Sizing an arbitrage with lopsided odds

Question

A two-outcome fight. Book A offers 2.30 on the favorite winning; Book B offers 2.00 on the underdog winning. The prices are lopsided, so an equal \500 / \500 split would leave you exposed to which side actually wins. Confirm an arbitrage exists, then size a \1000 bankroll so your profit is identical regardless of outcome. What is the return?

Accepted Answer

Arbitrage check: \frac{1}{2.30} + \frac{1}{2.00} = 0.4348 + 0.5000 = 0.9348 < 1, so a Dutch book exists. Equalizing payouts. If you stake sA on the favorite and sB on the underdog, the two payouts are 2.30\,sA and 2.00\,sB. Set them equal and use sA + sB = 1000. The clean way is to make each stake proportional to its implied probability: sA = \frac{0.4348}{0.9348}	imes 1000 \approx \465.1, \qquad sB = \frac{0.5000}{0.9348}	imes 1000 \approx \534.9. Check the payouts: 465.1 	imes 2.30 \approx \1069.7 and 534.9 	imes 2.00 \approx \1069.8. Both legs return about \1070 on a \1000 outlay: 	ex…

Sizing an arbitrage with lopsided odds

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