How an arbitrage bet is calculated: the 100% test and the stakes

An arbitrage exists when the implied probabilities of every outcome, taken at the longest price offered for each, add up to less than 100%. Stake each outcome in proportion to its implied probability. At 2.04 on one side and 2.02 on the other the total is 98.52%, a 1.50% margin, and either result returns the same amount.

Work any pair of prices through the arbitrage calculator to check the stakes below.

Step 1: put every price in decimal

A positive American price converts as 1 + price ÷ 100, so +104 is 2.04. A negative price converts as 1 + 100 ÷ the price without its sign, so -110 is 1.909. Decimal is the return on one unit including the stake, which is what the next step needs.

Step 2: the 100% test

Take 1 ÷ the decimal price for each outcome and add them. This sum is the book percentage of the market you have assembled from those prices. If it is below 1, the margin is 1 ÷ sum − 1. If it is 1 or above, no stake split gives the same result on every outcome.

Step 3: split the stake

For a total outlay T, the stake on each outcome is T × (1 ÷ its decimal price) ÷ the sum. With T = 1,000:

• Stake on the 2.04 side: 1,000 × 0.4902 ÷ 0.9852 = 497.54
• Stake on the 2.02 side: 1,000 × 0.4950 ÷ 0.9852 = 502.46
• If the first wins: 497.54 × 2.04 = 1,014.98 returned
• If the second wins: 502.46 × 2.02 = 1,014.98 returned

The return is the same on both sides, 14.98 above the 1,000 outlay. The shorter price takes the larger share of the money, because it has to return the same amount on a smaller multiple.

If you can only stake one side

When one stake is fixed, say 500 at 2.04, the other is 500 × 2.04 ÷ 2.02 = 504.95. That makes the total outlay 1,004.95 and either result returns 1,020, a 15.05 gain. The rule is: stake on the other outcomes whatever makes their return equal to the first one's return.

More than two outcomes

The same three steps apply to any number of outcomes: add 1 ÷ price for each, check the sum is below 1, and split in proportion. A three-way soccer market is handled in its own article.

Why the margin is small

A gap between two sets of prices exists because the prices were set at different times or from different models. It closes quickly once someone acts on it, and the typical size is covered in the article on why arbitrage is smaller than it sounds. A 1.5% margin on a 1,000 outlay is about 15, before any cost of moving money between accounts.

Check a pair. Enter the prices from both sides into the arbitrage calculator to see the sum, the margin and the stakes for any total.