In the equity space and between different institutions, options are usually not quoted in implied volatilities. For this to work, you'd additionally have to also quote at least a risk-free rate, forward and a measure of time-to-expiry (e.g. a day count convention) for European options. For American options, it get's even less feasible.
In the broker/OTC market, options are usually quoted as a price, spot reference and delta combination. The delta and spot reference determine how the price is adjusted to account for spot moves between quote and trade time to the first order. This is necessary as a trade usually does not happen immediately immediately but only after the broker collected a sufficient number of quotes and the initiating counterpart made a decision. Let $T_\text{quote}$ and $T_\text{trade}$ be the times of the quote and trade, respectively. The traded price is then given by
$$
P \left( T_\text{trade} \right) = P \left( T_\text{quote} \right) + \Delta \left( T_\text{quote} \right) \left[ S \left( T_\text{trade} \right) - S \left( T_\text{quote} \right) \right].
$$
Of course, you (as the one providing the quote) can always retract it while it hasn't been traded yet. E.g. when the too much time passed, the implied volatility changed or the spot price moved too much for the delta approximation to be valid.
On the screen, there is not need for this as you can update your quotes at all times.