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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.

Implied volatility is the volatility input of an option-pricing model that reproduces an observed option price. Black-Scholes implied volatility specializes this inversion to the Black-Scholes model.

The vega of an European option is the derivative of its price with respect to spot volatility, holding its other parameters fixed. A dividend-free European call option under the Black-Scholes formula has $\nu =S_0\sqrt{T}\varphi (d_1)>0$. Differentiation proves this because $S_0\varphi (d_1)=K{e}^{-rT}\varphi (d_2)$ and ${\partial }_{\sigma }(d_1-d_2)=\sqrt{T}$. Positivity proves uniqueness of Black-Scholes implied volatility for prices strictly between the limiting call-price bounds.

The Black-Scholes implied volatility of a traded call price $C$ is the volatility $\sigma$ which makes the Black-Scholes formula equal to $C$. Forthe price is a continuous strictly increasing function of $\sigma >0$, by its positive vega, with precisely these endpoint limits. The intermediate value theorem and strict monotonicity therefore give a unique positive implied volatility. The lower endpoint corresponds to the limiting value $\sigma =0$; the upper endpoint is reached only as $\sigma \to \infty$. Prices outside the bounds admit the arbitrages above and have no implied volatility.

For this square-root payoff, the time-zero Black-Scholes model price at volatility $\hat{\sigma }$ isParts (a)(i) and (a)(ii), together with $a\le [M{]}_T\le b$, giveThe exponential is strictly decreasing, so comparison with the defining Black-Scholes price givesThus the Black-Scholes implied volatility lies between the lower and upper realized-variance bounds.