Consider a random variable $X$ taking values in $[-1,1]$ with mean $0$ and variance $\sigma^2$. A quantity $V^2$ is a sub-Gaussian variance proxy if $\mathbb{E} \exp(\lambda X) \leq \exp(\lambda^2 V^2/2)$ for all $\lambda \in \mathbf{R}$; we may take $V^2$ to be "the" sub-Gaussian variance proxy if it is the smallest $V^2$ satisfying the condition.
I have verified that, in general, there no constant $C$ for which $V^2 \leq C \sigma^2$; but is there a non-trivial bound for $V^2$ that depends on $\sigma^2$? Note that an absolute bound on $V^2$ follows because $X$ is bounded; I'm looking for a bound that depends on the variance $\sigma^2$.
Question. Is there a bound of the form
$$ V^2 \leq f(\sigma^2) \quad \text{as } \sigma^2\to 0 \qquad \text{(i.e. for sufficiently small $\sigma^2$)} $$ for some function $f$ with $f(x) \to 0$ as $x \to 0$?