(\tau),
$$
so the average NO price in the basket becomes
$$
\bar{P}_N(\tau;m)\approx 1-\frac{1-D(\tau)}{m}.
$$
For the canonical largest basket,
$$
\bar{P}_N(\tau;n-1)\approx 1-\frac{1-D(\tau)}{n-1}.
$$
The wedge $1-D$ is therefore spread across $m$ legs, mechanically compressing the discount per leg; larger outcome counts increase the compression [2605.31431].

Empirically, near-par weekly $99.9$th-percentile price frontiers place negRisk markets closer to $1$ across horizons than non-negRisk markets, with larger-$n$ negRisk events sitting higher. The ordering is clearest at medium and long horizons. The mechanism is capital recycling through synthetic collateral, not an improvement in the underlying event certainty [2605.31431].

Yield-bearing collateral changes the curve differently. If locked collateral earns a yield $r_c$ while awaiting settlement, the opportunity cost falls, $D(\tau)$ rises for a given horizon, and the ASW curve flattens. The paper summarizes this relation interpretively as
$$
r_{\mathrm{PM}}(\tau)\approx r_{\text{opportunity}}(\tau)-r_c+\varphi(\tau),
$$
where $\varphi(\tau)$ captures residual platform, oracle, and liquidity risks. Thus higher collateral yield reduces the lock-up wedge [2605.31431].

A cross-platform comparative static supports this mechanism. Kalshi’s frontier remains closer to par than Polymarket’s across horizons, consistent with APY on cash and open positions, indicating a flatter ASW curve. By contrast, Polymarket’s later $4\%$ holding rewards did not induce a comparable upward break in ASW, because composition shifts toward shorter maturities dominated [2605.31431].

These design effects make the settlement-discount term structure endogenous to collateral productivity and capital-recycling rules. The object is therefore not merely a passive market-implied belief curve; it is partly engineered by market architecture [2605.31431].

## 6. Interpretation, neighboring literatures, and implications

Conceptually, the settlement discount can be decomposed as an increment to standard discounting:
$$
D_{\text{total}}(T)=D_{rf}(T)\times D_{\text{settlement}}(T).
$$
Here $D_{rf}(T)$ is the risk-free discount to date $T$, while $D_{\text{settlement}}(T)$ captures platform-specific opportunity cost, liquidity demand, and settlement frictions. In practice, the prediction-market quantity $r_{\mathrm{PM}}(\tau)$ is a reduced-form required return in the platform environment rather than a Treasury or OIS rate [2605.31431].

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 [1606.03899]. 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 [1404.0340]. 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 [1604.02237]. 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 [1702.04053]. 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 [2605.31431; 1702.04053].

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 [2605.31431]. 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 [2605.31431]. 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 [2605.31431].

Source: https://www.emergentmind.com/topics/settlement-discount-term-structure