I suggest you have a look at the paper:
- Schloegel, Erik (2010) "Option Pricing Where the Underlying Assets Follow a Gram/Charlier Density of Arbitrary Order", Journal of Economic Dynamics and Control, Vol. 37, No. 3, pp. 611-631
available on SSRN.
A random variable $Y$ that follows a Gram/Charlier Type A series distribution has the probability density function
\begin{equation}
f_Y(x) = \phi(x) \sum_{j = 0}^\infty c_j \mathrm{He}_j(x).
\end{equation}
Here, $\phi(x)$ is the standard normal density function and the $\text{He}_j(x)$ are the Hermite polynomials. The coefficients $c_j$ are related to the cumulants of the distribution. When you truncate the infinite sum at $j = 4$, set $c_0 = 1$, $c_1 = c_2 = 0$, $c_3 = \mathcal{S} / 6$ and $c_4 = \mathcal{K} / 24$, then the resulting distribution has zero mean, unit variance and an approximate skewness of $\mathcal{S}$ and excess kurtosis of $\mathcal{K}$.
We can now define the logarithmic price process $X$ as
\begin{equation}
X_t = X_0 + \gamma t + \sigma \sqrt{t} Y,
\end{equation}
where $\gamma$ is chosen such that $S_t = S_0 \exp \left\{ X_t \right\}$ is a martingale under the bank account numeraire. This approach can be extended to multi periods. Please see the above paper for details. I can also recommend the author's book
- Schloegel, Erik (2014) "Quantitative Finance - An Object-Oriented approach in C++", Chapman & Hall