Stabilizing Heston Calibration with Parameter Reparametrization

# Heston model reparametrisation


# Heston model reparametrisation












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).

## Answer by jherek (score 2)

https://quant.stackexchange.com/a/43552

Do you have a concrete example showing the issue?

I have calibrated Heston to many different equities and never really had the issue of disentangling $\kappa$ from $\sigma$ (the vol of vol). In general, the initial guess will strongly influence the $\kappa$. From there on it's a local search, and you won't end up too far.

The variance swap curve may also be included in the calibration at almost no additional computational cost and will help in stabilizing the $\kappa$.

If you calibrate regularly, it is common practice to add a regularization term to control the change in parameter values.