Relationship between Beta distribution and its inverse
Relationship between Beta distribution and its inverse
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Otto Winata · External communityPost link
External question — Quantitative Finance Stack Exchange
Author: Otto Winata
Original post: https://quant.stackexchange.com/questions/80382
License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/
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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
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Wei · External communityPost link
External answer — Quantitative Finance Stack Exchange
Author: Wei
Original post: https://quant.stackexchange.com/a/80385
License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/
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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.
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Quoted from Forex.com.bd-Editorial External question — Quantitative Finance Stack Exchange Author: Otto Winata Source score (net votes, not local likes): 0 Original post: https://quant.stackexchange.com/questions/80382 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. 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
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