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Math-Driven Edge

Arbitrage Calculator Pro

Exploit price discrepancies between market makers. Cover every outcome. Outcomes are mathematically defined — regardless of who wins.

01

Find Discrepancy

Scan odds across market makers for events where implied probabilities sum to less than 100%.

02

Calculate Stakes

Our engine calculates exactly how many units to place on each side to mathematically balance your position.

03

Capture the Edge

Execute the strategy. No matter who scores or wins, your mathematical edge is locked in before the game even starts.

The Mathematics Behind Arbitrage

Foundation

Arbitrage Detection Condition

An arbitrage exists when the sum of implied probabilities across all mutually exclusive outcomes is strictly less than 1:

$$\text{Arb Margin} = \sum_{i=1}^{n} \frac{1}{O_i} < 1$$

Where \(O_i\) = decimal odds for outcome \(i\), and \(n\) = total number of possible outcomes.

Core Formula

Optimal Stake Allocation

To equalize returns regardless of outcome, distribute your total investment \(T\) as follows:

$$S_i = \frac{T \cdot \dfrac{1}{O_i}}{\displaystyle\sum_{j=1}^{n} \dfrac{1}{O_j}}$$

Where \(S_i\) = stake on outcome \(i\), \(T\) = total bankroll to invest.

Profit Formula

Calculated Edge

Your identified statistical edge from any arbitrage opportunity is:

$$\text{ROI} = \left(\frac{1}{\displaystyle\sum_{i=1}^{n} \frac{1}{O_i}} - 1\right) \times 100\%$$

The smaller the margin sum, the larger your built-in edge. Values below 0.95 are excellent opportunities.

Worked Example: Step-by-Step Derivation

Tennis — ATP Final

Player A vs Player B

market maker 1 Player A @ 2.10
market maker 2 Player B @ 2.15
Step 1 Check for Arbitrage
$$\frac{1}{2.10} + \frac{1}{2.15} = 0.4762 + 0.4651 = 0.9413$$

0.9413 < 1.0 — Arbitrage Confirmed!

Step 2 Calculate Stakes (Total Investment: 1,000 units)
$S_A = \frac{1000 \times 0.4762}{0.9413} = 505.84 \text{ units}$
$S_B = \frac{1000 \times 0.4651}{0.9413} = 494.16 \text{ units}$
Step 3 Verify Calculated Payout
Scenario Calculation Payout Net Delta
Player A Wins 505.84 × 2.10 1,062.26 units +62.26 units
Player B Wins 494.16 × 2.15 1,062.44 units +62.44 units
Calculated Delta Positive Delta — Statistical Edge

Outcome-agnostic structure. Execution-dependent. Pure mathematics.

Note: Arbitrage depends on execution timing, market maker rules, and stake limits.

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PhD Level

Probabilistic Modeling Suite

A publication-grade probabilistic engine that identifies positive Expected Value (+EV) opportunities with mathematical precision. When the edge is in your favor, compounding does the rest.

Probability Modeling

Bivariate Poisson, xG regression, and Bayesian networks to estimate true outcome probabilities.

Edge Detection

Compare our model's probabilities against market maker odds to find exploitable value gaps.

Kelly Staking

Fractional Kelly Criterion dynamically sizes each position for optimal geometric bankroll growth.

Risk Control

CVaR95 Monte Carlo simulation protects against ruin and keeps drawdowns manageable.

Core Mathematical Framework

Core

Expected Value (EV)

An opportunity has positive expected value when your estimated probability multiplied by the payout exceeds your stake:

$$EV = \underbrace{P_{\text{win}} \cdot (O - 1)}_{\text{expected gain}} - \underbrace{(1 - P_{\text{win}}) \cdot 1}_{\text{expected loss}}$$

Simplified to the edge percentage:

$$\text{Edge} = P_{\text{win}} \times O - 1$$

We only execute when \(\text{Edge} > 0\). A 5% edge corresponds to an expected value of +5 units per 100 units over the long run.

Staking

Fractional Kelly Criterion

Determines the mathematically optimal fraction of your bankroll to wager on each position:

$$f^* = c \cdot \frac{b \cdot p - q}{b}$$
\(b\) = net decimal odds \((O - 1)\)
\(p\) = probability of winning
\(q\) = probability of losing \((1 - p)\)
\(c\) = fractional multiplier (typically 0.25)

Using \(c = 0.25\) (Quarter Kelly) reduces variance by 75% while retaining 50% of the growth rate of Full Kelly.

Prediction

Bivariate Poisson Model

Models the joint probability of two teams scoring \(x\) and \(y\) goals, accounting for correlation \(\theta\):

$$P(X\!=\!x, Y\!=\!y) = e^{-(\lambda+\mu+\theta)} \frac{\lambda^x}{x!}\frac{\mu^y}{y!} \sum_{k=0}^{\min(x,y)} \binom{x}{k}\binom{y}{k} k! \left(\frac{\theta}{\lambda\mu}\right)^{\!k}$$

Where \(\lambda\) = home attack, \(\mu\) = away attack, \(\theta\) = correlation coefficient (Dixon-Coles correction).

Adaptation

Bayesian Updating

Our system continuously updates probabilities as new evidence (injuries, weather, line movements) arrives:

$$P(H | E) = \frac{P(E | H) \cdot P(H)}{P(E)}$$
\(P(H)\) = prior probability (model's baseline)
\(P(E|H)\) = likelihood of evidence given hypothesis
\(P(H|E)\) = posterior probability (updated belief)

Real-time Bayesian updating ensures our probability estimates incorporate the latest market intelligence.

Worked Example: Finding a +EV Bet

Football — Premier League

Liverpool vs Manchester City

market maker Odds Liverpool Win @ 2.80
Our Model Liverpool Win Prob = 42%
Step 1 Calculate the Edge
$$\text{Edge} = 0.42 \times 2.80 - 1 = 1.176 - 1 = +0.176$$

Edge = +17.6% — Strong positive EV!

Step 2 Calculate Kelly Stake (Bankroll: 10,000 units, c = 0.25)
$$f^* = 0.25 \times \frac{1.80 \times 0.42 - 0.58}{1.80} = 0.25 \times \frac{0.756 - 0.58}{1.80} = 0.25 \times 0.0978 = 2.44\%$$

Optimal stake = 244 units (2.44% of bankroll)

Step 3 Expected Long-Term Value
$EV = 0.42 \times (2.80 - 1) \times 244 - 0.58 \times 244 = 184.46 \text{ units} - 141.52 \text{ units} = +42.94 \text{ units}$
Expected Value Per Event +42.94 units per occurrence

Over 100 similar occurrences, expected aggregate EV: Positive Yield. Variance shrinks. Math wins.

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