Question 5 [20 marks in total] Consider a one-period investment model where each investor chooses her position (holdings) at Date 0 and receives payouts at Date 1 . There are m1 states at Date 1 and n1 securities currently available for trading at Date 0 . Let A be the mn matrix whose (i,j) entry is the price of Security j in State i of Date 1 . The market is called complete if the rank of A is m. Now suppose that a new security is added with payout vector v, so the i th entry of the m1 matrix v is the price of the new security in State i. The new security is called redundant if its payout can be replicated by a holding of the existing securities; in other words, the security is redundant if v is a linear combination of columns of A. 1. [5 marks] (SF) Give an example where a new security is NOT redundant. Pick your own m,n,A and v. 2. [5 marks] (Medium) Show that if the market is complete without the new security, then the new security is necessarily redundant. 3. [10 marks] (Medium) Now assume that the following three securities already exist in the market: Security B pays 1+r in every state, Security S pays yi in State i for each i, and Security C pays max{yiK,0} (the larger number between yiK and 0) in State i for each i. Here r,y1,,ym and K are known positive numbers. To avoid uninteresting situations, we assume that some yi are greater than K while some are smaller than K. The market may contain other securities but may not be complete. Now a new security P is introduced, which pays max{Kyi,0} in State i for each i. Show that P is redundant: it can be replicated by some combination of B,S and C. The number of states m is large; for concreteness you may take m=100 even though the result we want to derive holds for every m. If you find it helpful, you may assume that y1y2ym Question 5 [20 marks in total] Consider a one-period investment model where each investor chooses her position (holdings) at Date 0 and receives payouts at Date 1 . There are m1 states at Date 1 and n1 securities currently available for trading at Date 0 . Let A be the mn matrix whose (i,j) entry is the price of Security j in State i of Date 1 . The market is called complete if the rank of A is m. Now suppose that a new security is added with payout vector v, so the i th entry of the m1 matrix v is the price of the new security in State i. The new security is called redundant if its payout can be replicated by a holding of the existing securities; in other words, the security is redundant if v is a linear combination of columns of A. 1. [5 marks] (SF) Give an example where a new security is NOT redundant. Pick your own m,n,A and v. 2. [5 marks] (Medium) Show that if the market is complete without the new security, then the new security is necessarily redundant. 3. [10 marks] (Medium) Now assume that the following three securities already exist in the market: Security B pays 1+r in every state, Security S pays yi in State i for each i, and Security C pays max{yiK,0} (the larger number between yiK and 0) in State i for each i. Here r,y1,,ym and K are known positive numbers. To avoid uninteresting situations, we assume that some yi are greater than K while some are smaller than K. The market may contain other securities but may not be complete. Now a new security P is introduced, which pays max{Kyi,0} in State i for each i. Show that P is redundant: it can be replicated by some combination of B,S and C. The number of states m is large; for concreteness you may take m=100 even though the result we want to derive holds for every m. If you find it helpful, you may assume that y1y2ym
