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The QuantLib pricing method AnalyticGJRGARCH may result to a negative price for a european option.
For example, negative prices have been observed for one-year european options where the following settings for the involved GJRGARCH model are used:
Omega = 0.000003
Alpha = 0.03
Beta = 0.93
Gamma = 0.059
Lambda = 0.1
DaysPerYear = 365
Discretization = Full Truncation
Click on download to download an xml file that contains an Excel formula that demonstrates this issue.
You may then reproduce the issue in spreadsheet by clicking on Go -> Load Excel Formula from XML or one of its variations and choose the xml file downloaded in the step above.