# How to calibrate models with unbounded parameter space

I am calibrating the Heston model with sequential quadratic programming algorithm. It turns out that the volatility surfaces I am calibrating to can be fit very well with extreme values of mean reversion $$\lambda$$ and vol of vol $$\xi$$, such as $$\lambda=11000$$ and $$\xi = 2000$$. The values are legitimate since the only constraints are $$\lambda > 0$$ and $$\xi > 0$$. If both increase simultaneously, large mean-reversion stretches convexity of the smile and the volatility surface is not that extreme at all.

However, such extreme values of parameters have nothing to do with the actual conditions of the market. I notice that this problem is also not unique to the Heston model. Hence I wanted to ask how to constrain parameters of Heston model during calibration to sensible limits (perhaps penalize large values of parameters might help)?

I see that Madan et. al. (2019, Figure 5b) observed the same behavior ($$\kappa$$ – mean reversion, $$\theta$$ – vol of vol):

• This sort of question is dealt with in literature by the term regularisation. I would suggest browsing ridge regression and lasso regression as an introduction to some of the techniques used - there are many
– Attack68
Jul 16 '20 at 8:02

Usually multidimensional objective function of calibration error of stochastic volatility models (Heston , bergomi etc) have many local minima, thus you would get similar calibration error for very different set of parameters.

Some ways to deal with it:

• specify paramter range your are comfortable with. let's say you want your vol of vol to be in the region of $$[ 0.1, 4]$$ , then you just add term of $$1000*1(volvol>4~or~ volvol<0.1)$$ to your objective function (thisis ok if you use simplex-based minimization procedure, in case you use gradient based, you need to used smoothed version of indicator function)

• once you perform first calibration, keep one of the parameter constant for some time (days, weeks). this restricts number of local minima (by making problem a lower dimension one)

• simplify your objective function by including fewer calibration options (i.e. weight by vega, set weights to 0 for very OTM options). This should improve parameter stability.