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There is a paper by Dragulescu and Yakovenko (DY) in 2002 proposing a pdf for the stock returns in the Heston model. However, in a paper by Daniel, Bree and Joseph, they actually perform statistical tests on DY's pdf and show it is not really any better than a log normal pdf.
Is anyone aware of more recent attempts at a closed-form solution for the distribution of returns under the Heston model?
The issue I have with these approaches is that they use the unconditional distribution to eliminate the latent volatility. However, when the volatility process has very weak mean reversion one would need a very long and clean sample to make robust parameter identification from the unconditional density. They just throw away all the information from the transition dynamics.
My preference is a filtering approach. There have been some older papers that did that, google for Heston together with Ghysels, Gallant, Renault, Chernov, Tauchen, Pan, Bates, Shephard, MCMC, unscented Kalman filter and you will get some references. It is still ugly, since volatility is unobserved, but at least you are looking at conditional transitions rather than the stationary distribution. Even better, you can implied vols and perform joint estimation. Some of the references above do that too.
Gatheral (Amazon) has a quite extensive discussion on that, and dives into calibration issues. In summary, what you describe appears to be less of a modeling issue, and more of a calibration problem. This is primarily because the model functions (such as the Heston model) are not by nature convex in their input parameters. This is simply result of the fact that they are designed with intuitive understanding of parameters in mind, and not functional properties. You can intuitively understand what variance of variance implies, but there's simply not technical reason it should be convex. This results in highly unstable optimizations, and thus ambiguous interpretations, such as described in the sources you give.
