spb/wp8_uqo

1% ============================================================================2% Section 7: The 2024 Election Market and Further Checks3% ============================================================================4\section{The 2024 Election Market and Further Checks}\label{sec:robustness}56\subsection{The complete set of seventeen candidate claims}\label{subsec:election}78The ``Presidential Election Winner 2024'' event is the most heavily traded market in the platform's history and provides a state space more than three times larger than the Fed group. The second column of Table~\ref{tab:completeness} reports the pricing statistics over 304 daily closes from 5 January to 5 November 2024. The sum of the seventeen prices averages 1.0146 with a standard deviation of 0.0313 and a median of 1.0105; the mean overround of 1.46 percentage points has a HAC standard error of 0.20. The overround exceeds one on 92 percent of days, is within two percentage points of one on 74 percent of days and within five on 99 percent. As in the Fed group it shrinks towards resolution---1.022 over the first thirty days, 1.0077 (standard deviation 0.0011) over the last thirty---and the slope on time to resolution is positive (0.020 per year, HAC $t=3.73$). Deviations are far less persistent than in the Fed group: the first-order autocorrelation is 0.06 and the Dickey--Fuller statistic $-16.4$, so that a deviation on one day is essentially gone the next. The maximum of 1.515 is a single day, 9 April 2024, on which the claim on one minor candidate printed a trade at 0.50 against surrounding closes of 0.0015 and 0.0020; it illustrates once more that a recorded daily price on an inactive claim is not an executable quote, and the median-based statistics in the table are unaffected by it.910The lower bound is further from one in this market than in the Fed group because the horizon is longer: $e^{-r_t\tau_t}$ averages 0.978 and is 0.957 in January 2024 when the effective rate was 5.33 percent and the election ten months away. The complete set therefore traded on average 3.7 percentage points above its discounted value, and the entire overround is again on the upper side of the band. With seventeen claims, fifteen of which traded at less than two cents for most of the sample, the sum of half-spreads alone accounts for about one percentage point at a tick of 0.001 and for far more at the one-cent tick that applies to most of the price range on the platform. The evidence from the two state spaces is therefore consistent: the complete set is priced at or marginally above one, the time value of money is not reflected in the prices, and the measured overround is of the order of the spread costs of the low-priced legs.1112Figure~\ref{fig:election} plots the five most traded candidate claims and the sum of all seventeen. The price of the Trump claim ranged from 0.405 to 0.705, the maximum of 0.705 being reached on 15 July 2024 after the attempted assassination of 13 July (0.595 on 12 July, 0.695 on 14 July). President Biden's withdrawal on 21 July moved his claim from 0.0695 to 0.0005 and Vice-President Harris's from 0.18 to 0.32 within a day, and the debate of 10 September moved the Trump claim from 0.5225 to 0.496 over two days. The Trump claim peaked at 0.6655 on 29 October and closed at 0.58 on the evening of 5 November, with the Harris claim at 0.4215 and the fifteen remaining claims summing to 0.0075. The earlier version of this paper described this final value as a post-election ``convergence'' to 0.58 and inferred a settlement delay; the description was wrong, since the sample ends before the result was known, and the 0.58 is simply the market's last pre-result price. The market resolved to the Trump claim after the election, outside our sample.1314\begin{figure}[htbp]15  \centering16  \includegraphics[width=\textwidth]{../figures/fig11_election_case_study.pdf}17  \caption{Upper panel: daily closing prices of the five most traded claims in the ``Presidential Election Winner 2024'' event, 4 January to 5 November 2024; dotted lines mark the attempted assassination (13 July), President Biden's withdrawal (21 July), the Harris--Trump debate (10 September) and election day (5 November). Lower panel: price of the complete set of seventeen claims with the discounted lower bound (dashed) and one (solid).}18  \label{fig:election}19\end{figure}2021\subsection{Sensitivity of the pricing tests}2223The pricing statistics of Table~\ref{tab:completeness} are insensitive to the treatment of the listing day except for the maximum and the standard deviation, as reported in the table note. Computing the lower bound with the upper bound of the target range (3.75 percent) instead of the effective rate shifts $\delta_t$ by at most 0.0006 and changes no conclusion. Replacing the sample mean by the median overround (0.93 percentage points for the Fed group, 1.05 for the election group) does not change the ordering of the two markets or the sign of the term-structure slope. Using a cost allowance of one cent on the complete set instead of half a cent raises the share of Fed days inside the executable band from 29 to 49 percent; using the contemporaneous Roll estimate of the effective spread of the modal claim as the allowance gives 42 percent. We have not attempted to reconstruct the executable bid-side sum over the whole sample, which would require historical order-book data that the public API does not provide; this is the most useful extension of the pricing analysis.2425\subsection{Sensitivity of the announcement and lead--lag results}2627The announcement-day comparisons use absolute changes and rank-based or median-centred tests, so that no single observation drives them. Dropping the two largest daily moves in the sample---18 March 2026 (FOMC) and 6 March 2026 (employment report)---leaves the pooled release-day variance ratio for the ``No change'' state price at 4.7 (Brown--Forsythe $p=0.005$, Mann--Whitney $p=0.023$) but reduces that for the expected policy change to 1.2 ($p=0.38$; Mann--Whitney $p=0.051$): the release-day effect on the modal state price is robust, whereas the effect on the first moment of the distribution rests on the two days on which the distribution shifted most. Valuing the open-ended states at $\pm 75$ rather than $\pm 50$ basis points rescales the expected-policy-change series by about 11 percent (correlation 0.99 between the two versions) and leaves the pooled tests unchanged (variance ratio 4.9, Brown--Forsythe $p=0.004$). The lead--lag tests are reported at three lag orders in Table~\ref{tab:granger}. Replacing the expected policy change by the change in the ``No change'' state price as the Polymarket variable produces one cell below the 10 percent level among the sixteen tests at one and two lags---the state price to SPY, with $p=0.07$ at one lag and $0.04$ at two---which is about what the null predicts for sixteen correlated tests; the seven other directions have $p$-values above 0.10 at both lag orders. Splitting the 110 trading days into two halves of 55 produces no rejection at one lag in either half in any direction.2829\subsection{Checks not performed}3031Several analyses that would strengthen the paper are not feasible with the data at hand and are recorded here as recommendations rather than results. Intraday price histories (the API offers minute and hourly fidelity for active markets) would allow the response to each release to be timed within the day and compared with the response of Treasury futures, which is the natural way to assess price discovery across the two venues; the daily data used here cannot speak to leads of minutes or hours. Federal funds futures or overnight index swap quotes for the June 2026 meeting would allow a direct comparison of the Polymarket-implied expected change with the futures-implied one and an estimate of the difference in risk premia, in the spirit of \citet{piazzesi2008futures}; we do not have those quotes. Historical order books would allow the executable bounds to be tested day by day. Finally, a cross-section of negative-risk groups---other FOMC meetings, other elections---would allow the term structure of the overround and its relation to depth to be estimated rather than described. None of these extensions requires new methods, only data we did not collect.32