hw12-1 - 4 Interest Rate Derivative Securities 261 V f t +...

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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- w ) V f r- rV f + 4 N k =2 max( V bk ( r, t k- 1 )- Q, 0) ( 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 " N X k =1 a ik ( S k- E [ S k ]) ! 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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hw12-1 - 4 Interest Rate Derivative Securities 261 V f t +...

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