Feller Condition Tradeoffs in Heston Model Calibration
Summary
The discussion asks whether the Feller condition, expressed through the Heston variance parameters, should be enforced when calibrating option prices. The motivating examples reportedly place the condition close to its boundary, with some parameter sets violating it. The accepted response advises against imposing it as a hard constraint or penalty when that would prevent a good fit to observed market data.
The replies say violations are common, particularly for options with maturities beyond a few weeks. They emphasize a separate implementation concern: the Heston characteristic function must remain continuous, since mishandling its branch cut can produce incorrect prices. A transformation described in the cited literature can address this issue. These are practical calibration recommendations rather than a universal theorem; calibration choices depend on the market fit, and the thread does not quantify the impact of constraint choices across datasets.
Key ideas
- Enforcing the Feller condition can prevent Heston parameters from fitting market option prices well.
- The condition is often violated in calibrations, especially for options with longer maturities.
- A calibration should separately ensure continuity of the Heston characteristic function.
- Branch cut handling matters because discontinuity can lead to erroneous option prices.
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# About the Feller Condition in Heston Calibration # About the Feller Condition in Heston Calibration I have noticed when reading (many) articles about Heston Calibration that not all (few actually) do care about the Feller condition. Below is a compilation of calibration results from some different authors (source): If we calculate the Feller condition $2 \kappa \theta - \sigma^2$ for the above we can see that in all cases it is extremely close to 0. For row 1, 3 and 6 it is in fact negative. So the Question is basically: In my own calibration code, should I use the Feller condition as a calibration constraint (e.g. as a penalty function) or should I skip the constraint since it doesn't always hold in the market? Looking forward to your input! ## Answer by q.t.f. (score 5, accepted) https://quant.stackexchange.com/a/22983 You should not use the Feller condition as a constraint. In many cases its violation will be required for a good fit to the market data. ## Answer by Andreas (score 3) https://quant.stackexchange.com/a/30279 As q.t.f stated, you shouldn't pay too much attention to the Feller Condition since it is often violated in the Heston model, especially for options with more than a few weeks until maturity. However, you should make sure that your Characteristic Function stays continuous, else you'll end up with "wrong" prices. This is caused by the branch cut along the negativ real axis and can be avoided through a simple transformation. There are several formulations out there that prevent that from happening, a good and simple start is Albrecher et al. (2006).
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