Implied volatility is the volatility that when inputted in the Black-Scholes model, it returns the theoretical market price of a European option value.
I understand that implied volatility is not observable. However, we have access to option value in real life. How is the option value calculated in real life? All the while I thought that we need all inputs for Black-Scholes model to calculate option value. But in real life, it seems that it is different.
I'm not sure if you're asking about listed market prices but if you are, then they are just that, . market prices, i.e. they're decided by the supply and demand of the market. The implied volatility is gotten from the market prices, not the other way around, hence the term "implied.
If you're asking about how the volatility is found to price options not traded on the market then you would typically use a volatility surface of the underlying to find the volatility to use when pricing your option. Essentially it means you interpolate your volatility from existing implied volatilities of the market. The implied volatility is viewed to be a function of the strike price and time to maturity in this case which is what you're interpolating between on your surface.
Everything in the market is a price. Stock price, bond prices, bond options, interest rates, caps, equity options, swaptions, etc. Models are used to try to explain reality.
The market prices of options are absolute amounts and that is what counts. You can express the prices in volatilities (Black vol or others) but that is just a representation of a market price.
