I am new to the topic of Asian options. Assume I want to price an Asian put (fixed strike, discrete average) in the Black Scholes world. I know implementations to calculate the value but what is the best way to find the implied volatility parameter? Is there a usual way to derive it from the option market of plain vanilla products, e.g. European calls or puts of a certain range of maturities?
The easiest way is to use single-expiry volatility that you would get from your volatility surface. It is usually good enough for government work (e.g. to get a sense if you are getting fleeced by a dealer or to understand your vega risk).
A better way is to use local volatility model and the whole volatility surface up to the date of expiry. There is also a bunch of semi-analytical approximations that use weighted volatilities up to there expiry date. Unless you are a dealer and trying to quote these in competition, you don't need to bother with these.
The approximation I mentioned earlier is that in order to price an Asian option with strike K and maturity T on an asset with spot price S0, one should use the implied volatility at the modified strike K'=S0*(K/S0)^(6/5) and the same maturity T. This assumes an asset with a flat forward curve, like a futures contract.
The derivation of this result is presented in this Risk magazine article What is the volatility of an Asian option?
The approach is based on a short maturity expansion for Asian options in the local volatility model (continuous time averaging) presented in an earlier paper with Lingjiong Zhu.