Why Resampling a Noisy Option-Implied Density Cannot Ensure a Better CDF

# How to fit KDE from existing probability density function values


# How to fit KDE from existing probability density function values












I am working with options data, and I am using Breeden-Litzenberger formula to derive the risk-neutral terminal stock price PDF. After applying the formula, here is a scatter plot of strike price vs Breeden-Litzenberger pdf:

At this stage, I would like to fit KDE using `statsmodels.nonparamteric.KDEUnivariate` and then use `scipy.interpolate.interp1d` to obtain a CDF function, ensuring the area under the fitted PDF = 1 etc.

How do I do this with PDF values and not with a sample from the distribution in question? Should I do bootstrapping? Should I fit the CDF with `GammaGam` to have values > 0 and imposing `constraints='monotonic_inc'`?

The overall goal is to obtain a CDF function, making sure it is actually statistically correct.

Any input would be appreciated!

## Answer by lehalle (score 0, accepted)

https://quant.stackexchange.com/a/73711

From a statistical viewpoint, it is not standard to be in a situation to

- have a "noisy CDF",

- sample points from it,

- deduce another CDF that would be "less noisy".

You can repeat point (2) and (3) to "bootstrap the CDF", but what would it means?

First, you have to know that bootstrap is not magic: it allows to "naively" obtain an unbiased estimate of the variance of an estimator, but nothing more. If you want to obtain quantiles (that is exactly what you aim for, since empirical CDF are made of quantiles), you have to apply some corrections. Efron (the author of bootstrap), has a nice paper on that topic "Bootstrap confidence intervals" by DiCiccio, Thomas J., and Bradley Efron (1996).

Qualitatively, it is clear that if you do not have enough sample points, you cannot obtain better estimates of quantiles, just reusing your points. It is only valid at the asymptotic limit (and if you have an infinity of points, you do not need bootstrap).

Second, starting from a noisy CDF cannot generate sample points that are not noisy. Your best hope is that if you sample "few enough" points, the method you use at point (3) would "regularize" the "secondary" CDF. The truth is that there is no good reason that for, except if the "true, not noisy, underlying CDF" is of the same family as the one use by method of your step (3). For instance, to make it very simple:

- if the underlying CDF is a Gaussian,

- and step (3) is assuming it is a Gaussian, hence it just computes its mean and variance.

- Then of course it may work.

Third, to be very practical, you should keep in mind that you are talking about derivatives, market prices and risk-neutral probabilities: if you change it, it will say something about market price. The cost of modifying your original distribution should be reflected on market prices (i.e. at given strikes), and should not only come from a statistical procedure. For instance you shouldn't modify points are strikes that are heavily traded, because market participants "strongly believe" in them.