FINM345A09Lecture7 - FinM 345/Stat 390 Stochastic Calculus,...

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Unformatted text preview: FinM 345/Stat 390 Stochastic Calculus, Autumn 2009 Floyd B. Hanson , Visiting Professor Email: fhanson@uchicago.edu Master of Science in Financial Mathematics Program University of Chicago Lecture 7 (from Chicago) Distributions and Financial Applications 6:30-9:30 pm, 09 November 2009 at Kent 120 in Chicago 7:30-10:30 pm, 09 November 2009 at UBS Stamford 7:30-10:30 am, 10 November 2009 at Spring in Singapore Copyright c 2009 by the Society for Industrial and Applied Mathematics, and Floyd B. Hanson. FINM 345/Stat 390 Stochastic Calculus Lecture7page1 Floyd B. Hanson FinM 345 Stochastic Calculus: 7. Compound Jump-Diffusion Distribution and Applications in Financial Engineering: 7.1. Compound Jump-Diffusion Distribution: 7.1.1 Distribution of Increment Log-Process: Theorem 7.1. Distribution of the State Increment Logarithm Process for Linear Mark-Jump-Diffusion SDE: Let the logarithm-transform jump-amplitude be ln(1+ ( t,q ))= q . Then the increment of the logarithm process Y ( t )=ln( X ( t )) , assuming X ( t )= x > and the jump-count increment, approximately satisfies Y ( t ) ' ld ( t ) t + d ( t ) W ( t )+ P ( t ; Q ) X j =1 Q j (7.1) for sufficiently small t , FINM 345/Stat 390 Stochastic Calculus Lecture7page2 Floyd B. Hanson where ld ( t ) d ( t )- 2 d ( t ) / 2 is the log-diffusion (LD) drift, d > and the Q j are pairwise IID jump marks for P ( s ; Q ) for s [ t,t + t ) , counting only jumps associated with P ( t ; Q ) given P ( t ; Q ) , with common density Q ( q ) . The Q j are independent of both P ( t ; Q ) and W ( t ) . Then the distribution of the log-process Y ( t ) is the Poisson sum of nested convolutions Y ( t ) ( x ) ' X k = p k ( ( t ) t ) G ( t ) ( * Q ) k ( x ) , (7.2) where G ( t ) ld ( t ) t + d ( t ) W ( t ) is the incremental Gaussian process and ( G ( t ) ( * Q ) k )( x ) denotes a convolution of one distribution with k identical FINM 345/Stat 390 Stochastic Calculus Lecture7page3 Floyd B. Hanson densities Q . The corresponding log-process density is Y ( t ) ( x ) ' X k = p k ( ( t ) t ) G ( t ) ( * Q ) k ( x ) . (7.3) Proof: By the law of total probability (B.92), the distribution of the log-jump-diffusion Y ( t ) ' G ( t )+ P ( t ) j Q j , dropping the Q parameter in P ( t ; Q ) to simplify, is Y ( t ) ( x ) Prob[ Y ( t ) x ] = Prob h G ( t )+ P ( t ) j =1 Q j x i ltp = k =0 Prob h G ( t ) + P ( t ) j =1 Q j x P ( t )= k i Prob[ P ( t )= k ] = X k =0 p k ( ( t ) t ) ( k ) ( x ) , (7.4) where p k ( ( t ) t ) is the Poisson distribution with parameter ( t ) t FINM 345/Stat 390 Stochastic Calculus Lecture7page4 Floyd B. Hanson and the k-jump distribution is ( k ) ( x ) Prob " G ( t ) + k X j =1 Q j x # ....
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FINM345A09Lecture7 - FinM 345/Stat 390 Stochastic Calculus,...

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