What is the arbitrage situation?

The theory.

The term:

Arbitrage situation - a situation in the market of reception of rates between two or more participants when their offers on separate event during the certain moment are included into the mutual contradiction.

For occurrence of a arbitrage situation performance simultaneously two obligatory conditions is necessary:

1. Offers participants of this market are exposed on mutually opposite offers (outcomes).
2. Odds for these offers are in the certain mathematical parity.

Odds show the player parity between the sum of the bet on outcome and possible win in case the rate is guessed. Today in the world of sports and bets on sport event there are three parallel systems display Odds. All the further reasonings which you will find on our site will be, to concern only in decimal system. In this system of the Odd are displayed in decimal number which shows, how many payment under the won bet is more than times the sum of the bet.

Let's look at the first obligatory condition of occurrence of arbitrage situation. What is the opposite outcomes? If we look in a bookies line, such outcomes usually adjoin with each other. For example, in tennis - K1 and K2, in hockey or soccer - K1, X, K2, etc.

Again the strict term.
Mutually opposite outcomes in an arbitration situation completely block set of probable outcomes of event. Therefore, the player on arbitrage situation, doing stakes on a concrete match should be assured, that any outcome of a match necessarily even one outcome will play, and payments from winnings bet with profit will pay back the lost rates on other outcomes of this match.

The second condition of occurrence in arbitrage situation is just responsible for reception of profit on bets in arbitrage situation. This mathematical parity counts value profit under with stakes on outcomes. Further the general technique of reception of this parity with pair examples for the most typical kinds of arbitrage situation is resulted. To practise to the reader drawing up of these parities for all variety of combinations of outcomes on which occurrence of arbitrage situation is possible, we leave it for personal leisure.

Next important term.
Profit(profitability) is a percentage expression of profit margin on the operation carried to the sum of expenses after this.

In our case, expense - that other, as the sum of stakes on all outcomes in arbitrage situation. Depending on quantity of those:

LOST = B₁ + B₂ + ….

Here B₁, B₂ - the sums of stakes on outcomes.

the sums of stakes on outcomes

Profit=(WIN-LOST)/ LOST * 100%

It is necessary for us, that in any case the result was positive, that is profitability> 0. Also it is generally that parity about which it is spoken in the second item. You may ask, where Odds? The matter is that value WIN depends on factors and set of opposite outcomes. For different types of arbitrage situation - outcomes on which these situations are caught is different. Profit is an important for us in any case, so in any outcome of event:

WIN= WIN1= WIN2=…,etc.

it is so much, how many possible outcomes participate in this arbitration situation. Further. Having expressed for each possible outcome value WIN through coefficient K and the sum of bet B, we shall receive incomplete system of the equations. From it parities between the sums of rates and factors are extracted. It looks approximately so:

B₂=B₁*(something correlation 1)
B₃=B₁*(something correlation 2)

Then we collect this all in to the parity for profit - and all business. So complicated? – Invite to help you anybody.

Time for example.

Tennis match.
We search for arbitrage situation on opposite outcomes.
We have, accordingly, odds K₁ and K₂.
Stakes on these outcomes - B₁ and B₂.
Then,

LOST=B₁ + B₂

WIN= WIN1=B₁ * K₁= WIN2=B₂ * K₂

Parity between the sums of stakes and coefficients:

B₁ * K₁ = B₂ * K₂

B₂ = B₁ * K₁ / K₂

Then we collect this to the formula for profit:

PROFIT = (WIN-LOST)/ LOST * 100% = (B₁ * K₁ / (B₁ + B₂))/ (B₁ + B₂) * 100%

Change B₂ на B₁ * K₁ / K₂:

PROFIT = (B₁ * K₁ / (B₁ + B₁ * K₁ / K₂))/ (B₁ + B₁ * K₁ / K₂) * 100%

reduce B₁

PROFIT = (K₁ * K₂/(K₁ + K₂) - 1) * 100% > 0

Therefore, if factors on victories of contenders over a tennis match satisfy to this inequality - means, we deal with arbitrage situation. Otherwise arbitrage situation is not present. Certainly, in this case K₁ and K₂ - offers different bookies, as in a line of one books on the same match of such Odds will not give, differently it not arbitrage situation, and this will be a “technical mistake of the personnel”.

Next example.

A soccer match.
We search for arbitrage situation on three opposite outcomes: Sprd1(0), Sprd2(-0,5), Sprd2(+0,5). Where Sprd1 and Sprd2 is spread.
Odds K₁, K₂, K₃.
We put B₁, B₂, B₃.

LOST=B₁ + B₂ + B₃

We consider a winnings for three outcomes accordingly
1. The match ended with a difference in 1 goal and more in favour of first team.

WIN1=B₁ * K₁

2. The match ended with a difference in 0 goals (draw)

WIN2=B₁ + B₃ * K₃

3. The match ended with a difference in 1 goal and more in favour of second team.

WIN3= B₂ * K₂ + B₃ * K₃

Next step,

WIN= B₁ * K₁ = B₁ + B₃ * K₃ = B₂ * K₂ + B₃ * K₃

In this case,

B₂ = B₁ / K₂

B₃ = B₁ * (K₁ - 1) / K₃

and last step, collect all:

PR = (WIN-LOST)/ LOST * 100% = (B₁ * K₁ / (B₁ + B₂ + B₃))/ (B₁ + B₂ + B₃) * 100% =

= K₁/(1 + 1/K₂ + K₁/K₃ - 1/K₃) * 100% - 100%>0

This is formula of profit calculation on arbitrage situation with participation of the basic whole spread odds.

On it, the given question I consider considered entirely and in details. In end I would be desirable to notice, that all variety of combinations of outcomes on which occurrence in arbitration situations easily is possible pays off by the technique resulted here.