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2

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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Everything (warning: I have not checked 3rd order greeks) that is not delta is in terms of ccy2 in the standard Garman Kohlhagen model. Gamma is not in CCY1 by default either (some vendors like Bloomberg display it like that to be consistent with Delta). Why Let's start by not looking at FX but equity to help build intuition. The actual price of an option is ...

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The best way is to start with definitions (instantaneous and their finite difference versions) of Greeks. For a currency pair $(FOR,DOM)$ with FX rate $S$, the number of $[DOM]$ (domestic, numeraire, right-side) units needed to buy one $[FOR]$ (foreign, asset, left-side) unit, let $V(S)$ be an option's price in $[DOM]$ units. Note that the unit of $S$ is: $$... 2 Not sure if you meant only short puts with "when option is ITM increase in volatility will decrease the delta, whereas for OTM option increase in volatility will increase the delta". Either way, you cannot generalize it like that as you figured out yourself. Adding a few remarks to @Kermittfrog's excellent answer. If you plot delta and the ... 5 In the Black-Scholes-Merton model, with model option price V as a function of underlying price S_t, strike price X, continuously compounded risk-free rate r, continuously compounded dividend yield y, time-to-maturity (in year fractions) \tau and implied volatility \sigma, our \Delta is defined as$$ \Delta\equiv \frac{\partial V}{\partial S_t}...

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