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Unformatted text preview: 4 Interest Rate Derivative Securities 261 ∂V f ∂t + 1 2 w 2 ∂ 2 V f ∂r 2 + ( u λw ) ∂V f ∂r rV f + 4 N ∑ k =2 max( V bk ( r, t k 1 ) Q, 0) δ ( t t k 1 ) = 0 , r l ≤ r ≤ r u , t * ≤ t ≤ t 4 N 1 , V f ( r, t 4 N 1 ) = 0 , r l ≤ r ≤ r u . Then V f ( r, t * ) gives the value of the floor and the premium of the floor is given by V f ( r * , t * ) , where r * is the spot interest rate at time t * . 19. a) S is a random vector and its covariance matrix is B . Let ¯ S = AS , A being a constant matrix, and its covariance matrix be C . Find the relation among A , B , and C . b) How do we choose A so that C will be a diagonal matrix? c) Suppose that ¯ S 1 , ¯ S 2 , ··· , ¯ S K are variables and ¯ S K +1 , ¯ S K +2 , ··· , ¯ S N are fixed numbers. Find the dependence of S K +1 , S K +2 , ··· , S N on S 1 , S 2 , ··· , S K . Solution : a) Let S = S 1 S 2 . . . S N , ¯ S = ¯ S 1 ¯ S 2 . . . ¯ S N , A = a 1 , 1 a 1 , 2 ··· a 1 , N a 2 , 1 a 2 , 2 ··· a 2 , N . . . . . . . . . . . . a N, 1 a N, 2 ··· a N,N , B = b 1 , 1 b 1 , 2 ··· b 1 , N b 2 , 1 b 2 , 2 ··· b 2 , N . . . . . . . . . . . . b N, 1 b N, 2 ··· b N,N , and C = c 1 , 1 c 1 , 2 ··· c 1 , N c 2 , 1 c 2 , 2 ··· c 2 , N . . . . . . . . . . . . c N, 1 c N, 2 ··· c N,N . Then c ij = Cov £ ¯ S i ¯ S j / = E £( ¯ S i E £ ¯ S i /)( ¯ S j E £ ¯ S j /)/ = E "ˆ N X k =1 a ik ( S k E [ S k ]) !ˆ N X l =1 a jl ( S l E [ S l ]) !# = N X k =1 N X l =1 a ik a jl Cov [ S k S l ] = N X k =1 N X l =1 a ik b kl a jl . 262 4 Interest Rate Derivative Securities Thus we have C = ABA T . b) Suppose that c 2 i and a i = a i, 1 a i, 2 . . . a i, N , i = 1 , 2 , ··· , N, be the eigenvalues and unit eigenvectors of the matrix B . Then Ba i = c 2 i a i , i = 1 , 2 , ··· , N , or put them together, we have B · [ a 1 a 2 ··· a N ] = [ a 1 a 2 ··· a N ] c 2 1 ··· c 2 2 ··· . . . . . . . . . . . . ··· c 2 N . Thus if we define A = a 1 , 1 a 1 , 2 ··· a 1 , N a 2 , 1 a 2 , 2 ··· a 2 , N . . . . . . . . . . . . a N, 1 a N, 2 ··· a N,N and C = c 2 1 ··· c 2 2 ··· . . . . . . . . . . . . ··· c 2 N , then we will have ABA T = C . Here we have used the fact A = ( A T ) 1 because B is a symmetric matrix and a i , i = 1 , 2 , ··· , N , are unit vectors....
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This note was uploaded on 02/11/2010 for the course MATH 6203 taught by Professor Zhu during the Spring '10 term at University of North Carolina Wilmington.
 Spring '10
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