Uniform Bounds for Black-Scholes Implied Volatility

The Black–Scholes implied total variance function is defined by VBS(k, c) = v ⇔ Φ ( − k/ √ v + √ v/2 ) − eΦ ( − k/ √ v − √ v/2 ) = c. The new formula VBS(k, c) = inf x∈R [ Φ−1 ( c + eΦ(x) ) − x ]2 is proven. Uniform bounds on the function VBS are deduced and illustrated numerically. As a by-product of this analysis, it is proven that F is the distribution function of a logconcave probability measure if and only if F (F−1(·) + b) is concave for all b ≥ 0. From this, an interesting class of peacocks is constructed.