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hw9-1 - 182 3 Exotic Options c Let V(S H = SW and = H/S...

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182 3 Exotic Options c) Let V ( S, H ) = SW ( η ) and η = H/S . Then ∂V ∂S = W + S dW · - H S 2 = W - η dW , 2 V ∂S 2 = d W - η W · - H S 2 = η 2 S d 2 W 2 , ∂V ∂H = S dW 1 S = dW , V ( S, H ) - H = S ( W ( η ) - η ) . Thus 1 2 σ 2 S 2 2 V ∂S 2 + ( r - D 0 ) S ∂V ∂S - rV = S σ 2 η 2 2 d 2 W 2 + ( r - D 0 ) W - η dW - rW = S σ 2 η 2 2 d 2 W 2 + ( D 0 - r ) η dW - D 0 W and ∂V ∂H ( S, S ) = dW (1) . Consequently, from min - σ 2 η 2 2 d 2 W 2 - ( D 0 - r ) η dW + D 0 W , W - η = 0 , 1 η, dW (1) = 0 , for any S > 0 we can have min - 1 2 σ 2 S 2 2 V ∂S 2 + ( r - D 0 ) S ∂V ∂S - rV , V ( S, H ) - H = 0 , S H, ∂V ∂H ( S, S ) = 0 . That is, the function SW ( H/S ) is a solution of the LC problem given at the beginning.
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