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Inverting a Beta CDF Calibration Between Real-World and Risk-Neutral Distributions

Article Quant Q&A · Author: Otto Winata
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Summary

The document considers a calibration that maps a real-world cumulative distribution function into a fitted risk-neutral cumulative distribution by applying a beta CDF to the real-world CDF. It asks whether reversing that mapping uses the inverse beta distribution.

The answer states that, for the calibrated distribution, the reverse mapping applies the inverse beta CDF: the original real-world CDF is recovered by applying it to the fitted risk-neutral CDF. The answer limits this conclusion to the risk-neutral distribution produced by the calibration. It does not claim that the fitted mapping reverses a true risk-neutral measure, nor does it discuss calibration quality, parameter estimation, or conditions beyond the stated mapping.

Key ideas

  • The proposed calibration maps a real-world CDF through a beta CDF to produce a fitted risk-neutral CDF.
  • Applying the inverse beta CDF to the fitted risk-neutral CDF recovers the original real-world CDF.
  • The inversion statement applies to the calibrated risk-neutral distribution described in the document.
  • The answer does not extend the result to a true risk-neutral measure or assess the fit.

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Full text
# Relationship between Beta distribution and its inverse


# Relationship between Beta distribution and its inverse












I am attempting to transform a real world density into risk-neutral density via calibration through the beta distribution.

Calibration in this context is transforming the rw density into the rn density by applying the beta distribution onto the rw cdf in a way that maximises the fit. So I have a rn cdf and rw cdf, and I want to obtain the parameters of the beta distribution that would transform the rw cdf into a new cdf that best matches the rn cdf. Say I obtain the distribution that does the above transformation. Would the inverse of this beta distribution = the beta distribution that is obtained when transforming risk-neutral into real-world aka the opposite approach to the initial? If not, why?

Approach is outlined on this paper for reference: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2093397

## Answer by Wei (score 0)

https://quant.stackexchange.com/a/80385

So if I understand correctly, you have a "real-world" CDF $F$. Then you calibrate a beta CDF $B$ to give you a "risk-neutral" CDF $G$ with $G(x) = B(F(x))$.

Then yes, it is the case that $F(x)=B^{-1}(G(x))$.

Note that this only holds for the calibrated risk-neutral measure, not the "true" risk-neutral measure that you were fitting to.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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