 Black-Scholes implied volatility

For a standard European call option, the Black-Scholes implied volatility is the value $\hat{\sigma }$ solving $C_{\mathrm{BS}}(\hat{\sigma })=C_{\mathrm{observed}}$, with spot, strike, remaining maturity, interest rate and dividend convention held fixed. In the non-dividend model with $S,K,\tau >0$,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.