low delta always means OTM, intuitively

(Low delta means low absolute magnitude. The sign of delta is a separate feature.)

The more OTM, the less sensitive to underlier moves — low delta.

The more ITM, the more stock-like — high delta. This holds for both calls and puts.

For a call holder, the most stock-like is a delta of 100%
For a put holder, the most stock-like is a delta of -100% i.e. a short stock

For a put Writer, the most stock-like is a delta of +100% i.e. long stock. This put is so deep ITM it will certainly be exercised (unload/put to the Writer), so the put writer effectively owns the underlier.

On FX vol smile curve, people quote prices at low-strike points and high strike points both using low deltas like 25 delta or 10 delta. (The 50 delta point is ATM).

– On the low-strike side, they use an OTM Put. Eg a put on USD/JPY struck@55. Such a put is clearly OTM since as of today option holder will not “unload” her USD (the silver) at a dirt cheap price of 55 yen.

– On the high-strike side, they use an OTM Call. Eg a call on USD/JPY @140. Such a call is clearly OTM, since as of today option holder will not buy (“call in”) USD (the silver) at a sky high price of 120 yen.

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