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Agree with @Brian B. With BS, you cannot have the issue in (1). Tree, grid, Monte Carlo could all result in errors though. (2) is a likely reason. I just tried in Julia for ATM, 0 div and rates plus 0.2 vol and 1 year tenor. Shifts smaller than ~ 0.00008 result in an error for Gamma. Delta seems to be less sensitive for this, and it is fine for at least 1e-7 ...

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