Section 24.2

Betting Odds and Dutch Books

Locking in profit across mutually exclusive outcomes: bookmaker odds, binary contracts, hedging a two-state stock.

Toolkit

Arbitrage across mutually exclusive, exhaustive outcomes exists whenever quoted prices (odds, contract prices) do not correspond to a valid set of state prices summing to the riskless discount factor. Convert decimal odds $o_i$ (payout per $1 staked) to implied probabilities $q_i=1/o_i$ ; a Dutch book exists iff $\sum_i q_i \neq 1$ (bookmaker overround) or, for state/binary contracts with price $p_i$ per unit payoff, iff $\sum_i p_i \neq B$ for the relevant bundle of outcomes, where $B$ is the price of a sure $1 (bond).

Construction: stake amounts proportional to each $q_i$ (or buy/sell the mispriced basket against bonds) so that the payoff is the same constant $C$ regardless of which outcome occurs; profit per dollar staked is

\frac{1}{\sum_i q_i}-1.

For a two-state asset versus an even-money bet, match the hedge ratio so up- and down-state payoffs coincide, eliminating risk and leaving a sure gain equal to the probability mismatch.

Problems