True Probability Interval
The True Probability Interval is the range where the estimated true probability of the event lies, but neither buying "Yes" nor buying "No" would offer a higher expected return than a risk-free asset.
Upper Bound
The upper bound represents the probability at which buying a Yes position would provide the same expected return as investing in a risk-free asset.
P(upper) is the probability of the event occurring for which buying a “Yes” position would provide a sufficient return.
P(yes) is the current Polymarket price of the “Yes” outcome.
r is the yield of the U.S. Treasury maturity that is closest to the market’s resolution date.
T is the remaining time until market resolution, measured in years.
Lower Bound
The lower bound represents the probability at which buying a No position would provide the same expected return as investing in a risk-free asset.
P(lower) is the probability of the event occurring for which buying a “No” position would provide a sufficient return.
P(no) is the current Polymarket price of the “No” outcome.
r is the yield of the U.S. Treasury maturity that is closest to the market's resolution date.
T is the remaining time until market resolution, measured in years.
Example
Suppose a market has a Yes price of 80%, a No price of 20%, 2.5 years remaining until resolution, and a risk-free annual return of 4%.
The resulting True Probability Interval is therefore 77.9%–88.2%.
Interpretation
The interval can be interpreted as a no-trade range. Within the interval, neither side of the prediction market offers a higher expected return than the risk-free alternative.
Limitations
Despite its name, the True Probability Interval does not prove that the event's actual probability lies inside the displayed range.
The calculation takes into account only the risk-free rate, and does not account for other factors such as bid-ask spreads, execution risk, resolution uncertanty, platform risk, investor risk preferences, favorite-longshot bias, or other market ineffeciencies.