What is the volatility of an Asian option?

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Dan Pirjol and Lingjiong Zhu propose a simple approximation for pricing Asian options on an underlying asset with an implied volatility smile by substituting an appropriately adjusted volatility into a Black-Scholes pricer that takes a constant volatility. For Asian options with strikes close to the at-the-money point, the adjustment is independent of the shape of the volatility smile at leading order in an expansion in log-strike, which makes it robust against

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