Extrapolating Missing Tails in Option-Implied Risk-Neutral Densities

Summary

The document raises a practical problem in recovering a stock’s risk-neutral distribution from option prices. The author first derives implied volatilities and constructs densities, but observes that the plotted right tail becomes less complete for longer expirations. They ask how to model or extend the missing tails when there is no sample of observations on which to fit a conventional density estimate.

The question refers to a MATLAB example that fits a linear relationship to approximated call-price densities, but provides no answer or comparison of tail models. It therefore identifies a calibration and extrapolation challenge rather than establishing a recommended method. Any inferred tail depends on assumptions beyond the observed strike range, and the author notes that the displayed densities have not yet been normalized. The document does not specify the option inputs, pricing assumptions, or a validation procedure, so it cannot establish whether a proposed extension is stable or economically plausible.

Key ideas

• Option prices can be used to infer implied volatilities and construct risk-neutral densities.
• The question concerns extrapolating density tails beyond the range supported by available option prices.
• The referenced MATLAB approach fits a linear relationship to approximated call-price densities.
• The document supplies no answer or evidence comparing alternative tail specifications.
• Normalization and assumptions about unobserved tails remain unresolved in the question.

Tags

# R: How do i finish the tails in the risk neutral density, obtained from option prices


# R: How do i finish the tails in the risk neutral density, obtained from option prices












Im currently working on constructing the risk neutral probability distribution of a stock, based on the option prices. In doing so, i calculate the implied volatilities from the option prices, and then construct densities.

I have not yet normalized the densities in the plot below.

As the expiration date increases, more of the right-tail disappears.

My Question: How do i fit/model the "missing" tails in these densities?

When looking for solutions online, i only encounter examples where we have some dataset X, and then fit a density() to that, but given the problem here, i dont have a dataset, and therefore have to "extend" the density, by fitting some relationship, to extrapolate.

The work i am trying to replicate from matlab is: https://se.mathworks.com/company/newsletters/articles/estimating-option-implied-probability-distributions-for-asset-pricing.html where in matlab the author uses:

```
for k = 1:numel(T0)
    pdfFitsCall{k} = fit(pdfK, approxCallPDFs(:, k), 'linear');
end
```

This is my first question, so hopefully there is enough information in my question.

Thank you in advance

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.