hw6-2 - 3 Exotic Options Problems 1. Consider the following...

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Unformatted text preview: 3 Exotic Options Problems 1. Consider the following problem: V t + 1 2 2 S 2 2 V S 2 + ( r- D ) S V S- r V = 0 , S, t T, V ( S, T ) = ( 1 ( S ) , S B, 2 ( S ) , B < S, where 1 ( S ) and 2 ( S ) are continuous functions and 1 ( B ) = 2 ( B ) may not hold. a) Try to find such a relation between 1 ( S ) and 2 ( S ) that V ( B, t ) = . b) Based on the result in a), show that for the problem V t + 1 2 2 S 2 2 V S 2 + ( r- D ) S V S- rV = 0 , B l S, t T, V ( S, T ) = V T ( S ) , B l S, V ( B l , t ) = 0 , t T, the solution is V ( S, t ) = e- r ( T- t ) Z B l V T ( S ) G 1 ( S , T ; S, t, B l ) dS , where G 1 ( S , T ; S, t, B l ) = G ( S , T ; S, t )- ( B l /S ) 2( r- D- 2 / 2) / 2 G ( S , T ; B 2 l /S, t ) . 114 3 Exotic Options Here G ( S , T ; S, t ) = 1 S p 2 ( T- t ) e- [ ln( S /S )- ( r- D- 2 / 2 ) ( T- t ) ] 2 / 2 2 ( T- t ) . c) The value of a European down-and-out call option is the solution of the problem: c o t + 1 2 2 S 2 c o S 2 + ( r- D ) S c o S- rc o = 0 , B l S, t T, c o ( S, T ) = max ( S- E, 0) , B l S, c o ( B l , t ) = 0 , t T. Based on the result in part b), show that for the case B l E , the expression of c o is c o ( S, t ) = c ( S, t )- B l S 2( r- D- 2 / 2) / 2 c B 2 l S , t ; and for the case B l E , its expression is c o ( S, t ) = S e- D ( T- t ) N d 1 ( B l ) - E e- r ( T- t ) N d 1 ( B l )- T- t - ( B l /S ) 2( r- D- 2 / 2) / 2 " B 2 l S e- D ( T- t ) N ( d 1 ( B l ) )- E e- r ( T- t ) N d 1 ( B l )- T- t # , where d 1 ( B l ) = ln S e ( r- D )( T- t ) B l + 1 2 2 ( T- t ) , T- t , d 1 ( B l ) = ln B l e ( r- D )( T- t ) S + 1 2 2 ( T- t ) , T- t . d) Based on the result in a), show that for the problem V t + 1 2 2 S 2 2 V S 2 + ( r- D ) S V S- rV = 0 , S B u , t T, V ( S, T ) = V T ( S ) , S B u , V ( B u , t ) = 0 , t T, 3 Exotic Options 115 the solution is V ( S, t ) = e- r ( T- t ) Z B u V T ( S ) G 1 ( S , T ; S, t, B u ) dS , where G 1 ( S , T ; S, t, B u ) = G ( S , T ; S, t )- ( B u /S ) 2( r- D- 2 / 2) / 2 G ( S , T ; B 2 u /S, t ) . Here G ( S , T ; S, t ) = 1 S p 2 ( T- t ) e- [ ln( S /S )- ( r- D- 2 / 2 ) ( T- t ) ] 2 / 2 2 ( T- t ) . e) Based on the result in part d), find the closed-form solution of a Eu- ropean up-and-out put option for both the case B u E and the case B u E ....
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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.

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hw6-2 - 3 Exotic Options Problems 1. Consider the following...

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