Heston Calibration with Feller Constraints and Levenberg–Marquardt
Summary
The document considers fitting Heston model parameters to market option prices through least squares, using multiple initial guesses to explore local optima. It focuses on how to incorporate the Feller condition, which restricts the parameters to keep the variance process well behaved, into an optimization workflow that combines an active-set method with Levenberg–Marquardt.
The response argues that calibrations close to violating the condition can produce unrealistic forward volatility surfaces, and recommends enforcing a positive margin rather than relying on a barely strict inequality. It notes two possible practices described for the referenced solver: use a constrained Levenberg–Marquardt variant, or run unconstrained and reject solutions that violate the condition. The answer does not give a complete implementation or comparison of these methods, and says that unconstrained runs may be faster.
Key ideas
- Heston calibration fits model parameters to market option prices using a least-squares objective.
- Multiple starting points can be used to locate different local optima.
- The Feller condition can be enforced with a positive margin to avoid near-violations.
- A constrained Levenberg–Marquardt variant is one possible way to handle inequalities.
- Another approach is to run unconstrained optimization and reject solutions that violate the condition.
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# Calibrating Heston model parameters using the Active-set method and Levenberg–Marquardt
# Calibrating Heston model parameters using the Active-set method and Levenberg–Marquardt
Background: We're estimating the parameters of the Heston model from current market data of options. This is to be implemented using the active-set method (see section 16.5 here) and the Levenberg-Marquardt method (see here). The resulting optimization problem is a least squares problem (the same as that used in the Heston article, equation number (4)), subject to the Feller condition $2 \kappa \eta - \theta^2 > 0$ as an additional condition by us. The solving algorithm will be executed for multiple starting points to determine a set of local optima of the Heston model.
We fail to understand the conditions used for the active sets, as the article referenced for Levenberg-Marquardt states the optimization problem is chosen to be unconstrained. With the Feller condition being a strict inequality, it will not be used in an active set.
Our understanding: We're assuming to use the LM method to solve a sub-problem given by the active-set method. However, without any (non-strict) inequality conditions we fail to understand how our sub-problems will be formed.
Are we able to alter the strict inequality of the Feller condition by adding a slack variable? Say, to allow for a slack of magnitude $10^{-10}$.
If so, if the condition can be made active, how would the LM method be adapted to account for this condition? If not, are we missing something entirely?
## Answer by Brian B (score 1)
https://quant.stackexchange.com/a/50079
From the point of view of reasonable Heston model calibrations, coming even close to violating the Feller condition provides very unrealistic forward volatility surfaces. Therefore, you should feel perfectly comfortable enforcing $2 \kappa \eta - \theta^2 \geq \epsilon$ for some fairly nontrivial $\epsilon$.
The authors of your linked paper do not mention how they treat inequality constraints. They write that they used LEVMAR as their Levenberg-Marquardt solver, which has a variant allowing for such constraints, but it's also possible they ran unconstrained and then rejected any local minima that violated the Feller condition. That's a common approach.
Since the Levenberg-Marquardt algorithm is basically an interpolation between gradient descent and Gauss-Newton, the treatment of constraints is fairly well understood, but its always faster to just run without them.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.