# 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).

• It will depend on how you're actually calibrating - are you just fitting the asymptotic expansion to vanilla options? Or are you solving using a 2d pde? – will Oct 12 '17 at 15:12
• The usual thing is to fix $\kappa$ outside the calibration; then the effect of the remaining parameters is nondegenerate. – q.t.f. Oct 12 '17 at 19:50
• Quantuple, many thanks =) @will Let's assume I'm fitting the full fledged model to vanilla options. To be specific as far as the pricing method is concerned, I'm using a Fourier method (Attari's single integrand), with caching of the characteristic function (speed boost) and a smart control variate (accuracy boost). Note that I'm using the asymptotic expansion though but only to obtain a "decent" initial guess. Actually I'm fitting many such "expansions" (Forde et Al., Bergomi, Gauthier&Rivaille) and I take my initial guess as the best one. – Catherine Janssen Oct 13 '17 at 7:53
• @q.t.f., yes I've indeed seen this in many papers (Bergomi - Smile Dynamic I, Buehler's presentations). In any case you're making a tradeoff between the quality of the fit and the stability of kappa in that case. Assuming I'm ready to go with a fixed mean rev, how would you fix it? From the tests I ran, you cannot pick any value... but indeed a good value can be inferred from the variance curve (TS of variance swap par rates). Also, when fixing mean rev, the instability will translate to correlation and/or vol-of-vol. – Catherine Janssen Oct 13 '17 at 7:57

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 $$\kappa$$.