There are some more possibilities for people who are more interested in improving stochastics part of the project. Since we know the value of various moments and cross moments of the integrals of both correlated and uncorrelated parts of the log of Asset price diffusion along all the variance grid points, it could be possible to locally construct the characteristic function of the desired log of asset price in terms of powers of transform parameter, and then add different terms weighted with their respective probabilities along all of the grid points. And then we could have a characteristic function in the form of a polynomial in the transform parameter. I have not written anything on the lines on the paper so there might be some hurdles but this seems a much better approach to the problem than the edgeworth expansions and may become perfectly precise with a bit of research. This is a thought for people who want to improve the stochastics part of the work.

Since we know the value of various moments and cross moments of the integrals of both correlated and uncorrelated parts of the log of Asset price diffusion along all the variance grid points, it could be possible to locally construct the characteristic function of the desired log of asset price in terms of powers of transform parameter, and then add different terms weighted with their respective probabilities along all of the grid points. And then we could have a characteristic function in the form of a polynomial in the transform parameter. This seems a much better approach to the problem than the edgeworth expansions and may become perfectly precise with a bit of research.

The methodology can also be used for Hybrids and other reduced factor high dimensional models. Let us say we have a hybrid depending upon two/three correlated assets with their own stochastic volatilities (with these SV's uncorrelated with each other but possibly correlated with their own assets) and we can write the evolution of the total process in the form of a logarithm independent of the asset price itself but depending on the variances. We can orthogonalize the total process in terms of two/three independent factors each depending simultanously on two/three uncorrelated stochastic volatilities. We could do one dimensional evolution of each SV process using the one dimensional transition probabilities and find different values of moments of the related stochastic integrals. There are terms in the form of the cross product of the square root of variances but they can easily be handled without any two/three dimensional transition densities since the SV's are uncorrelated. We can finally construct a single characteristic function in the form of sum of the independent eigenvectors. Please note that each eigenvector has simultaneous dependence on all of the SV processes. Reduced factor LIBOR Market Model also falls in the same category after the freezing of the drift recipe. It is also possible to apply the method on LIBOR Market model with stochastic basis.

We could also work with other interesting stochastic differential equations for variance other than standard mean reverting or normal/lognormal as long as we could find the moments of the integral of SDE of variance using one dimensional transition probabilities which could be possible for a very large class of variance SDEs. Since most models in vogue today are limited due to analytical tractability of Fourier/Laplace transform methods, the Lahore method of solution of SV equations would help to work with much larger class of variance SDEs.

With a bit of hack, you could add other features like jumps in the model while retaining the framework of transition densities and the moments of various stochastic integrals leading to a characteristic function.

If somebody wants to discuss the possibility of the use of this method in more complicated problems, please feel free to discuss it with me on skype. My skype id is ahsan.amin2999 I know that the method could be used with a bit of effort in more complicated and sophisticated models in higher dimensions.