Arbitrage and odds

Idea: Profit in every outcome

An arbitrage makes money whatever happens. The simplest: one bid above another maker's ask.

• A: $99$ bid, $100$ ask. B: $101$ bid, $102$ ask
• Buy at A's ask, sell at B's bid: $5\times (101-100)=5$ on $5$ lots
• You end flat, so the price no longer matters

Formula: Check both directions

$\text{profit}=n\times (b-c),ifb>c$

Compare each bid b with the other's ask c. Fair value plays no part.

Common mistake: Taking the gap between mids

Mids $99.5$ and $101.5$ suggest $101.5-99.5=2$ per lot.

Instead You buy at an ask and sell at a bid: $101-100=1$ per lot.

Formula: Odds are prices

$\text{price}=\frac{1}{d}$

Decimal odds d return d per stake of $1$, so the bet breaks even at probability 1 over d.

Example: Remove the margin

Odds $1.5$ and $2.4$, exactly one outcome happens. Fair chance of the first?

1. Prices: $\frac{1}{1.5}+\frac{1}{2.4}=\frac{13}{12}$, the margin is $\frac{1}{12}$
2. Divide by the sum: $\frac{\frac{1}{1.5}}{\frac{13}{12}}=\frac{8}{13}$

Idea: When prices add to less than 1

Best odds $2.5$ on A and $2$ on B: $\frac{1}{2.5}+\frac{1}{2}=0.9$, less than the sure payoff.

• Stake in proportion to $\frac{1}{d}$: $400$ on A, $500$ on B
• Either way: $400\times 2.5=1000$ and $500\times 2=1000$
• Stake limits and voided bets can break the lock

Quick check: The locked profit

Total stake $900$ split as above on odds $2.5$ and $2$. What is the sure profit?

Idea: Say it out loud

For quotes, say which bid you hit and which ask you lift, and check both directions.

• For odds: convert to prices, add, compare with $1$
• Above $1$ is margin, below is an arbitrage
• Name what could break the lock