Approximate inversion of the Black–Scholes formula using rational functions

The Black–Scholes formula is often used in the backward direction to invert the implied volatility, usually with some solver method. Solver methods, being aesthetically unappealing, are also slower than closed-form approximations. However, closed-form approximations in previous works lack accuracy, often providing option pricing errors well exceeding the bid–ask spreads. We develop a new closed-form method based on the rational approximation. The rational approximation is much faster than typical solver methods and very accurate for both at-the-money and away-from-the-money options. Its accuracy can be further improved by one or two steps of Newton–Raphson iterations.