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FINM345A09Lecture5Corrected - FinM 345/Stat 390 Stochastic...

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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 5 (from Chicago) More Jump-Diffusion Calculus, Simple and Compound 6:30-9:30 pm, 26 October 2009, Kent 120 in Chicago 7:30-10:30 pm, 26 October 2009 at UBS Stamford 7:30-10:30 am, 27 October 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 Lecture5–page1 Floyd B. Hanson
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FinM 345 Stochastic Calculus: 5. More Jump-Diffusion Calculus — Simple & Compound: 5.1. Jump-Diffusion Rules and SDEs Continued: (finishing Chapt. 4 of the textbook) 5.1.1. Solution Simulations for Linear Jump-Diffusion SDEs with Constant Coefficients: Upon merging and modifying the simulation algorithms for small time increments in Figure 7 on L1-p32 and using the cumulative sum of normal RNG Wiener increment approximations, together with the cumulative sum of uniform RNG Poisson increment approximations with the binomial RNG with n=1 to model the “Bernoulli” zero-one jump law, we show simulations of the financially relevant linear jump-diffusion process with constant parameters solution (4.39) on L4-51 in Figure 5.1. FINM 345/Stat 390 Stochastic Calculus Lecture5–page2 Floyd B. Hanson
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The basic simulation is performed on the approximate exponent increment Δ Y i ( μ 0 - σ 2 0 / 2)Δ t + σ 0 Δ W i +ln(1+ ν 0 P i , (5.1) corresponding to SDE (5.1), where Δ t = 0 . 001 for this MATLAB-generated figure, Δ W i DW( i ) , Δ t = Dt , where DW= sqrt (Dt) * randn (1 , N ) ; and Δ P = binornd (1 , λ 0 Dt, 1 , N ) , approximating the zero-one jump law through Bernoulli form of the binomial RNG when the first parameter is n = 1 and the second is the Poisson parameter ΔΛ = λ 0 Δ t . { In place of calls by random states to randn (1 , N ) , use serial calls to normrnd (0 , sqrt (Dt) , 1 , N ) . } FINM 345/Stat 390 Stochastic Calculus Lecture5–page3 Floyd B. Hanson
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The sample path of the state exponent, YS , starting from a zero initial condition YS(1)=0 rather than ln( x 0 ) , for i =1: N , is approximated by YS( i + 1)=YS( i )+( μ 0 - σ 2 0 / 2) * Dt+ σ 0 * DW( i ) + log (1+ ν 0 ) * DP ( i ) , with t ( i + 1) = i * Dt . The sample path of the desired state, XS , is approximated by X ( t ( i + 1)) XS( i + 1) = x 0 * exp(YS( i + 1)) . The mean trajectory, XM , is given by E[ X ( t ( i + 1))] XM( i + 1) = x 0 * exp(( μ 0 + λ 0 * ν 0 ) t ( i + 1)) . FINM 345/Stat 390 Stochastic Calculus Lecture5–page4 Floyd B. Hanson
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The mean trajectory is also displayed in the figure along with the upper XT exponential standard deviation estimate E[ X ( t ( i +1))] * V ( i +1) XT( i +1) =XM( i +1) * V ( i +1) and lower XB exponential standard deviation estimate E[ X ( t ( i +1))] /V ( i +1) XB( i +1) =XM( i +1) /V ( i +1) , where the factor V ( i +1)=exp Var[ Y ( t ( i +1))] =exp ( σ 2 0 + λ 0 * log 2 (1+ ν 0 )) t ( i +1) is the exponential of the standard deviation of the exponent process Y ( t ) in discrete form.
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