Hong, Zheng HW9 - CNETzhengh HW9 Zheng Hong November 30,...

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Unformatted text preview: CNETzhengh HW9 Zheng Hong November 30, 2009 Contents 1 mean-reverting, square-root diffusion 2 2 Simulate a solution to the SV-SDE 2 1 1 mean-reverting, square-root diffusion Set X ( t ) = V + 0 . 5 R t e v ( s ) / 2 ( v dW v )( s ) Then, dX ( t ) = 0 . 5 e v ( t ) / 2 v ( t ) dW v ( t ). Set Y ( t ) = X 2 ( t ), g = 0 . 5 e v ( t ) / 2 v ( t ), f = 0, Then, according to the Chain Rule for Diffusion, dY ( t ) dt = 1 2 g 2 * 2( X ( t ) ,t ) dt + ( g * 2 X )( X ( t ) ,t ) dW v ( t ) That is, dY ( t ) dt = 1 4 e v ( t ) 2 v ( t ) dt + e v ( t ) / 2 v ( t ) X ( t ) dW v ( t ) Next, let V ( t ) = F ( Y ( t ) ,t ) = e- v ( t ) Y ( t ), applying the Chain Rule for Diffusion, dV ( t ) dt = ( F t + f F y + 1 2 g 2 F yy )( Y ( t ) ,t ) dt + ( g F y )( Y ( t ) ,t ) dW v ( t ) =- v Fdt + 1 4 e v ( t ) 2 v ( t ) e- v ( t ) dt + e v ( t ) / 2 v ( t ) X ( t ) e- v ( t ) dW v ( t ) =- v V ( t ) dt + 1 4 2 v ( t ) dt + e- v ( t ) / 2...
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Hong, Zheng HW9 - CNETzhengh HW9 Zheng Hong November 30,...

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