How to avoid having negative volatility when applying Heston model?

When applying the Heston model to generate the sample volatility surface, some of the volatility value will be negative. I am just wondering what do practioners normally do with these negative value. Do you

1. simply ignore it;



2. set negatives to 0; or


3. square it, take absolute values, or something else?

Answer

It is not necessarily something that must be wrong with your model. Inherent in the Heston discretization methods of its continuous time dynamics is the possibility of negative values in the variance process.

Here are couple solutions you can look at in order to "fix" your problem:

• Usage of different Euler schemes, such as the Full Truncation scheme.


• Making the discretization grid smaller.


• approximate Fourier inversions needed to simulate the integrated variance process.



• Moment-matching techniques (for example, approximating the non-centrally chi-squared distribution by a related distribution whose moments are (locally) matched with those of the exact distribution).


• Using drift interpolation instead of Fourier inversion