I'm working on a Monte Carlo function for valuing several assets with partially correlated returns. Currently, I just generate a covariance matrix and feed to the the rmvnorm() function in R. (Generates correlated random values.)However, looking at the distributions of returns of an asset, it is not normally distributed.This is really a two part question:1) How can I estimate some kind of PDF or CDF when all I have is some real-world data without a known distribution?2) How can I generate correlated values like rmvnorm, but for this unknown (and non-normal) distribution?

First of all, to fit a one-dimensional distribution you usually start with a parametric form (e.g. normal, t, generalized hyperbolic) and fit the parameters. It is the modeller's job to choose a suitable form.Secondly, for dependence there are two common approaches: elliptical distributions and copulas. Read up on both methods. Elliptical distributions are more intuitive but less flexible.

Hi,The distributions do not appear to fit any known distribution. I think it would be very dangerous to assume a parametric and then use that for monte carlo estimation.Isn't there some kind of bootstrap or "empirical monte carlo" method I can look at?

QuoteThe distributions do not appear to fit any known distributionThis is a strong statement.QuoteIsn't there some kind of bootstrap or "empirical monte carlo" method I can look at?Bootstrapping is certainly an option. Beware that this can give biased results with small samples.

Hi noah977, About the distribution, take a look at kernel density estimation (aka parzen windows). Is going to allow you to fit a density without any assumption on the distribution (the data has to be i.i.d though) . You still are going to have to calibrate the so called bandwidth, but if you have enough data there are a lot of straight forward methods to do it (http://en.wikipedia.org/wiki/Kernel_density_estimation)I dont use R, but in scipy (python) you can use (http://www.scipy.org/doc/api_docs/SciPy ... n_kde.html).Hope this helps.

QuoteOriginally posted by: acaneteHi noah977, About the distribution, take a look at kernel density estimation (aka parzen windows). Is going to allow you to fit a density without any assumption on the distribution (the data has to be i.i.d though) . You still are going to have to calibrate the so called bandwidth, but if you have enough data there are a lot of straight forward methods to do it (http://en.wikipedia.org/wiki/Kernel_density_estimation)I dont use R, but in scipy (python) you can use (http://www.scipy.org/doc/api_docs/SciPy ... n_kde.html).Hope this helps.Is this method multivariate?

yeshttp://en.wikipedia.org/wiki/Multivariate_kernel_density_estimation

QuoteOriginally posted by: acaneteyeshttp://en.wikipedia.org/wiki/Multivariate_kernel_density_estimationInteresting.I have used kernel estimation for univariate fitting, but I had be careful at the tails of the distribution. Because I was using a Gaussian kernel, the fitted tails were too thin. I ended up using the kernel estimation for the body of the distribution, and separately fitting the tails with a generalized Pareto distribution.

QuoteOriginally posted by: ehQuoteOriginally posted by: acaneteyeshttp://en.wikipedia.org/wiki/Multivariate_kernel_density_estimationInteresting.I have used kernel estimation for univariate fitting, but I had be careful at the tails of the distribution. Because I was using a Gaussian kernel, the fitted tails were too thin. I ended up using the kernel estimation for the body of the distribution, and separately fitting the tails with a generalized Pareto distribution.This Extreme Value Theory (EVT) fit with the Generalized Pareto Distribution is probably the simplest but the best way to fit a tailed distribution.

Thats a good idea. I'll give it a try. I tried something similar but using kernels with different bandwidths instead of the GPD.