An Exact and Explicit Implied Volatility Inversion Formula

Abstract

In this paper, we develop an exact and explicit (model-independent) Taylor series representation of the Black-Scholes implied volatility in terms of market-observed option prices. Based on an extended Faa di Bruno formula under the operator calculus setting, we first derive the Taylor series of the Black-Scholes formula with respect to the volatility around a pre-chosen positive initial value. Then we apply the Lagrange inversion theorem to explicitly invert the Taylor series to obtain the implied volatility formula. We rigorously establish that our formula converge to the true implied volatility as the truncation order increases, and choose the initial value as the model-independent upper bound of the true implied volatility. Numerical examples illustrate the remarkable accuracy of the formula. Our formula distinguishes from previous literature in that it converges to the true exact implied volatility, is a closed-form formula whose coefficients are explicitly determined and do not involve numerical iterations, and is extremely efficient, which makes it suitable for industrial implementation and adoption.