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I see these terms being used on the floor, but don't really understand precisely what is being referred to. One colleague asserted that an 'off term' volatility is for an option whose delivery date doesn't match the expiry date, but this doesn't make sense given that the delivery date is irrelevant for volatility - but rather the expiry date determines volatility.

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I think what your colleague meant is that the expiry date of the product doesn't match THE expiry date of the market. For instance a volatility product that would expire at 10:42 am on a random day would be off term. One that expires at the same time than a major listed contract would be term vol.

Your desk will quote off term products with a higher margin/spread because they are harder to hedge.

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