purchased and r is the cost of borrowing the funds to buy the stock?The minimum discount that the dealer would set is such that the expected utility of the optimal portfolio without buying the stock equals the expected utility of the portfolio with the unwanted inventory: EUm]=EU[W+(1+产2-(1+r)Q-C】 (1) Applying a Taylor series expansion to both sides,taking expectations,assuming ry is small enough to be ignored,and solving for c=C/O,yields (2) where z is the dealer's coefficient of relative risk aversion,Wo is the dealer's initial wealth,o2 is the variance of return of the stock.The bid price for depth of Q dollars must be below the consensus stock value by the proportion c to compensate the dealer for his inventory costs.These costs arise because the dealer loses diversification and because he assumes a level of risk that is inconsistent with his preferences.The discount of the bid price is greater the greater dealer's risk aversion,the smaller his wealth,the greater the stock's return variance,'and the larger the quoted depth. The proportional discount,c,is affected by the initial inventory of the dealer, which was assumed to be zero in the above derivation.If the dealer enters the period with inventory of I dollars in one or more stocks,the proportional discount for depth of can be shown to be C= (3) where oro is the covariance between the return on the initial inventory and the return on the stock in which the dealer is bidding.If /<0 and cto>0,the dealer may be willing to pay a premium to buy shares because they hedge a short position in the initial inventory. On the other hand,the dealer's asking price will be correspondingly higher with an initial short position because the dealer will be reluctant to sell and add to the short position. 2Q is valued at the consensus price in the absence of a bid-ask spread.The loan is collateralized by the dealer's stock position. The variance,not the beta,is relevant because the inventory position is not diversified. 1212 purchased and rf is the cost of borrowing the funds to buy the stock. 2 The minimum discount that the dealer would set is such that the expected utility of the optimal portfolio without buying the stock equals the expected utility of the portfolio with the unwanted inventory: [ °] [ ° (1 ) (1 )( )] EU W EU W f = + + r% Q - + - rQC . (1) Applying a Taylor series expansion to both sides, taking expectations, assuming rf is small enough to be ignored, and solving for c=C/Q, yields 1 2 2 0 z c Q W = s (2) where z is the dealer’s coefficient of relative risk aversion, W0 is the dealer’s initial wealth, s 2 is the variance of return of the stock. The bid price for depth of Q dollars must be below the consensus stock value by the proportion c to compensate the dealer for his inventory costs. These costs arise because the dealer loses diversification and because he assumes a level of risk that is inconsistent with his preferences. The discount of the bid price is greater the greater dealer’s risk aversion, the smaller his wealth, the greater the stock’s return variance,3 and the larger the quoted depth. The proportional discount, c, is affected by the initial inventory of the dealer, which was assumed to be zero in the above derivation. If the dealer enters the period with inventory of I dollars in one or more stocks, the proportional discount for depth of Q can be shown to be 1 2 2 0 0 IQ z z c I Q W W = + s s , (3) where sIQ is the covariance between the return on the initial inventory and the return on the stock in which the dealer is bidding. If I<0 and sIQ > 0, the dealer may be willing to pay a premium to buy shares because they hedge a short position in the initial inventory. On the other hand, the dealer’s asking price will be correspondingly higher with an initial short position because the dealer will be reluctant to sell and add to the short position. 2 Q is valued at the consensus price in the absence of a bid-ask spread. The loan is collateralized by the dealer’s stock position. 3 The variance, not the beta, is relevant because the inventory position is not diversified