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Unformatted text preview: shares of the asset. Here you sell shares at the bid price, up to the quantity q buy (or −q , whichever
is smaller); if needed, you sell shares at the price pbuy , and so on, until all −q shares are sold. Here
2
buy
buy
we assume that −q ≤ q1 + · · · + qM , i.e., you are not selling more shares than the total quantity
of oﬀers to buy. Let A denote the amount you receive from the sale. Here we deﬁne the transaction
cost as
T (q ) = −pmid q − A,
the diﬀerence between the amount you would have received had you sold the shares at the midprice,
and the amount you received. It is always positive. We set T (0) = 0.
(a) Show that T is a convex piecewise linear function.
(b) Show that T (q ) ≥ (s/2)q , where s is the spread. When would we have T (q ) = (s/2)q  for
all q (in the range between the total shares oﬀered to purchase or sell)?
(c) Give an interpretation of the conjugate function T ∗ (y ) = supq (yq − T (q )). Hint. Suppose you
can purchase or sell the asset in another market, at the price pother . 103 13.2...
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 Fall '13
 F.Borrelli
 The Aeneid

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