= Solution
<Implied volatility> is the volatility parameter that makes a specified option-pricing model reproduce an observed option price. For a call, it solves
$$
\boxed{C_{\rm BS}(S,K,T-t,r;\sigma_{\rm imp})=C_{\rm market}.}
$$
For 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.
Back to article page
