Worked Example: An Arbitrage Window From Open to Close

What is an arbitrage window

An arbitrage window exists when the combined implied probabilities of all outcomes in a market sum to less than 100%. The gap between 100% and the sum is the arbitrage edge.

A simple example:

Outcome	Book A	Book B	Implied Prob
Team X win	1.50	—	66.7%
Draw	—	3.80	26.3%
Team Y win	—	2.20	45.5%
Total			138.5%

This total exceeds 100%, so there is no arbitrage. Both books are overround (they keep margin).

But if Team Y's price were 2.25 instead of 2.20:

Outcome	Book A	Book B	Implied Prob
Team X win	1.50	—	66.7%
Draw	—	3.80	26.3%
Team Y win	—	2.25	44.4%
Total			137.4%

Still overround. No arbitrage.

But if Team Y were 2.86:

Outcome	Book A	Book B	Implied Prob
Team X win	1.50	—	66.7%
Draw	—	3.80	26.3%
Team Y win	—	2.86	35.0%
Total			128.0%

Still overround, still no arbitrage.

True arbitrage would require total under 100%, which almost never happens in mainstream markets. Instead, semi-professionals look for positive-value windows where one outcome is mispriced relative to fair value, even if true arbitrage does not exist.

But let us work through a scenario where a brief technical arbitrage opens and closes.

The scenario: A mispriced set

Setup: Monday evening, League Cup match. Team P (strong favorite) vs Team Q (weak underdog).

Two books, BookAlpha and BookBeta:

Opening prices:

Outcome	BookAlpha	BookBeta	Fair value (assumed)
P win	1.50	1.52	1.50
Draw	3.40	3.50	3.40
Q win	5.50	5.50	5.50

Both books are close to fair value, carrying roughly 5% overround.

Implied probabilities:

• P win: 66.7% (BookAlpha), 65.8% (BookBeta).
• Draw: 29.4% (BookAlpha), 28.6% (BookBeta).
• Q win: 18.2% (both books).

Total probabilities:

• BookAlpha: 114.3% (normal overround).
• BookBeta: 114.4% (normal overround).

Both books are standard recreational pricing with no arbitrage window.

3:15 PM — News breaks (90 minutes to kickoff)

An injury to Team P's key midfielder is confirmed. This is material negative news for Team P.

What might happen: Team P's price may lengthen. The draw and Team Q prices may also shorten, though the exact repricing depends on the market's assessment.

What actually happens:

• BookAlpha: Moves P to 1.70, Draw to 3.20, Q to 4.80 within 5 minutes. These prices are fast and decisive. The book is repricing in real-time.
• BookBeta: Still at opening prices (1.52, 3.50, 5.50) as of 3:18 PM. The book is slower to react or has not yet processed the news.

3:18 PM market state:

Outcome	BookAlpha	BookBeta	Implied (BA)	Implied (BB)
P win	1.70	1.52	58.8%	65.8%
Draw	3.20	3.50	31.3%	28.6%
Q win	4.80	5.50	20.8%	18.2%
Total			110.9%	112.6%

BookAlpha has repriced tighter (lower overround). BookBeta is stale.

The arbitrage check:

Can you bet these three outcomes at the stated prices and guarantee profit?

Let us say you want to stake money such that you net the same payoff regardless of outcome.

If you assume true probability is:

• P win: 62%
• Draw: 28%
• Q win: 10%

Then:

• P at 1.70 (58.8% implied) is undervalued. You want to take it.
• Draw at 3.50 (28.6% implied) is fair. You are indifferent.
• Q at 4.80 (20.8% implied) is overvalued. You do not want it.

This is not a true arbitrage; it is a value play. You have an opinion (P is underpriced even after the repricing) and you take it.

But what if the window is tighter?

3:20 PM — Trying for true arbitrage

You notice the imbalance and decide to check if a true arbitrage exists within these four book combinations available to you (you have access to 4 books, but we will focus on two for this example).

Alternative prices you can access:

Outcome	Best price for back	Best price for lay
P win	1.70 (BookAlpha)	1.52 (BookBeta)
Draw	3.50 (BookBeta)	3.20 (BookAlpha)
Q win	5.50 (BookBeta)	4.80 (BookAlpha)

Now check: can you back all three at the best available and come out ahead?

Scenario: Back €100 on each outcome at the best prices:

• Back P at 1.70: return €170 if P wins.
• Back Draw at 3.50: return €350 if Draw.
• Back Q at 5.50: return €550 if Q wins.

Total stake: €300. Total returns: Min(€170, €350, €550) = €170 (if P wins).

Expected value if true probability is the fair line (assumed):

• 62% × €170 + 28% × €350 + 10% × €550 = €105.4 + €98 + €55 = €258.4.
• On €300 staked, that is -13.9% return (you lose value).

This is not an arbitrage. You are still risking losing money because the combined implied probability still exceeds fair value.

An arbitrage calculation requires all accepted legs to produce the same positive settlement after total stakes. That condition is uncommon and can disappear before every leg is accepted.

3:22 PM — A different angle: Lay and back the extreme

Instead of trying all three outcomes, let us try a two-leg spread.

You believe:

1. P's repricing at 1.70 (BookAlpha) is good value — P is unlikely to lose given the injury news might be overreacted.
2. Q's price at 5.50 (BookBeta) is bad value — Q is given too much chance.

Bet 1: Back P at 1.70 for €500 (BookAlpha). Return if P wins: €850.

Bet 2: Lay Q at 5.50 for €100 (on an exchange or bet against Q winning for €500 at 1.20 equivalent). Return if Q does not win: €500 × 1.20 = €600. If Q wins, loss on this leg is €(500 × 4.50) = -€2,250.

