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Leftwing asymptotics of the implied volatility in the presence of atoms, available at arXiv:1311.6027
, 2013
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UNIFORM BOUNDS FOR BLACK–SCHOLES IMPLIED VOLATILITY
"... Abstract. The Black–Scholes implied total variance function is defined by VBS(k, c) = v ⇔ Φ ( − k/√v +√v/2) − ekΦ( − k/√v −√v/2) = c. The new formula VBS(k, c) = inf x∈R Φ−1 c + ekΦ(x)) − x]2 is proven. Uniform bounds on the function VBS are deduced and illustrated numerically. As a byproduct of ..."
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Abstract. The Black–Scholes implied total variance function is defined by VBS(k, c) = v ⇔ Φ ( − k/√v +√v/2) − ekΦ( − k/√v −√v/2) = c. The new formula VBS(k, c) = inf x∈R Φ−1 c + ekΦ(x)) − x]2 is proven. Uniform bounds on the function VBS are deduced and illustrated numerically. As a byproduct 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. 1.