= Black-Scholes implied volatility
{c}
{title2=$\widehat\sigma$}
For a standard <European call option>, the <Black-Scholes implied volatility> is the value $\widehat\sigma$ solving $C_{\rm BS}(\widehat\sigma)=C_{\rm observed}$, with spot, strike, remaining maturity, interest rate and dividend convention held fixed. In the non-dividend model with $S,K,\tau>0$,
$$
\lim_{\sigma\downarrow0}C_{\rm BS}(\sigma)=(S-Ke^{-r\tau})^+,\qquad \lim_{\sigma\to\infty}C_{\rm BS}(\sigma)=S.
$$
Positive <option vega> makes the call price continuous and strictly increasing in $\sigma>0$. Thus there is exactly one finite positive <Black-Scholes implied volatility> when the observed call price lies strictly between these bounds. The lower boundary corresponds to zero volatility; the upper boundary is approached only as volatility tends to infinity. Outside the bounds there is no solution in this model.
Monotonicity makes bracketed <root finding> a direct numerical inversion. This is a model-implied parameter; calls with different strikes can imply different values even though the basic <Black-Scholes model> assumes one constant volatility. For more general claim payoffs, uniqueness requires a separate monotonicity argument.
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