Hw9-1 - 182 3 Exotic Options c Let V(S H = SW and = H/S Then V S 2V S 2 V H V(S H H Thus 1 2 2 2V V S(r D0 S rV 2 2 S S dW 2 2 d 2 W rW(r D0 W =S 2

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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 µ 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 ) µ 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 µ - 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 µ - 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
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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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Hw9-1 - 182 3 Exotic Options c Let V(S H = SW and = H/S Then V S 2V S 2 V H V(S H H Thus 1 2 2 2V V S(r D0 S rV 2 2 S S dW 2 2 d 2 W rW(r D0 W =S 2

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