Arbitrage Theorem

Let X=(X_1, \dots, X_n) be the amount bet on each of the n experiment. Each experiment has possible outcome j=1,2, \dots, m, r_i(j) = return function of bet i when outcome is j the expected return when outcome is j: = E[X| j]= \sum_{i=1}^{n} X_i r_{i}(j)

There must be a set of probabilities P=(P_1, \dots, P_n) for which, the expected return = \sum_{j=1}^m P_j r_i(j)=0, for all i=1, \dots, n.

Otherwise, an arbitrage would exists: \sum_{i=1}^n X_i r_i(j) >0 for all j=1, \dots, m.

The set of P are called risk-neutral probabilities, resulting in all bets being fair.

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Arbitrage Theorem

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