Consider a GBM : $$S(t) = S(0)\exp\left({(\mu-\frac{1}{2}\sigma^2) t + \sigma W_t}\right)$$ $$d\log S(t) = (\mu-\frac{1}{2}\sigma^2) t + \sigma dW_t$$ $$\frac{d S(t)}{S(t)} = \mu t + \sigma dW_t$$ where $W_t$ is the value of random Brownian Motion with difference distributed standard normal.
I have seen that $$\mathbb{E}(S_t)= S_0e^{\mu t}$$ $$\operatorname{Var}(S_t)= S_0^2e^{2\mu t} \left( e^{\sigma^2 t}-1\right)$$ And I know that the median is $$S(0)e^{(\mu-\frac{1}{2}\sigma^2)t}$$, But I'm looking for an expression for the skewness (and kurtosis) if there is one.
I think the probability density function of a ''St'' is: $f_{S_t}(s; \mu, \sigma, t) = \frac{1}{\sqrt{2 \pi}}\, \frac{1}{s \sigma \sqrt{t}}\, \exp \left( -\frac{ \left( \ln s - \ln S_0 - \left( \mu - \frac{1}{2} \sigma^2 \right) t \right)^2}{2\sigma^2 t} \right).$
Many thanks!