 Solution

Implied volatility is the volatility parameter that makes a specified option-pricing model reproduce an observed option price. For a call, it solvesFor positive time to expiry and positive asset and strike prices, the Black-Scholes call price increases with $\sigma$, so an interior arbitrage-consistent price gives a unique finite positive value. It is an inverse model parameter, not the sample standard deviation of past returns or a guaranteed forecast of future realised volatility. Different implied volatilities across strikes or maturities form smiles or surfaces, signalling that one constant volatility does not fit all prices simultaneously.