It is well-known that calibrating Heston to the vanilla market is not as easy as it seems: some parameters are "interdependent" and the objective function exhibit plateaus in the parameter space (at least in some dimensions of the parameter space, typically mean-reversion). A good reference on this is this 2017 paper by Cui et Al.
The authors mention
There are two possible approaches that one can seek to deal with this: the first is to scale the parameters to a similar order and search on a better-scaled objective function; the second is to decrease the tolerance level for the optimisation process, meaning to approach the very bottom of this objective function
I am particularly interested in the first approach and was wondering the parameterisations that you experts tend to use for your daily Heston calibrations? Is there a sound way to disentangle $\kappa$ and $\rho$ for instance?
For instance, the variance process being CIR the asymptotic variance of variance computes as $$\lim_{t \to \infty} \text{Var}_0^\Bbb{Q}[v_t] = \theta \frac{\xi^2}{\kappa} $$ To disentangle the effects of $\kappa$ and $\xi$ on smile convexity, one could therefore reparametrise the Heston variance process as $$ dv_t = \kappa (\theta-v_t) dt + \xi^* \sqrt{\kappa} \sqrt{v_t} dW_t $$ where we have defined a new parameter $\xi^*$ such that $\xi = \xi^* \sqrt{\kappa}$. This parameter looks more natural since it would eventually lead us to: $\lim_{t \to \infty} \text{Var}_0^\Bbb{Q}[v_t] = \theta \xi^*$.
Actually, I've found that this parametrisation was already proposed by Hans Buehler (see here, section 1.1.1. for a small discussion and equation (2) for the result). In some other presentations he mentions another reparametrisation where vol-of-vol appears in the drift (but the idea is the same IMO).
