Stochastic and local volatility models, volatility surface, term structure, and break-even volatility

Implied volatility of an option contract is the volatility of that implied (calculated) from the price of the option with an equation, like Black and Scholes (BS), which contains the market view of the contract holders about the future asset movement. For a put option, such implied volatility indicates the level of the down risk that reflects in the cost of purchasing such hedging instruments. For a call option, it indicates the upside opportunity or how bullish the underlying asset is.