ADGSTUDIOS | Arbitrage Profitability Calculator

We begin by assuming a total stake \( T \) distributed between two outcomes: \[ x + y = T. \]

For outcome A with odds \( a \) and outcome B with odds \( b \), the returns are: \[ \text{Return A} = a \cdot x, \quad \text{Return B} = b \cdot y. \]

Setting the returns equal for an arbitrage opportunity: \[ a \cdot x = b \cdot y = R. \]

This gives: \[ x = \frac{R}{a}, \quad y = \frac{R}{b}. \]

Substituting back into the total stake: \[ \frac{R}{a} + \frac{R}{b} = T \quad \Rightarrow \quad R \left(\frac{1}{a} + \frac{1}{b}\right) = T. \]

Hence, the common return is: \[ R = \frac{T}{\frac{1}{a} + \frac{1}{b}}. \]

For an arbitrage to be profitable, we require \( R > T \), which simplifies to: \[ \frac{1}{a} + \frac{1}{b} < 1. \]

Lemma (Arbitrage Condition)

Lemma: For any two odds \( a \) and \( b \), if \[ \frac{1}{a} + \frac{1}{b} < 1, \] then a two‑way arbitrage opportunity exists.

Theorem (Profitability Theorem)

Theorem: Given a total stake \( T \) and odds \( a \) and \( b \), the arbitrage profit margin \( P \) is: \[ P = 1 - \left(\frac{1}{a} + \frac{1}{b}\right), \] and an arbitrage opportunity exists if and only if \( P > 0 \).

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In a market with \( n \) outcomes and \( k \) bookmakers, let the odds provided for outcome \( i \) by bookmaker \( j \) be denoted as \( o_{ij} \). Define the best odds for outcome \( i \) as: \[ o_i^* = \max_{j=1,\dots,k} o_{ij}. \] An arbitrage opportunity exists if: \[ \sum_{i=1}^{n} \frac{1}{o_i^*} < 1, \] with profit margin: \[ 1 - \sum_{i=1}^{n} \frac{1}{o_i^*}. \]

Advanced Lemma (Best Odds Selection)

Lemma: For a given outcome \( i \) and a set of \( k \) bookmakers, the optimal odds is: \[ o_i^* = \max_{1 \leq j \leq k} o_{ij}. \]

Advanced Theorem (Matrix Arbitrage Theorem)

Theorem: Given a matrix of odds \( \{ o_{ij} \} \) for \( n \) outcomes and \( k \) bookmakers, an arbitrage opportunity exists if: \[ \sum_{i=1}^{n} \frac{1}{\max_{1 \leq j \leq k} o_{ij}} < 1, \] with profit margin: \[ 1 - \sum_{i=1}^{n} \frac{1}{\max_{1 \leq j \leq k} o_{ij}}. \]

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For a series of \( k \) independent matches (or games), assume each match \( g \) has \( n_g \) outcomes with best odds \( o_{i}^{*(g)} \) for outcome \( i \) in match \( g \). Consider a combined bet covering every possible outcome combination across the \( k \) matches. The compound odds for a specific combination \( (i_1, i_2, \dots, i_k) \) is: \[ O_{i_1,i_2,\dots,i_k} = \prod_{g=1}^{k} o_{i_g}^{*(g)}. \]

If a total stake \( T \) is distributed among all \( N = \prod_{g=1}^{k} n_g \) combinations, with each combination receiving a bet \( x_{i_1,i_2,\dots,i_k} \) such that: \[ \sum_{(i_1,\dots,i_k)} x_{i_1,i_2,\dots,i_k} = T, \] and aiming for a common return \( R \), we set: \[ x_{i_1,i_2,\dots,i_k} = \frac{R}{O_{i_1,i_2,\dots,i_k}}. \]

Summing over all combinations gives: \[ R \left( \sum_{(i_1,\dots,i_k)} \frac{1}{O_{i_1,i_2,\dots,i_k}} \right) = T. \]

Thus, the common return is: \[ R = \frac{T}{\sum_{(i_1,i_2,\dots,i_k)} \frac{1}{O_{i_1,i_2,\dots,i_k}}}. \]

For the arbitrage to be profitable, we require \( R > T \), which implies: \[ \sum_{(i_1,i_2,\dots,i_k)} \frac{1}{O_{i_1,i_2,\dots,i_k}} < 1. \]

The profit margin for the multi-match arbitrage is then: \[ P = 1 - \sum_{(i_1,i_2,\dots,i_k)} \frac{1}{O_{i_1,i_2,\dots,i_k}}. \]

Multi-Match Arbitrage Theorem

Theorem: For \( k \) independent matches, with each match \( g \) having \( n_g \) outcomes and corresponding best odds \( o_{i}^{*(g)} \), an overall arbitrage opportunity exists if: \[ \sum_{(i_1,i_2,\dots,i_k)} \frac{1}{\prod_{g=1}^{k} o_{i_g}^{*(g)}} < 1. \] The total arbitrage profit margin is given by: \[ P = 1 - \sum_{(i_1,i_2,\dots,i_k)} \frac{1}{\prod_{g=1}^{k} o_{i_g}^{*(g)}}. \]