what's the relationship between forecasted stock volatility and implied volatility? I know that implied volatility is the volatility calculated by BS formula, is there any relationship between implied volatility and underlying asset volatility?(forecasted or historical)
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One is computed from historical stock time-series, that is, from observed past, the other is computed from traded option prices (prices of bets made on the stock with payoff at a future time), that is, from a view on the future paid for with cash.
They are both equally important and useful (if one has enough data to compute them). Loosely speaking, historical one tends to be lower than the implied one, but the comparison needs to be done carefully, given their specific definition elements. A priori there is no expectation for an actual "relationship" between the two. They should rather be viewed as complementing each other.
BS formula is the way to relate traded option prices and implied volatility. This is done assuming a specific geometric Brownian motion dynamics for stock, and many other things. That's why one refers to implied volatility as BS-implied volatility not to be confused with possibly other model implied volatility (like say, Bachelier model, whose dynamics are assumed arithmetic Brownian motion). Implied volatilities can be viewed as expectations of future realized volatilities (see also trading of variance/volatility swaps).
