Re run-times, later on in Sec. 7, I say that it takes me 0.5 - 2 minutes to compute my (Case 1) option price examples. This is the 'physical case', in the sense that calibrated parameters will fall under Case 1.
Case 2, which is 'unphysical' (as there is no stationary vol distribution), takes much longer --- that one is for hardcore math finance types only!
Of course, these times are for the exact formulas evaluated in Mathematica. Pure numerics, such as a PDE approach, will be very similar for Heston vs. XGBM, and can be *very* fast. An expert/reference on very fast PDEs for SV models (including XGBM) is Yiannis Papadopoulos and his project site here
Oct 8, 2018 update: I just added a blog post here
, which provides a little more motivation for the model based upon fits of the stationary vol density [$]\psi(\sigma)[$].