In that case we lose rij xj ie our net is gain rij xj

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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 mid-price, 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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