Why is volatility (so) important?
Of all the inputs to theoretical pricing models such as Black-Scholes, volatility is the hardest to get right and of the greatest significance. It's the future realized volatility that every trader wants to know but most, if not all, traders have no crystal ball so the volatility input is based on a best possible estimate using available information and judgement.
While historical realzied volatility is associated with underlying contracts, implied volatility is derived from the price of an option in the marketplace so in a way it's what marketplace thinks the future realzied volatility will be over the life of an option. Because of this, in practice traders often use premium and implied volatility interchangeably. What's truly special about implied volatility is that it's a more effective measurement unit of options across different strikes and maturities. Just like bond traders use yield to compare bonds, option traders use implied volatility for relative comparsion as it strips away strike, maturity and underlying level. When you're trading options, you're really expressing a view on volatility: you'd buy an option when you expect the underlying will move more than an implied volatility suggests!
Gold Futures (GC=F) — Oct 2008 to Jul 2012 Full-period annualised historical volatility
Historical volatility is the annualised standard deviation of log returns. For most underlying contracts, a fundamental property is that the result is roughly the same regardless of the sampling interval — daily, weekly, or monthly — as long as the data covers the same period and you use the correct annualisation factor.
Rolling Annualised Volatility Daily 40-day vs Weekly 5-week vs Monthly 6-month windows
Weekly and monthly series are noisier because each window has fewer observations than the daily series, but all three track the same underlying volatility level.