Midterm Solutions_2005

Midterm Solutions_2005 - Ha gf r i o g str fr!i f j r m lam...

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Hagfræ ! i og stær ! fræ ! i fjármálamarka ! a 04.07.50 Lausnir á mi ! annarprófi 1) Setjum upp ver ! bréfasafn me ! ! hluti í B og " hluti í S sem líkir eftir söluréttinum p . Me ! " ví a ! sko ! a grei ! slu vegna söluréttarins " egar ver ! hækkar/lækkar fáum vi ! jöfnurnar 1 , 05 ! + 11 " = [10 , 5 ! 11] + = 0 1 , 05 ! + 9 " = [10 , 5 ! 9] + = 1 , 5 sem hafa lausnina ! = 7 , 85714 og " = ! 0 , 75 . Ver ! söluréttarins vi ! tímann t = 0 er jafnt ver ! mæti safnsins sem líkir eftir honum, " .e. 7 , 85714 + ( ! 0 , 75) · 10 = 0 , 36 . 2) # etta lei ! ir beint af "put-call parity", c ( t ) + Ke ! r ( T ! t ) = p ( t ) + S ( t ) , me ! t = 0 og K = S (0) e rT . 3) M er martingali ef E [ M ( t + ! t ) | M ( t )] = M ( t ) fyrir öll t " 0 og ! t > 0 . Ef vi ! beitum reglu Itos á W 4 " á fáum vi ! dW 4 = 4 W 3 dW + 1 2 · 12 W 2 dt. # etta "$! ir a ! 1 2 W 4 ( t ) = 2 ! t 0 W 3 ( u ) dW ( u ) + 3 ! t 0 W 2 ( u ) du e ! a 1 2 W 4 ( t ) ! 3 ! t 0 W 2 ( u ) du = 2 ! t 0 W 3 ( u ) dW ( u ) sem er martingali, " ví a ! slembiheildi ! " t 0 W 3 ( u ) dW ( u ) er martingali. M er martingali og " ví gildir E [ M ( t )] = M (0) ( " etta gildir fyrir alla mar- tingala). Í tilvikinu a ! ofan er M (0) = 0 . # ví gildir fyrir öll t , og " .m.t. t = 1 , a ! E [ M ( t )] = 0 .
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