Challenges in Approximating the Black and Scholes Call Formula with Hyperbolic Tangents

In this paper, we introduce the concept of standardized call function and we obtain a new approximating formula for the Black and Scholes call function through the hyperbolic tangent. Differently from other solutions proposed in the literature, this formula is invertible; hence, it is useful for pricing and risk management as well as for extracting the implied volatility from quoted options. The latter is of particular importance since it indicates the risk of the underlying and it is the main component of the option's price. That is what trading desks focus on. Further we estimate numerically the approximating error of the suggested solution and, by comparing our results in computing the implied volatility with the most common methods available in the literature, we discuss the challenges of this approach