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by: Alvena Marks IV


Alvena Marks IV

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This 3 page Study Guide was uploaded by Alvena Marks IV on Thursday October 29, 2015. The Study Guide belongs to FNCE 7550 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 56 views. For similar materials see /class/231810/fnce-7550-university-of-colorado-at-boulder in Finance at University of Colorado at Boulder.

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Date Created: 10/29/15
FNCE 7550 notes Garland B Durham Leeds School of Business University of Colorado October 11 2008 HW4 Consider the model y My 039Y eXP39Ut2 1t 1 H39Util O39VEQt with My 0 O39y 008 H 96 UV 5 en N N0l for i 12 corr 611621 0 and v0 N N0a17 H2 Suppose that we observe yl 02 and that it is known that v0 i2 and v2 3 We wish to compute py1lv0 v2 First note that Pltyllv07v2 py17v1lv07v2dvl plty1v1gtpltv1vov2gtdv1 Pyllvlwdvl 10010702 py1 v110v2lv1v1lvo dvl 10v2lv0 where 10y1lv1 y1 W705 eXPv1 10v2lv1 MW 1703 10v1lv0 MW 0703 10v2lv0 v2 H2v07 H2 003 and M 02 is the pdf of a NW 02 random variable evaluated at z 1 Plot py1lv1 Ew across the range v1 6 712 using the parameters given above 2 Now we will compute the integral above using straightforward deterministic nu merical integrationi Using 5 1000 a 71 and b 2 let v15 a b 7 asS s 1HiS and compute S 5 5 bia pv2lv1 pv1 lvo v v m E v i pyll 0 2 S 51pyll 1 pltv2 v0 31 Now try computing the integral using Monte Carlo integration with no importance F sampling Using 5 1000 a 71 and b 2 let v55 3 11 11 S be random draws from unifa b and compute S 5 5 bia v v v v py1 v07v2 m ZPy1 vlP2l 1 P 1 l 0 51 5 10v2lv0 Repeat this a few times to get an idea of the simulation error1 Also try it a few times with S 1001 Finally we will compute the integral using Monte Carlo integration using importance sampling with 4 based on the Laplace approximation to the target density a First we need to obtain the mode of the log target density 9011 10gPy1lv1 10g10v2lv1 10g10v1lv0 10gPU2lv0 Do this using Newton7s method ie something like this it should converge within several iterations v g vg v Let f argmaxu 91 denote the mode and H g be the second derivative evaluated at the mode The importance sampler will be Na 7H 11 Let 111 451 1 iH l denote the pdf of the sampler1 Plot 5x555 over the range 1 6 ab where a 71 and b 2 to see how good the importance sampler is Also plot 91 7 logpyllv0 1 together with log 111 on a single set of axes Note that logpyllv0v2 is just a scaling factor to make the two densities comparable use one of your estimates computed above1 Now using 5 1000 let v55 3 1111 5 be random draws from N iH l and compute 5 5 5 1 PU2lv gt100 lv0Pv2lv0 pmvm m E Zpltyllvlgt1 51 41 1 Repeat this several times to get an idea of the simulation error1 Also try it with S 1001


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