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Unformatted text preview: i are λi to one,
then the return matrix R ∈ Rm×m is given by
rij = λi if j = i
rij = −1 otherwise.
107 Show that there is no sure-win betting scheme (or arbitrage opportunity) if
1 + λi In fact, you can verify that if this equality is not satisﬁed, then the betting strategy
xi = 1/(1 + λi )
1 − m 1/(1 + λi )
i=1 always results in a proﬁt.
The common situation in real life is
1 + λi because the bookmakers take a cut on all bets.
13.8 Log-optimal investment. We consider an instance of the log-optimal investment problem described
in exercise 4.60 of Convex Optimization. In this exercise, however, we allow x, the allocation vector,
to have negative components. Investing a negative amount xi W (t) in an asset is called shorting
the asset. It means you borrow the asset, sell it for |xi W (t)|, and have an obligation to purchase it
back later and return it to the lender.
(a) Let R be the n × m-matrix with columns rj :
R= r 1 r 2 · · · rm . We assume that the elements rij of R are all pos...
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- Fall '13
- The Aeneid