I hear that in practice, traders quote options prices in terms of volatility.
What is this convention, and what is the motivation? How do they think about and manage vega risk?
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Volatility is in effect what you are trading in options and the one unknown quantity in options valuation.
The other inputs in option valuation: (1) Spot Price: observed in the markets, (2) Strike: defined by contract terms, (3) time: defined by maturity of contract, and (4) interest rates: also observed in the markets.
So in effect volatility is therefore the price of the option, when put into the option pricing model where all the other inputs are known.
Since the Spot Price is constantly moving and therefore the value of the option. When both parties can agree on the volatility, they can then re-price the option at the prevailing Spot Price at the time of the trade.
With respect to managing vega risk, they will both buy and sell options to manage the vega exposure.