Stabilizing Heston Calibration with Parameter Bounds and Simpler Objectives
Summary
The document addresses unstable Heston model calibration: very large mean-reversion and volatility-of-volatility parameters can fit a volatility surface well even when the values lack a plausible market interpretation. It notes that stochastic-volatility calibration objectives can have many local minima, allowing substantially different parameter sets to produce similar fitting errors.
Suggested remedies include setting acceptable parameter ranges and penalizing values outside them, with a smoothed penalty needed for gradient-based optimizers. Another approach is to hold a parameter fixed temporarily, reducing the optimization dimension and potentially limiting local-minimum variation. The objective can also be simplified by using fewer calibration options, such as weighting by vega or excluding very out-of-the-money options. These are practical stabilization ideas, not guarantees of a unique or economically correct calibration; the choice of bounds and option weights remains a modeling judgment.
Key ideas
- Heston calibration can return extreme parameter values while still fitting observed volatility surfaces closely.
- Different parameter sets may have similar errors because the objective has multiple local minima.
- Set plausible parameter bounds and penalize values outside the chosen ranges.
- For gradient-based optimizers, smooth a discontinuous bound penalty.
- Holding a parameter fixed or reducing the calibration option set can improve stability.
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# How to calibrate models with unbounded parameter space # 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): Thank you in advance. ## Answer by alexprice (score 4, accepted) https://quant.stackexchange.com/a/55719 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.
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