This is not a complete hedge. Its settlement changes with the outcome.

Correct hedged approach:

Use stake splitting to guarantee the same return on any outcome.

Illustrative target: the same return on any outcome. This is a calculation check, not an instruction to place a wager.

Let us use:

• Stake on P at 1.70 = x.
• Stake on Q at 5.50 = y.

We want:

• If P wins: €x × 0.70 (profit part of 1.70) = €50.
• If Q wins: €y × 4.50 (profit part of 5.50) = €50.
• If Draw: €50 (from a separate draw bet or hedge).

From the first two equations:

• x = €50 / 0.70 = €71.43
• y = €50 / 4.50 = €11.11

Hedged bet:

• Back P at 1.70 for €71.43 (BookAlpha).
• Back Q at 5.50 for €11.11 (BookBeta).
• Back Draw at some price for €17.46 (remaining amount to reach €100 stake).

Total stake: €100.

Outcome 1 - P wins:

• P leg: €71.43 × 1.70 = €121.43. Profit: €50.
• Q leg: €11.11 × 0 (loses). Loss: €11.11.
• Draw leg: €17.46 × 0 (loses). Loss: €17.46.
• Net: €50 - €11.11 - €17.46 = €21.43.

This is not hedged properly. To achieve true hedging, we need to use the Draw price as well.

3:25 PM — Proper stake splitting with three legs

Goal: Net €30 profit on any outcome, €100 total stake.

Outcome	Bet	Amount	Odds	Return if win	Net if win
P win	Back at BookAlpha	€58.82	1.70	€100	€41.18
Draw	Back at BookBeta	€30.77	3.50	€100	€69.23
Q win	Back at BookBeta	€18.18	5.50	€100	€81.82
Total stake		€107.77

This does not work because stake total is €107.77, not €100.

Let me recalculate correctly.

Proper stake split for guaranteed €30 payout on any outcome:

We want:

• Stake_P × 1.70 = €130 (€100 initial + €30 profit).
• Stake_D × 3.50 = €130.
• Stake_Q × 5.50 = €130.

So:

• Stake_P = €130 / 1.70 = €76.47.
• Stake_D = €130 / 3.50 = €37.14.
• Stake_Q = €130 / 5.50 = €23.64.

Total stake: €76.47 + €37.14 + €23.64 = €137.25.

Expected return: €130 (any outcome).

Profit: €130 - €137.25 = -€7.25 loss on any outcome.

This is the reality: Even with BookAlpha and BookBeta's prices, you cannot create a true arbitrage because the combined overround is still positive (~12–13% total).

3:28 PM — What actually happens

You check your available books one more time:

New data point: BookGamma (a third book you have access to) has just opened the match:

Outcome	BookGamma
P win	1.75
Draw	3.10
Q win	4.50

These prices are tighter than BookAlpha on two outcomes!

Outcome	Best price now
P win	1.75 (BookGamma)
Draw	3.50 (BookBeta)
Q win	5.50 (BookBeta)

New stake split calculation:

• Stake_P = €130 / 1.75 = €74.29.
• Stake_D = €130 / 3.50 = €37.14.
• Stake_Q = €130 / 5.50 = €23.64.

Total: €135.07.

Profit: €130 - €135.07 = -€5.07 loss on any outcome.

Still a loss, but smaller. The prices are slowly aligning.

3:32 PM — BookBeta catches up

BookBeta adjusts to match the market:

Outcome	BookBeta (new)
P win	1.68
Draw	3.30
Q win	4.95

The window is closing. BookBeta is repricing and the opportunities are shrinking.

New best prices:

Outcome	Best price
P win	1.75 (BookGamma)
Draw	3.50 (old BookBeta, but it just moved to 3.30)
Q win	5.50 (old BookBeta, still there?)

Actually, you need to act before BookBeta moves, not after.

By 3:32, if you had not already placed bets, the window has closed.

What to check instead

Before describing any price set as arbitrage, check:

1. Opening lines across 3–4 key books.
2. Repricing lags when news breaks.
3. Value plays where one outcome is clearly mispriced relative to fair value.

In the scenario above:

At 3:18 (right after news): BookBeta has not moved, but the selected best prices still sum to more than 100% implied probability. The set is therefore not an arbitrage.

The older 1.52 price on Team P is shorter than the newer 1.70 price. After news that weakens Team P, that difference does not make 1.52 a better price for Team P. It is a reason to stop, verify the market logic, and avoid treating a stale quote as a conclusion.

Key lessons from the scenario

1. True arbitrage is rare. In modern regulated markets with sophisticated bookmakers, guaranteed-profit arbitrage almost never exists under normal conditions.

2. A price difference is not enough. It may reflect a different margin, market rule, timing, or an error in the comparison.

3. Stake splitting is arithmetic. Use one common target payout and verify that the total stake is lower than that payout before calling the set arbitrage.

4. Verification matters. A timestamped quote can change, and the comparison should be rechecked before any conclusion.

5. Record the context. Track the selected prices, settlement rules, and time of observation so the calculation can be audited later.

For details on the underlying framework, see Line Shopping Across 50+ Books: A Worked Example and Pinnacle vs. Soft Books: Why the Same Match Prices Differently.

The discipline is to check the arithmetic, identify whether the prices describe an arbitrage at all, and record the context. A movement or price difference is not a bet instruction.

Configure this in PhotonOdds

Use Strategies to review comparisons you define. Read Arbitrage Betting Explained and Arbitrage Stake Split Maths before relying on any calculation.

18+ only. Betting carries risk. PhotonOdds provides analytical and educational tools, not a promise of profit or a recommendation to place a bet. If gambling is causing harm, see Responsible Gambling.