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Unformatted text preview: vector p in the second condition, so that 1T p = 1, we can interpret it as
a probability vector on the outcomes. The condition RT p = 0 means that
E Rx = pT Rx = 0
for all x, i.e., the expected return is zero for all betting strategies. In economics, p is called a
risk neutral probability.
We can therefore rephrase the arbitrage theorem as follows: There is no surewin betting
strategy (or arbitrage opportunity) if and only if there is a probability vector on the outcomes
that makes all bets fair (i.e., the expected gain is zero).
(b) Betting. In a simple application, we have exactly as many wagers as there are outcomes
(n = m). Wager i is to bet that the outcome will be i. The returns are usually expressed as
odds. For example, suppose that a bookmaker accepts bets on the result of the 2000 European
soccer championships. If the odds against Belgium winning are 14 to one, and we bet $100 on
Belgium, then we win $1400 if they win the tournament, and we lose $100 otherwise.
In general, if we have m possible outcomes, and the odds against outcome...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.
 Fall '13
 F.Borrelli
 The Aeneid

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