Final exam Zheng - CNETzhengh final exam Zheng Hong December 7 2009 Contents 1 JD SDE Transformations 3 2 Stochastic Calculus Example from Forward

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Unformatted text preview: CNETzhengh final exam Zheng Hong December 7, 2009 Contents 1 JD SDE Transformations 3 2 Stochastic Calculus Example from Forward Contracts 5 3 Stock-Variance Covariance 7 4 Very Heuristic Model of American Option Smooth Contact to Put Payoff Function 9 5 RGW Approximations for American Call Option Prices with Early Exercise Only Opti- mal on the Final Discrete Dividend 10 6 BAW Modified Quadratic Approximation for American call(put) Option Pricing with Constant Yield Dividend 12 7 Merton-Like Optimal Portfolio and Consumption Problem for Multiple Assets and SVJD 15 1 2 1 JD SDE Transformations (a) Set F ( S ( t ) ,t ) = 1 S ( t ) ; We have known that dS ( t ) = S ( t )( μ ( t ) dt + σ ( t ) dW ( t )) + Σ dP ( t ) j =1 S ( T- j ) ν ( T- j ,Q j ); d ( F ( S ( t ) ,t )) = d ( cont ) F ( S ( t ) ,t ) + d ( jump ) F ( S ( t ) ,t ); Using the Stochastic Chain Rule for State-Dependent SDEs, d ( cont ) F ( S ( t ) ,t ) dt = F t ( S ( t ) ,t ) dt + F s ( S ( t ) ,t ) d ( cont ) S ( t ) + 1 2 F ss ( S ( t ) ,t )( S ( t ) σ ( t )) 2 dt = 0 + [- 1 S 2 ( t ) · S ( t ) μ ( t )] dt + [- 1 S 2 ( t ) · S ( t ) σ ( t )] dW ( t ) + 1 2 [ 2 S 3 ( t ) · S ( t ) σ 2 ( t )] dt = 1 S ( t ) ( σ 2 ( t )- μ ( t )) dt + 1 S ( t ) (- σ ( t )) dW ( t ) (1) d ( jump ) F ( S ( t ) ,t ) dt = Σ dP ( t ) j =1 ( 1 S ( T- j ) + S ( T- j ) ν ( T- j ,Q j )- 1 S ( T- j ) ) = Σ dP ( t ) j =1- ν ( T- j ,Q j ) S ( T- j )(1 + ν ( T- j ,Q j )) (2) So, set f ( s,t ) = σ 2 ( t )- μ ( t ), g ( s,t ) =- σ ( t ), h ( s,t,ν ( t,q )) =- ν ( t,q ) s (1 + ν ( t,q )) , then d ( 1 S ( t ) ) = ( 1 S ( t ) ) · ( f ( S ( t ) ,t ) dt + g ( S ( t ) ,t ) dW ( t )) + Σ dP ( t ) j =1 h ( S ( T- j ) ,T- j ,ν ( T- j ,Q j )) (b) Similar process as (a): Set K ( Y ( t ) ,t ) = exp ( Y ( t )); We have known that dY ( t ) = μ ( Y ( t ) ,t ) dt + σ ( Y ( t ) ,t ) dW ( t ) + Σ dP ( t ) j =1 ν ( Y ( T- j ) ,T- j ,Q j ); d ( K ( Y ( t ) ,t )) = d ( cont ) K ( Y ( t ) ,t ) + d ( jump ) K ( Y ( t ) ,t ); Using the Stochastic Chain Rule for State-Dependent SDEs, d ( cont ) K ( Y ( t ) ,t ) dt = K t ( Y ( t ) ,t ) dt + K y ( Y ( t ) ,t ) d ( cont ) Y ( t ) + 1 2 K yy ( Y ( t ) ,t ) σ 2 ( Y ( t ) ,t ) dt = 0 + exp ( Y ( t )) · μ ( Y ( t ) ,t ) dt + exp ( Y ( t )) · σ ( Y ( t ) ,t ) dW ( t ) + 1 2 exp ( Y ( t )) · σ 2 ( Y ( t ) ,t ) dt = exp ( Y ( t )) · [ μ ( Y ( t ) ,t ) + 1 2 σ 2 ( Y ( t ) ,t )] dt + exp ( Y ( t )) σ ( Y ( t ) ,t ) dW ( t ) (3) 3 d ( jump ) K ( S ( t ) ,t ) dt = Σ dP ( t ) j =1 ( exp ( Y ( T- j ) + ν ( Y ( T- j ) ,T- j ,Q j ))- exp ( Y ( T- j ))) = Σ dP ( t ) j =1 exp ( Y ( T- j ))( exp ( ν ( Y ( T- j ) ,T- j ,Q j ))- 1) (4) So, set F ( y,t ) = exp ( y )( μ ( y,t ) + 1 2 σ 2 ( y,t )), G ( y,t ) = exp ( y ) σ ( y,t ), H ( y,t,ν ( y,t,q )) = exp ( y )( exp ( ν ( y,t,q ))- 1) , then d ( exp ( Y ( t ))) = F ( Y ( t ) ,t ) dt + G ( Y ( t ) ,t ) dW ( t ) + Σ dP ( t ) j =1 H ( Y ( T- j ) ,T- j ,ν ( Y ( T- j ) ,T- j ,Q j )) 4 2 Stochastic Calculus Example from Forward Contracts...
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This note was uploaded on 04/19/2010 for the course FIN 390 taught by Professor Hansen during the Fall '09 term at University of Illinois, Urbana Champaign.

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Final exam Zheng - CNETzhengh final exam Zheng Hong December 7 2009 Contents 1 JD SDE Transformations 3 2 Stochastic Calculus Example from Forward

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