According to the saddlepoint approximation, if the cumulant generating function $K(t) = \log E[e^{tX}]$ of the distribution of the random variable $X$ exists and is known, then the density $f(x)$ of $X$ can be approximated as follows: $$ f(x) \approx \sqrt{\frac{1}{2\pi K''(\hat t(x))}} \exp\{K( \hat t(x)) - \hat t(x) x \} $$ where $\hat t(x)$ is the solution to $K'(\hat t(x)) = x$. Primes denote differentiation wrt $t$.
Now suppose the CGF is not known but approximated $$ \tilde K(t) \approx \sum_{n=1}^4 \frac{\tilde \kappa_n}{n!} t^n $$ Then I don't think there is always a unique and real solution to $$ \tilde K'(\hat t(x)) = x $$ which is a $3$-rd order polynomial equation, such that $$ \tilde K''(\hat t(x)) > 0 $$
Are there `regularization' methods to ensure the approximated CGF has a proper valid region?
I am not sure otherwise what the applicability is of the saddlepoint method, at least in options pricing, if you cannot use an approximated CGF since many times the CGF is unknown.
Background to my question
If the skew is generated by a stoch vol model, I have found a way (actually years ago already) how to turn the skew into an approximately symmetric smile that respects the distribution of realized volatility.
Once I have this smile I can estimate the moments (and thus the cumulants) of realized volatility. For various values of correlation up to say $\pm 0.4$ I see that the estimated first four moments are quite stable, i.e. doesn't vary wildly with correlation which is how it should be.
Under this smile I can then in principle use the Hull and White mixing formula to approximate the distribution of realized volatility by assuming for example the CGF I wrote above.
I've tried estimating the density of realized vol using Gram-Charlier/Edgeworth as well, but those methods appear to be unstable.
