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% Section 3: Conceptual Framework
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\section{Conceptual Framework}\label{sec:theoretical}

\subsection{State claims and the price of the complete set}

Consider a single decision with a finite set of mutually exclusive and exhaustive outcomes $\mathcal{S}=\{s_1,\ldots,s_N\}$, resolved at a known date $T$. A state claim on $s_j$ pays one unit of account at $T$ if $s_j$ obtains and nothing otherwise; its payoff is $a_j(s)=\mathbf{1}\{s=s_j\}$. Let $p_{j,t}$ be the price of the claim at date $t<T$ and $\boldsymbol{p}_t=(p_{1,t},\ldots,p_{N,t})'$.

\begin{definition}[Spanning]\label{def:completeness}
The market spans $\mathcal{S}$ if every payoff $c:\mathcal{S}\to\mathbb{R}$ can be replicated by a portfolio of traded claims. With one state claim per outcome the payoff matrix is the identity and spanning holds by construction.
\end{definition}

Spanning is therefore not an empirical property of a negative-risk group: the platform lists one claim per outcome and its settlement rules guarantee that exactly one claim pays. What the data can test is how the complete set is priced.

\begin{proposition}[Absence of arbitrage]\label{prop:no_arb}
If there is no arbitrage, there exist a strictly positive state-price vector $\boldsymbol{\psi}_t\in\mathbb{R}^N_{++}$ and, equivalently, a discount factor $\delta_t\in(0,1]$ and a probability measure $\mathbb{Q}_t$ on $\mathcal{S}$ such that $p_{j,t}=\psi_{j,t}=\delta_t\,q_{j,t}$ for every $j$. The price of the complete set is the price of a riskless claim to one unit of account at $T$:
\begin{equation}\label{eq:sp_sum}
  \Sigma_t \equiv \sum_{j=1}^{N} p_{j,t} = \delta_t = \frac{1}{(1+r_t)^{\tau_t}},
\end{equation}
where $r_t$ is the riskless rate and $\tau_t=T-t$ the remaining time to resolution.
\end{proposition}

Equation~\eqref{eq:sp_sum} is the Arrow--Debreu restriction tested in this paper. Its two ingredients deserve separate statements. Positivity, $p_{j,t}>0$, is required for $\mathbb{Q}_t$ to be equivalent to the physical measure; on an order book with a minimum tick it is satisfied mechanically whenever a claim is quoted, so that it is a necessary condition without empirical content. The restriction with content is on the level of $\Sigma_t$.

\subsection{Executable bounds}

In a frictionless market $\Sigma_t=\delta_t$. On Polymarket the complete set can be traded in two directions with different settlement timing, which makes the no-arbitrage band asymmetric. If $\Sigma_t<\delta_t$, a trader buys one unit of each claim for $\Sigma_t$ and holds the set to resolution, when exactly one claim pays one unit; the profit is $1-\Sigma_t$ at $T$, or $\delta_t-\Sigma_t$ in present value, and the capital is tied up for $\tau_t$. If $\Sigma_t>1$, a trader splits one unit of collateral into $N$ YES--NO pairs (one per outcome), sells the $N$ YES tokens for $\Sigma_t$, and converts the $N$ NO tokens through the negative-risk adapter into $N-1$ units of collateral immediately; the round trip costs one unit and returns $\Sigma_t+N-1$, a riskless profit of $\Sigma_t-1$ realised at $t$ rather than at $T$. Allowing for a cost $c_t$ of executing the $N$ legs at the quotes, the executable bounds are
\begin{equation}\label{eq:bounds}
  \delta_t - c_t \;\le\; \Sigma_t \;\le\; 1 + c_t .
\end{equation}
The lower bound depends on the riskless rate and on the horizon, the upper bound does not. With $r_t\approx 3.6$ percent and $\tau_t$ up to half a year, $\delta_t$ ranges from 0.98 to 1 over the Fed sample, so that the band is about two percentage points wide at the start of the sample and collapses to the width of the spread costs at the end. The transaction-cost term $c_t$ is naturally measured by the sum across claims of half the quoted spread: for five contracts quoted at a tick of 0.001 with one-tick spreads, $c_t=0.0025$, and we use $0.005$ as a conservative value in the tables.

