  This note improves the lower and upper bounds of the Black-Scholes implied
volatility (IV) in Tehranchi (SIAM J. Financial Math., 7 (2016), p. 893). The
proposed tighter bounds are systematically based on the bounds of the option
delta. While Tehranchi used the bounds to prove IV asymptotics, we apply the
result to the accurate numerical root-finding of IV. We alternatively formulate
the Newton-Raphson method on the log price and demonstrate that the iteration
always converges rapidly for all price ranges if the new lower bound found in
this study is used as an initial guess.
