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Let $$\ln\left(S_T/S_t\right) $$ have mean $\mu_\tau$ and standard deviation $\sigma_\tau$, where $\tau=T-t$, and density of its standardized form $$ X= \frac{\ln(S_T/S_t)-\mu_\tau}{\sigma_\tau} $$ approximated by Gram-Charlier expansion $$ f_X(x) = \phi(x) - \gamma_{1\tau} \frac{1}{3!} D^3 \phi(x) + \gamma_{2\tau} \frac{1}{4!} D^4 \phi(x), $$ with $\phi$ ...


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