Quantiles of Lognormal Variables Under Exponential Transformations
Summary
The document explains how to obtain a quantile of an exponentiated normally distributed variable. If the logarithm of a positive variable is normal, its quantile is found by taking the exponential of the corresponding normal quantile. Because the exponential function is increasing, it preserves the ordering of values and therefore maps each normal quantile directly to the matching lognormal quantile.
The discussion corrects a proposed adjustment of the normal mean by a volatility-related term. That adjustment may arise in a separate stochastic-process setting, but it is not needed when transforming the quantile of a specified normal random variable. The note gives a probability argument showing that the transformed threshold has the same cumulative probability. It focuses narrowly on the mathematical transformation; it does not cover estimating distribution parameters, validating a lognormal model for stock prices, or deriving a price distribution from a particular financial model.
Key ideas
- An increasing transformation maps a distribution quantile to the same transformation applied to that quantile.
- For a lognormal variable, exponentiate the corresponding quantile of its normally distributed logarithm.
- A volatility adjustment to the mean is not required solely because the variable is exponentiated.
- The result assumes the logarithm has the stated normal distribution.
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# Quantile normal and lognormal
# Quantile normal and lognormal
Let's assume we have a normal distribution $X\sim \mathcal{N}(\mu,\sigma^2)$. In a normal distribution the quantile can be calculated as follows:
\begin{equation} \Phi_X ^{-1}(p)=\mu +\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(2p-1) \end{equation}
If we want to calculate the value in the future of a stock we map it as:
\begin{equation} Y=\exp(X) \end{equation} Which means: \begin{equation} \log(Y)\sim \mathcal{N}(\mu,\sigma^2) \end{equation}
I would like to know that if the function of the quantile can be calculated based directly on:
\begin{equation} \Phi_Y ^{-1}(p)=\exp(\mu -\sigma/2+\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(2p-1)) \end{equation}
The part of the equation $-\sigma/2$ is extracted from îto calculus, however, I cannot find anywhere the correctness of this equation (I deduced it). I think the function $exp$ is monotonic, so, it should preserve the value for the quantiles, but I'm not certain. One of my certainties is that $\mu$ changed to $\mu-\sigma/2$, I have no idea if that modifies in some way the calculation of $\Phi_Y ^{-1}(p)$, or if $\sigma$ also changed.
## Answer by Quantuple (score 6, accepted)
https://quant.stackexchange.com/a/35319
Quantiles are preserved under monotonic transformations, hence the quantile for $Y$ is simply the exponential of the quantile of $X$, no need for corrections whatsoever (see here for instance).
Put otherwise, let $q$ denote the quantile $\alpha$ of $X$ i.e. $$\Bbb{P}(X \leq q) = \alpha$$ then \begin{align} \Bbb{P}( X \leq q ) &= \Bbb{P}( \underbrace{\exp(X)}_{Y} \leq \underbrace{\exp(q)}_{Q} ) = \alpha \end{align}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.