Hong, Zheng HW5 - CNETzhengh HW5 Zheng Hong November 2 2009...

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Unformatted text preview: CNETzhengh HW5 Zheng Hong November 2, 2009 Contents 1 Theorem Proof 2 2 Simulate for linear simple jump-diffusion 3 3 log-double-uniform jump distribution 4 4 log-normally distributed jump amplitude case 4 1 1 Theorem Proof (a) Proof: Let t i = t + i * Δ t for i = 0 : n + 1 be a proper partition of [ t ,t ] with Δ t = ( t- t ) / ( n + 1), Δ W i = W ( t i +1 )- W ( t i ), Δ P i = P ( t i +1 )- P ( t i ), and σ i = σ ( t i ), λ i = λ ( t i ), ν i = ν ( t i ) for i = 0 : n . hspace 3 exE [ exp ( Z t t (( μ ( s )- σ 2 ( s ) / 2) ds + σ ( s ) dW ( s ) + ln (1 + ν ( s )) dP ( s )))] ims w E [ exp ( n X i =0 (( μ i- σ 2 i / 2)Δ t + σ i Δ W i + ln (1 + ν i )Δ P i ))] law = exp E [Π n i =0 exp (( μ i- σ 2 i / 2)Δ t + σ i Δ W i + ln (1 + ν i )Δ P i )] ind = inc Π n i =0 E [ exp (( μ i- σ 2 i / 2)Δ t ] E [ exp ( σ i Δ W i )] E [ exp ( ln (1 + ν i )Δ P i )] = Π n i =0 exp (( μ i- σ 2 i / 2)Δ t Π n i =0 Z ∝-∝ exp (- ω 2 2Δ t + σ i ω ) √ 2 π Δ t dω...
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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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Hong, Zheng HW5 - CNETzhengh HW5 Zheng Hong November 2 2009...

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