Q:

Why use implied volatility

First I'll describe the way I understood things so far from the literature, feel free to correct me here, and then I formulate some questions. I'd search through QSE, but haven't found so far similar question.
The BS model assumes the stock to follow an SDE with a linear diffusion term, hence constant volatility:
$$
  \mathrm dS_t = \mu S_t \mathrm dt + \sigma S_t\mathrm dW_t.
$$
Although model showed reasonable match with market prices back in early 70s, in a decade it was noticed that when assuming constant volatilities calibrated e.g. with ATM options, options on the same stock with different maturities/strikes differ in value from what BS model suggests.
Implied volatility $\sigma_{\mathrm{imp}}(K,T)$ hence was defined as a value of constant volatility which is when put in BS formula returns the market price for the option with maturity $T$ and strike $K$. Since the plot of $\sigma_{\mathrm{imp}}$ appeared far from being flat and showed infamous skew/smile effects, it was clear that assuming any constant level of volatility does not lead to observed market prices. Due to this reason, Derman and others in mid-90s introduced the local volatility model
$$
  \mathrm dS_t = \mu S_t\mathrm dt + \sigma_{\mathrm{loc}}(S_t,t)\mathrm dW_t
$$
and suggested calibrating $\sigma_{\mathrm{loc}}$ with $\sigma_{\mathrm{imp}}$ rather than e.g. with historical volatilities. Further, there were even models of dynamics for the entire volatility surface - I think there are some people doing this now with SPDEs.

Now, the implied volatility is a model-dependent concept. Even if that would be a very good estimate of market's expectation about the future volatility (which some people don't think to be true), that means assuming that everybody on the market is using the same model. E.g. classical implied volatility is computed via BS formulae, so taking it into account requires assuming most people on the market are using BS to price e.g. vanillas. Does it make sense?
Implied volatility necessary contains some noisy information - e.g. demand/supply of OTM options make their prices more volatile than those of ATM options, which is not an intrinsic property of the stock (which volatility is supposed to be). Even if one decides to use local deterministic/stochastic volatility model, why don't calibrate this on the historical data? I do know the criticism of the latter method, but since most of the models are Markovian anyways, that seem to justify historical approach way better than the implied one, doesn't it?

A:

My try to answer this question with some other questions:

Is the BS model right? No. Is it useful: yes. Taking a traded price and the BS Model there is only one input factor that is not given by the market: the implied volatility. It is a measure to compare options across time and strike. 
Are there better models? yes. Those that you mention: The local vol models. There are stochastic vol models, Lévy models and if calibrated correctly they perform much better than the BS model. In fact the BS model often is a trivial special case of the more complicated models. How are those models evaluated? One way is to see if the BS-implied vol of the prices calculated by these models fits the BS-implied vol that can be observed on the market. Again the BS-implied vol can be used to do comparisons. 
Why not use historical data? The volatility that you use for an option that expires in the future has only limited connection to past/observed volatility. E.g. if I trade an option on NIKKEI and the Fokushima catastrophy happens then implied vol will rise immediately. Historical vol will rise only gradually. In short: implied vol is forward looking and reactive whereas historical vol is backward looking and slow.

A:

we use implied vol for similar reasons why we use duration. we know that security prices are not linear functions of rates, yet we look at the duration, because it gives us an idea of sensitivity to a rate. implied vol gives you a measure of volatility, it doesn't perfectly describe it, but as long as we know this, it's still a valuable metric.

