3. Suppose a trader wishes to find a strategy to unwind a position with size of 1 million units that minimize the Liquidity-adjusted VAR. The dollar bid-ask spread and the quantity traded are associated in the way that p(q) = 0.01e59, where q is measured in millions of units. The standard deviation of the price change per day is $0.5 (i.e.o = 0.5) and the position should be unwound in 2 days. (i.e. n = 2) Define qi = units traded on day i Xi = size of trader's position at the end of day i Liquidity-adjusted VAR= 102(x;)2 + =1 9i p(qi) a) What is 2 for 95% confidence limit in Liquidity-adjusted VAR? b) Let the quantity traded in day 1 and day 2 be 91 and 192 respectively. Given the size of position 91 +92 = 1, find the first order derivative of liquidity-adjusted VAR with respect to q1. c) Find the trade quantity 91 such that the liquidity-adjusted VAR is minimized (at 95% confidence level). [Hint: A plot of the function f(x) = (5x + 1)e 5x (6 5x)e5(1-x) is given] 2 f(x) = (5x+1)e5x-(6-5x)e5(1-x) f(x) 1040 960 880 800 720 640 560 480 400 320 240 160 80 0 -8058 -160 -240 -320 -400 -480 -560 -640 -720 -800 -880 -960 0.16 0.21 26 0.31 9.36 0.41 0.46 0.51 0.56 1990 99.0 0.71 0.76 18.b 98.b 16:0 960 X 3. Suppose a trader wishes to find a strategy to unwind a position with size of 1 million units that minimize the Liquidity-adjusted VAR. The dollar bid-ask spread and the quantity traded are associated in the way that p(q) = 0.01e59, where q is measured in millions of units. The standard deviation of the price change per day is $0.5 (i.e.o = 0.5) and the position should be unwound in 2 days. (i.e. n = 2) Define qi = units traded on day i Xi = size of trader's position at the end of day i Liquidity-adjusted VAR= 102(x;)2 + =1 9i p(qi) a) What is 2 for 95% confidence limit in Liquidity-adjusted VAR? b) Let the quantity traded in day 1 and day 2 be 91 and 192 respectively. Given the size of position 91 +92 = 1, find the first order derivative of liquidity-adjusted VAR with respect to q1. c) Find the trade quantity 91 such that the liquidity-adjusted VAR is minimized (at 95% confidence level). [Hint: A plot of the function f(x) = (5x + 1)e 5x (6 5x)e5(1-x) is given] 2 f(x) = (5x+1)e5x-(6-5x)e5(1-x) f(x) 1040 960 880 800 720 640 560 480 400 320 240 160 80 0 -8058 -160 -240 -320 -400 -480 -560 -640 -720 -800 -880 -960 0.16 0.21 26 0.31 9.36 0.41 0.46 0.51 0.56 1990 99.0 0.71 0.76 18.b 98.b 16:0 960 X