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# 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 σ 2 S 2 ∂ 2 V ∂S 2 + ( r- D ) S ∂ V ∂S- r V = 0 , ≤ S, t ≤ T, V ( S, T ) = ( ϕ 1 ( S ) , ≤ S ≤ B, ϕ 2 ( S ) , B < S, where ϕ 1 ( S ) and ϕ 2 ( S ) are continuous functions and ϕ 1 ( B ) = ϕ 2 ( B ) may not hold. a) Try to find such a relation between ϕ 1 ( S ) and ϕ 2 ( S ) that V ( B, t ) = . b) Based on the result in a), show that for the problem ∂V ∂t + 1 2 σ 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- σ 2 / 2) /σ 2 G ( S , T ; B 2 l /S, t ) . 114 3 Exotic Options Here G ( S , T ; S, t ) = 1 S σ p 2 π ( T- t ) e- [ ln( S /S )- ( r- D- σ 2 / 2 ) ( T- t ) ] 2 / 2 σ 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 σ 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 )- µ B l S ¶ 2( r- D- σ 2 / 2) /σ 2 c µ 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 )- σ √ T- t ·- ( B l /S ) 2( r- D- σ 2 / 2) /σ 2 " B 2 l S e- D ( T- t ) N ( ¯ d 1 ( B l ) )- E e- r ( T- t ) N ‡ ¯ d 1 ( B l )- σ √ T- t · # , where ˜ d 1 ( B l ) = • ln S e ( r- D )( T- t ) B l + 1 2 σ 2 ( T- t ) ‚ , ‡ σ √ T- t · , ¯ d 1 ( B l ) = • ln B l e ( r- D )( T- t ) S + 1 2 σ 2 ( T- t ) ‚ , ‡ σ √ T- t · . d) Based on the result in a), show that for the problem ∂V ∂t + 1 2 σ 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- σ 2 / 2) /σ 2 G ( S , T ; B 2 u /S, t ) . Here G ( S , T ; S, t ) = 1 S σ p 2 π ( T- t ) e- [ ln( S /S )- ( r- D- σ 2 / 2 ) ( T- t ) ] 2 / 2 σ 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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hw6-2 - 3 Exotic Options Problems 1 Consider the following...

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