= Square-root stock implied volatility
{title2=$\Sigma(t,T)^2=-\frac8{T-t}\log\frac{C(t,T)}{\sqrt{S_t}}$}
For a <square-root stock claim> priced by $C(t,T)=\mathbb E[\sqrt{S_T}\mid\mathcal F_t]$, the constant-volatility <geometric Brownian motion> price is $\sqrt{S_t}\exp(-\sigma^2(T-t)/8)$. If $t<T$, $S_t>0$ and $0<C(t,T)\leq\sqrt{S_t}$, its unique nonnegative inverse volatility is the displayed $\Sigma(t,T)$. For bounded predictable volatility in $dS=S\sigma\,dW$, bounds $0\leq a\leq\sigma\leq b$ give $a\leq\Sigma\leq b$, without independence from the <Brownian motion>. At zero stock price the claim price is also zero and no volatility is identified by this equation; a separate convention is needed.
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