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FINM345A09Lecture7

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

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FinM 345/Stat 390 Stochastic Calculus, Autumn 2009 Floyd B. Hanson , Visiting Professor Email: [email protected] 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 Lecture7–page1 Floyd B. Hanson

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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+ ν ( t, q ))= q . Then the increment of the logarithm process Y ( t )=ln( X ( t )) , assuming X ( t 0 )= x 0 > 0 and the jump-count increment, approximately satisfies Δ Y ( t ) μ ld ( t t + σ d ( t W ( t )+ Δ P ( t ; Q ) j =1 Q j (7.1) for sufficiently small Δ t , FINM 345/Stat 390 Stochastic Calculus Lecture7–page2 Floyd B. Hanson
where μ ld ( t ) μ d ( t ) - σ 2 d ( t ) / 2 is the log-diffusion (LD) drift, σ d > 0 and the Q j are pairwise IID jump marks for P ( s ; Q ) for s [ t, t t ) , counting only jumps associated with Δ P ( t ; Q ) given P ( t ; Q ) , with common density φ Q ( q ) . The Q j are independent of both Δ P ( t ; Q ) and Δ W ( t ) . Then the distribution of the log-process Y ( t ) is the Poisson sum of nested convolutions Φ Δ Y ( t ) ( x ) k = 0 p k ( λ ( t t ) Φ Δ G ( t ) ( * φ Q ) k ( x ) , (7.2) where Δ G ( t ) μ ld ( t t + σ d ( t W ( t ) is the incremental Gaussian process and Δ G ( t ) ( * φ Q ) k )( x ) denotes a convolution of one distribution with k identical FINM 345/Stat 390 Stochastic Calculus Lecture7–page3 Floyd B. Hanson

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densities φ Q . The corresponding log-process density is φ Δ Y ( t ) ( x ) k = 0 p k ( λ ( t t ) φ Δ G ( t ) ( * φ Q ) k ( x ) . (7.3) Proof: By the law of total probability (B.92), the distribution of the log-jump-diffusion Δ Y ( t ) Δ G ( t )+ Δ P ( t ) j Q j , dropping the Q parameter in Δ P ( t ; Q ) to simplify, is Φ Δ Y ( t ) ( x ) Prob[Δ Y ( t ) x ] = Prob Δ G ( t )+ Δ P ( t ) j =1 Q j x ltp = k =0 Prob Δ G ( t ) + Δ P ( t ) j =1 Q j x Δ P ( t )= k · Prob[Δ P ( t )= k ] = k =0 p k ( λ ( t t ( k ) ( x ) , (7.4) where p k ( λ ( t t ) is the Poisson distribution with parameter λ ( t t FINM 345/Stat 390 Stochastic Calculus Lecture7–page4 Floyd B. Hanson
and the k -jump distribution is Φ ( k ) ( x ) Prob Δ G ( t ) + k j =1 Q j x . For each discrete condition Δ P ( t ) = k , Δ Y ( t ) is the sum of k + 1 terms, the normally distributed Gaussian diffusion part Δ G ( t ) = μ ld ( t t + σ d ( t W ( t ) and the Poisson counting sum k j =1 Q j , where the marks Q j are assumed to be IID but otherwise distributed with density φ Q ( q ) , while independent of the diffusion and the Poisson counting differential process Δ P ( t ) . Using the fact that Δ W ( t ) is normally distributed with zero-mean and Δ t -variance, FINM 345/Stat 390 Stochastic Calculus

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

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