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final exam Zheng Hong - CNETzhengh nal exam Zheng Hong...

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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
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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
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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
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2 Stochastic Calculus Example from Forward Contracts (a) dF ( t, T ) /F ( t, T ) = σ ( t, T ) dW ( t ) Set Y ( t, T ) = ln ( F ( t, T )), then using the Stochastic Chain Rule dY ( t, T ) = ln ( F ( t, T ) + dF ( t, T )) - ln ( F ( t, T )) dt = 1 F ( t, T ) dF ( t, T ) - 1 2 F 2 ( t, T ) ( dF ) 2 ( t, T ) ( dW ) 2 ( t ) dt = dt = σ ( t, T ) dW ( t ) - 1 2 σ 2 ( t, T ) dt (5) So, R t 0 dY ( s, T ) = R t 0 ( σ ( s, T ) dW ( s ) - 0 . 5 σ 2 ( s, T ) ds ); and we know that R t 0 dY ( s, T ) = Y ( t, T ) - Y (0 , T ) = ln [ F ( t, T ) /F (0 , T )]; then F ( t, T ) = F (0 , T ) exp ( R t 0 ( σ ( s, T ) dW ( s ) - 0 . 5 σ 2 ( s, T ) ds )) (b) S ( t ) = F ( t, t ) F ( t, T ( t )); Y ( t ) = ln ( S ( t )); F ( t, T ( t )) = F (0 , T ( t )) exp ( R t 0 ( σ ( s, T ( t )) dW ( s ) - 0 . 5 σ 2 ( s, T ( t )) ds )) dY ( t ) = Y ( t + dt ) - Y ( t ) = ln ( S ( t + dt )) - ln ( S ( t )) = ln ( F (0 , t + dt )) + Z t + dt 0 ( σ ( s, t + dt ) dW ( s ) - 0 . 5 σ 2 ( s, t + dt ) ds ) - ln ( F (0 , t )) - Z t 0 ( σ ( s, t ) dW ( s ) - 0 . 5 σ 2 ( s, t ) ds ) = ∂ln ( F ) ∂T (0 , t ) dt + Z t + dt t ( σ ( s, t + dt ) dW ( s ) - 0 . 5 σ 2 ( s, t + dt ) ds ) + Z t 0 ([ σ ( s, t + dt ) - σ ( s )] dW ( s ) - 0 . 5[ σ 2 ( s, t + dt ) - σ 2 ( s, t )] ds ) dt = ∂ln ( F ) ∂T (0 , t ) dt + σ ( t, t ) dW ( t ) - 0 . 5 σ 2 ( t, t ) dt + Z t 0 ( ∂σ ∂T ( s, t ) dt · dW ( s ) - 0 . 5 ( σ 2 ) ∂T ( s, t ) dt · ds ) = ∂ln ( F ) ∂T (0 , t ) dt + σ ( t, t ) dW ( t ) - 0 . 5 σ 2 ( t, t ) dt + dt · Z t 0 ∂σ ∂T ( s, t ) · ( dW ( s ) - σ ( s, t ) ds ) (6) (c) use dt with dt -precision, ( dt ) 2 dt = ms 0, ( dW ) 2 ( t ) dt = ms dt , dW ( t ) dt
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