# The math behind arbitrage

Arbitrage opportunities considering two way markets are easy to understand and to detect. For experienced arbers, just a fleeting glimpse at the following odds is enough to detect these arbitrage opportunities.

Djokovic 2.20 (+120)

Nadal 1.91 (-110)

Federer 1.55 (-181)

Gasquet 3.00 (+200)

Almost all arbers will recognise these situations as arbitrage opportunities. How do we do it?

## American odds.

Well, it’s quite easy to detect two way arbitrage opportunities on American odds; Positive odd must be higher than negative odds (considering its absolute value). Why is that?

Positive odds represent the profit for a 100 stake. +120 means our profit is 120 for a 100 stake.

Negative odds represent the stake necessary to win 100. -110 means we must bet 110 to win 100.

If the stake necessary to win 100 is less than the profit for a 100 stake on the opposite outcome, we have a surebet.

## Decimal odds.

The situation is not so obvious on Decimal odds, but still easy. We know an odd 2 requires at least another 2, an odd 3 requires at least 1.5, an odd 4 requires at least 1.333, an odd 5, requires 1.25, an odd 6 requires 1.2, etc. We know it by experience, but the formula to detect an arbitrage opportunity is easy. You simply must add the reciprocal of both odds (multiplicative inverse: 1/odd). If the sum is less than 1, there is an arb.

Let’s prove it. Assuming a two way market with odds O1 and O2 for both outcomes, we want to know if these odds represent an arbitrage opportunity and how much to bet (both stakes, S1 and S2). The equations we want are:

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