A positive overround, $\Sigma_t>1$, therefore cannot be explained by the time value of money, which pushes $\Sigma_t$ below one. It can be explained by the convention that the daily price is the last trade or midpoint of each contract: when four of five contracts trade at a few tenths of a cent with one-tick spreads, the sum of midpoints exceeds the sum of bids by $N/2$ ticks, and the last trades of thinly traded claims tend to occur on the ask side. It can also reflect a genuine but small premium for the convenience of the conversion mechanism, or, in the first days of a market, quotes seeded by the platform before trading begins.

\subsection{The risk-neutral distribution and the expected policy change}

Normalising prices by their sum removes the discount factor:
\begin{equation}\label{eq:rn_density}
  q_{j,t}=\frac{p_{j,t}}{\Sigma_t}, \qquad \sum_{j=1}^{N} q_{j,t}=1 .
\end{equation}
For the Fed group, assign to each state the policy change $x_j\in\{-50,-25,0,25,50\}$ basis points, with the two open-ended states valued at $\pm 50$. The risk-neutral expected policy change and its dispersion are
\begin{equation}\label{eq:eq}
  \mathbb{E}^{\mathbb{Q}}_t[\Delta r] = \sum_{j=1}^{N} q_{j,t}\,x_j, \qquad
  \mathrm{SD}^{\mathbb{Q}}_t[\Delta r] = \Big(\sum_{j=1}^{N} q_{j,t}\,(x_j-\mathbb{E}^{\mathbb{Q}}_t[\Delta r])^2\Big)^{1/2}.
\end{equation}
$\mathbb{E}^{\mathbb{Q}}_t[\Delta r]$ is the Polymarket analogue of the futures-implied expected rate change used in the monetary-policy literature; like it, it embeds a risk premium \citep{piazzesi2008futures}. Under $\mathbb{Q}_t$ each $p_{j,t}$ is a martingale up to the discount factor, so that daily changes in $\mathbb{E}^{\mathbb{Q}}_t[\Delta r]$ should be unpredictable from past information and their variance should reflect the arrival of news. This is the property exploited in the announcement-day analysis.

\subsection{Hypotheses}

The framework yields the following hypotheses, stated so that each can be rejected by the data.

\begin{hypothesis}[Law of one price for the complete set]\label{hyp:completeness}
$\Sigma_t$ lies within the executable bounds \eqref{eq:bounds} on most days; deviations are small relative to quoted spreads, decline as the lower bound rises towards one, and are transitory (stationary with a half-life of a few days).
\end{hypothesis}

\begin{hypothesis}[Information on scheduled release days]\label{hyp:info}
Absolute daily changes in $\mathbb{E}^{\mathbb{Q}}_t[\Delta r]$ and in the modal state price are larger on days of scheduled FOMC statements, CPI releases and employment reports than on other days; changes on weekend days, when conventional markets are closed, are smaller than on weekdays.
\end{hypothesis}

\begin{hypothesis}[Lead--lag with conventional markets]\label{hyp:leadlag}
Daily changes in $\mathbb{E}^{\mathbb{Q}}_t[\Delta r]$ neither Granger-cause nor are Granger-caused by daily changes in the ten-year Treasury yield, the S\&P~500, the VIX or Bitcoin, once the series are aligned on the same trading calendar.
\end{hypothesis}

Hypothesis~\ref{hyp:leadlag} is stated as a null of no predictability because it is the prediction of informational efficiency in both markets; a rejection in either direction would be the interesting finding, and we show in Appendix~\ref{app:reconciliation} how easily spurious rejections arise. A fourth hypothesis on the conditional volatility of the modal state price, which played a prominent role in the earlier version of this paper, is examined in Appendix~\ref{app:volatility}.

\subsection{Caveats}

The framework is a single-period abstraction applied to a market that trades for six months. Treating each daily close as a snapshot of the state-price vector is innocuous for the pricing restriction, which must hold at every date, but it means that the risk-neutral distribution is conditional on the information of that day and that its evolution mixes news with changes in risk premia. The finite state space is natural for an FOMC decision and for an election winner; it would not be for a continuous outcome. The ``$\pm 50$'' valuation of the open-ended states is a convention that matters little while those states carry a few percent of risk-neutral mass, as they do here. Finally, the mapping from the contract to the economic event depends on the resolution rules and on the oracle; the possibility of a disputed resolution adds a basis risk that we cannot measure in these two markets but discuss in Section~\ref{sec:conclusion}.