Importance Sampling in Monte carlo Simulation

[Raj_S]

Hi David et. al.,

Not sure whether my question is relevant for this topic ( P1 T2). Could you please explain in relatively simple terms 'Importance Sampling in Monte Carlo Simulations'. I understand that its used for reducing variance in tail but it will be good to know a bit more.

I was trying to read 'Glasserman and Li' but that appears to be more than I can chew atleast as of now.


Regards

 

[ShaktiRathore]

Sampling is done in MCS aka monte carlo simulations to reduce the standard error.
Suppose we have n samples then the standard error of the predicted value is given by sigma/sqrt(n). If you increase the number of samples from n1 to n2 then our standard error will change from SE1=sigma/sqrt(n1) to SE2=sigma/sqrt(n2) so that n2>n1=> sigma/sqrt(n2)<sigma/sqrt(n1) . IN this way the confidence interval will narrow so that our ability to predict the true value will increase at a given confidence level. So that interval for n1 is mean+-z*SE1 with interval width 2zSE1 and for n2 is mean +- z*SE2 with interval width 2zSE2 . As SE2<SE1=>2zSE2 < 2zSE1 . So interval width narrows so that our confidence in predicting the true value increases with increase in number of samples from n1 to n2 at a given confidence level. The more the number of samples the closer the mean as predicted will inch up closer to the actual population value. As predicted by central limit theorem that as the number of samples increase to predict a population mean the accuracy increases in predicting the actual population mean as the number of samples increases. So the ability to predict the expected mean value increase with the sampling and thereby accuracy. So accuracy increases in predicting the outcome in MCS.

thanks

 

[Backwardation]

Or in really simple terms: it is basically used to reduce the number times a particular model needs to be solved in an simulation. You focus on a particular important (-> importance sampling) part on the distribution. In risk management for example, it might be a good idea to focus on tail events and therefore focus you samples on this "area of the curve"

 

[Raj_S]

Thanks . I understand importance sampling is applied to tail - in my company it is applied to the tail 1 % in EC - they try to get more number of simulation in the tail of the distribution and hence try to reduce the variations between simulation (simulation noise). Our interest is in EC 99.9 and EC 99.97 .

My question was more on the methodology for imprtance sampling . Apologies if it was not clear earlier . The modellers use a weighting factor Lamda or something for giving more weight to the distributions in tail !!!!. Not really clear on this .