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hw7-1 - 126 3 Exotic Options Consequently noticing Co(S tK...

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126 3 Exotic Options Consequently, noticing ˜ C o ( S, t K ; B * l ) = ˜ C o ( S, t K ; B ** l ) = G c ( S ), we obtain ˜ C o ( S, t k ; B * l ) = max ˆ e - rΔt Z B * l ˜ C o ( S 0 , t k +1 ; B * l ) G 1 ( S 0 , t k +1 ; S, t k , B * l ) dS 0 , G c ( S ) ! max ˆ e - rΔt Z B ** l ˜ C o ( S 0 , t k +1 ; B ** l ) G 1 ( S 0 , t k +1 ; S, t k , B ** l ) dS 0 , G c ( S ) ! = ˜ C o ( S, t k ; B ** l ) , k = K - 1 , K - 2 , · · · , 0 . Letting K → ∞ and noticing that ˜ C o ( S, t ; B l ) generates C o ( S, t ; B l ) as K → ∞ , we arrive at the conclusion C o ( S, t ; B * l ) C o ( S, t ; B ** l ) if B * l B ** l . 3. Show that a European up-and-out put option with B u > E plus a Euro- pean up-and-in put option with the same parameters is equal to a vanilla European put option. Solution : For S B u , the value of the European up-and-out put option satisfies ∂p o ∂t + 1 2 σ 2 S 2 2 p o ∂S 2 + ( r - D 0 ) S ∂p o ∂S - rp o = 0 , 0 S B u , t T, p o ( S, T ) = max( E - S, 0) , 0 S B u , p o ( B u , t ) = 0 , t T and the value of the European up-and-in put option is the solution of the problem:
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