If $S_t$ is a GBM with $$S_t \sim LN\left(S_0 e^{\mu t + \frac{1}{2} \sigma^2 t} , \quad S_0e^{2 \mu t + \frac{1}{2} \sigma^2 t}\left(e^{\sigma^2 t} -1\right)\right) $$
Then if $x=log\left(\frac{S}{K}\right)$, is the distribution: $$x \sim N\left(\mu t + \frac{1}{2} \sigma^2 t , \quad log\left(\sigma^2_{S_t} \sqrt{K}\right)\right)$$
Basically, I am trying to replicate this density, but my variance is never small enough
