Shift brackets up for fraction in superscript

Question

The code

\documentclass{article}
\begin{document}
\[
E \left[ e^{\frac{\sigma^2 \lambda^2}{2}} \right]
\]
\end{document}

produces brackets that symmetric(-ish) around the x-height, so they include the exponent properly, but they look way too low on the bottom. Is there a way to correct this?

Please don't suggest to use exp() instead.

Answer 1

To make the math expression look better, I would not bother with either shifting up the large square brackets or with pulling down the exponent term. Instead, I would use inline-fraction notation to reduce the needlessly large vertical size of the exponent term; this adjustment will also let you use less-prominent square brackets. And, it will make the exponent term easier to read, since TeX will be in script-style instead of in scriptscript-style when it processes the letters \sigma and \lambda in the exponent.

By the way, assuming that "E" denotes the (unconditional) expectations operator, I would typeset the letter in the upright font shape.

Incidentally, I don't get your apparent aversion to \exp notation. It was created for sound typographic reasons.

\documentclass{article}
\usepackage{amsmath} % for "\DeclareMathOperator" macro
\DeclareMathOperator{\E}{E} % expectations operator
\begin{document}
\[
E \left[ e^{\frac{\sigma^2 \lambda^2}{2}} \right] 
\quad\text{vs.}\quad
\E \bigl[ e^{\sigma^2 \lambda^2/2} \bigr] 
\quad\text{vs.}\quad
\E [ \exp(\sigma^2 \lambda^2/2) ]
\]
\end{document}