odds and overround

Converting a price

The implied probability of a decimal price is one divided by the price. Do this for every outcome in a market and the results should, in a fair market, sum to exactly one. They never do. The excess above one is the overround, and it is the market equivalent of a house edge.

The margin as a share of turnover is one minus the reciprocal of that sum, which for a typical two-outcome market prices in the low single figures.

implied p(A) = 1 / 1.909 = 0.5238
implied p(B) = 1 / 1.909 = 0.5238
               ---------------------
sum          =            1.0476   <- overround 4.76%

margin on turnover = 1 - (1 / 1.0476) = 4.55%

a fair book on the same event would price both sides at 2.00

Balancing a book

A book is balanced when the stakes taken on each outcome are in proportion to the prices, so that the same amount is retained whichever outcome occurs. Prices move as money arrives, not because an opinion has changed but because the liability distribution has, and a price that shortens is a price that has attracted stake.

Perfect balance is unusual in practice, so a book normally carries some exposure to particular results and manages it by moving prices, limiting stakes or laying off elsewhere. The arithmetic of the margin is unaffected: it is set at the moment the prices are quoted.

Where the margin sits

Overround is not distributed evenly across a market. Long-shot outcomes conventionally carry a larger share of it than short-priced ones, which means the effective cost of backing an unlikely result is higher than the headline margin suggests. Markets with many outcomes carry a larger total overround than two-way markets, simply because there are more prices each carrying a deduction.

The practical consequence is that comparing the headline margin of a two-way market with that of a twenty-outcome market compares two different quantities.

What a price is not