  This paper introduces a martingale that characterizes two properties of
evolving forecast distributions. Ideal forecasts of a future event behave as
martingales, sequen- tially updating the forecast to leverage the available
information as the future event approaches. The threshold martingale introduced
here measures the proportion of the forecast distribution lying below a
threshold. In addition to being calibrated, a threshold martingale has
quadratic variation that accumulates to a total determined by a quantile of the
initial forecast distribution. Deviations from calibration or to- tal
volatility signal problems in the underlying model. Calibration adjustments are
well-known, and we augment these by introducing a martingale filter that
improves volatility while guaranteeing smaller mean squared error. Thus,
post-processing can rectify problems with calibration and volatility without
revisiting the original forecast- ing model. We apply threshold martingales
first to forecasts from simulated models and then to models that predict the
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