Intuitively, it seems that a lognormal distribution with a tiny $\sigma/\mu$ ratio might look quite a bit like a normal distribution. Can this be formalized in any way (e.g., by stating upper bounds on the size of the error margin in the cdf from using this approximation, etc.)?
Also, any intuition behind why this approximation works well would be appreciated.
For some reason, I could not find any references on the web, apart from this (unsupported) rule of thumb in a random risk software manual.
The lognormal random variable with parameters $\mu, \sigma$ is $e^{\mu + \sigma Z}$, where $Z$ is standard normal, and so its CDF is $F(x) = \Phi\left(\frac{\ln(x) - \mu}{\sigma}\right)$ for $x > 0$, where $\Phi$ is the standard normal CDF. When $x = e^{\mu} (1+ t)$ is near $e^{\mu}$, $$\ln(x) = \mu + \ln(1 + t) = \mu + t + O(t^2)$$ so that $$F(x) \approx \Phi\left(\frac{t }{\sigma}\right)$$ In fact $$F(x) - \Phi\left(\frac{t }{\sigma}\right) = \Phi\left(\frac{\ln(1+t )}{\sigma}\right) - \Phi\left(\frac{t}{\sigma}\right) $$
This will be a very good approximation for $t$ near $0$. For fixed $t$, it goes to $0$ as $\sigma \to 0$ because both $\Phi$'s go to $0$ if $t < 0$ and both $\Phi$'s go to $1$ if $t > 0$. When $\sigma$ is small there appear to be two local minima of $F(x) - \Phi\left({t }/{\sigma}\right) $, at approximately $t = \pm \sqrt{2} \sigma$, where $F(x) - \Phi\left(t/\sigma\right) \approx -\sigma/(\sqrt{2\pi} e) \pm \sigma^2/(6 \sqrt{\pi} e)$. Thus the maximum error in $F$ is approximately $$ \frac{\sigma}{\sqrt{2\pi} e} + \frac{\sigma^2}{6 \sqrt{\pi} e}$$
For example, for $\sigma = 1/2$, the actual maximum error is approximately $0.0817231495769266$, and the above approximation to the maximum error is $0.08202940444$. As $\sigma$ gets smaller, the approximation should get better.