The Margin in a Three-Way Line

Question

A book quotes a soccer match in decimal odds: home 2.10, draw 3.40, away 3.80. Compute each implied probability and the book's overround (the amount by which the implied probabilities exceed 100%).

Accepted Answer

Implied probabilities are reciprocals of decimal odds: frac12.10 = 47.62%, qquad frac13.40 = 29.41%, qquad frac13.80 = 26.32%. Sum = 103.35%, so the overround is mathbf3.35% (often quoted as "a 103.3% book"). What it means: the book has sold probability adding to more than certainty. If it balances its liabilities across the three outcomes in these proportions, it keeps roughly 1 - frac11.0335 approx 3.24% of total stakes no matter who wins, the three-way version of a bid-ask spread. Each quoted implied probability overstates the true chance of its outcome by just enough to fund that margin. Why a trading firm asks this: a multi-outcome book is exactly a market maker quoting several legs of one event. Implied probabilities summing over 100% across outcomes is the same picture as bid and ask implying overlapping probabilities on one binary; the overround is where the edge lives, and any de-vig, arbitrage, or fair-value work starts by computing it. The trap: comparing the overround to the margin per unit staked without noticing they differ: 3.35% is the excess probability sold; the guaranteed take as a fraction of handle is 3.35/103.35 approx 3.24%. Small gap here, but at bigger marg

The Margin in a Three-Way Line

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