# Arbitrage Betting

There is a number of potential arbitrage deals. Below is an explanation of some of them including formulas and risks associated with these arbitrage deals. The table below introduces a number of variables that will be used to formalise the arbitrage models.

Variable | Explanation |

s_{1} |
Stake in outcome 1 |

s_{2} |
Stake in outcome 2 |

o_{1} |
Odds for outcome 1 |

o_{2} |
Odds for outcome 2 |

r_{1} |
Return if outcome 1 occurs |

r_{2} |
Return if outcome 2 occurs |

This type of arbitrage takes advantage of different odds offered by different bookmakers. Assume the following situation:

We consider an event with 2 possible outcomes (e.g. a tennis match - either Federer wins or Henman wins), the idea can be generalized to events with more outcomes, but we use this as an example.

The 2 bookmakers have different ideas of who has the best chances of winning. They offer the following Fixxed odds gambling on the outcomes of the event

Bookmaker 1 | Bookmaker2 | |

Outcome 1 | 1.25 | 1.43 |

Outcome 2 | 3.9 | 2.85 |

For an individual bookmaker, the **sum of the inverse of all outcomes** of an event will always be greater than 1. 1.25 ^{? 1} + 3.9 ^{? 1} = 1.056 and 1.43 ^{? 1} + 2.85 ^{? 1} = 1.051

The fraction above 1, is the bookmakers return rate, the amount the bookmaker earns on offering bets at some event. Bookmaker 1 will in this example expect to earn 5.6% on bets on the tennis game. Usually these gaps will be in the order 8 - 12%.

The idea is to find odds at different bookmakers, where the sum of the inverse of all the outcomes are below 1. Meaning that the bookmakers disagree on the chances of the outcomes. This discrepancy can be used to obtain a profit.

For instance if one places a bet on outcome 1 at bookmaker 2 and outcome 2 at bookmaker 1:

1.43 ^{? 1} + 3.9 ^{? 1} = 0.956

Placing a bet of 100$ on outcome 1 with bookmaker 2 and a bet of $100 * 1.43 / 3.9 = 36.67 on outcome 2 at bookmaker 1 would ensure the bettor a profit.

In case outcome 1 comes out, one could collect *r*_{1} = $100 * 1.43 = $143 from bookmaker 2. In case outcome 2 comes out, one could collect *r*_{2} = $36.67 * 3.9 = $143 from bookmaker 1. One would have invested $136.67, but have collected $143, a profit of $6.33 (%4.6) no matter the outcome of the event.

So for 2 odds *o*_{1} and *o*_{2}, where . If one wishes to place stake *s*_{1} at outcome 1, then one should place *s*_{2} = *s*_{1} * *o*_{1} / *o*_{2} at outcome 2, to even out the odds, and receive the same return no matter the outcome of the event.

Or in other words, if there are two outcomes, a 2/1 and a 3/1, by covering the 2/1 with $500 and the 3/1 with $333, one is guaranteed to win $1000 at a cost of $833, giving a 20% profit. More often profits exists around the 4% mark or less.

Betting exchanges have opened up a new range of arbitrage possibilities since on the exchanges it is possible to *lay* (i.e. to bet against) as well as to back an outcome. Arbitrage using only the back or lay side might occur on betting exchanges. It is in principle the same as the arbitrage using different bookmakers. Arbitrage using back and lay side is possible if a lay bet on one exchange provides shorter odds than a back bet on another exchange or bookmaker. However, the commission charged by the bookmakers and exchanges must be included into calculations.

Back-lay sports arbitrage is often called *scalping* or *trading*. Scalping is not actually arbitrage, but short term trading. In the context of sports arbitrage betting a scalping trader or scalper looks to make lots of small profits, which in time can add up. In theory a trader could turn a small investment into large profits by re-investing his earlier profits into future bets so as to generate exponential growth. Scalping relies on liquidity in the markets and that the odds fill flucuate around a mean point. A key advantage to scalping on one exchange is that most exchanges charge commission only on the net winnings in a particular event, thus ensuring that even the smallest favourable difference in the odds will guarantee some profit.