bv_cvxbook_extra_exercises

Give the optimal total initial investment and compare

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Unformatted text preview: ctor x ∈ Rn , with as components xj the amounts we bet on each wager. If we use a betting strategy 106 Country Holland Italy Spain France Germany England Belgium Sweden Odds 3.5 5.0 5.5 6.5 7.0 10.0 14.0 16.0 Country Czech Republic Romania Yugoslavia Portugal Norway Denmark Turkey Slovenia Odds 17.0 18.0 20.0 20.0 20.0 33.0 50.0 80.0 Table 1: Odds for the 2000 European soccer championships. x, our total return in the event of outcome i is equal to vector Rx. n j =1 rij xj , i.e., the ith component of the (a) The arbitrage theorem. Suppose you are given a return matrix R. Prove the following theorem: there is a betting strategy x ∈ Rn for which Rx ≻ 0 if and only if there exists no vector p ∈ Rm that satisfies RT p = 0, p 0, p = 0. We can interpret this theorem as follows. If Rx ≻ 0, then the betting strategy x guarantees a positive return for all possible outcomes, i.e., it is a sure-win betting scheme. In economics, we say there is an arbitrage opportunity. If we normalize the...
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