A generalized derivation of the Black-Scholes implied volatility through hyperbolic tangents

Argumenta Oeconomica. 2022. No. 2(49). P. 23–57.

Mininni M., Orlando G., Taglialatela G.

This article extends the previous research on the notion of a standardized call function and how to
obtain an approximate model of the Black-Scholes formula via the hyperbolic tangent. Although the
Black-Scholes approach is outdated and suffers from many limitations, it is still widely used to derive
the implied volatility of options. This is particularly important for traders because it represents the risk
of the underlying, and is the main factor in the option price. The approximation error of the suggested
solution was estimated and the results compared with the most common methods available in the
literature. A new formula was provided to correct some cases of underestimation of implied volatility.
Graphic evidence, stress tests and Monte Carlo analysis confirm the quality of the results obtained.
Finally, further literature is provided as to why implied volatility is used in decision making.

Keywords: financial derivatives, Monte Carlo, options implied volatility, Black and Scholes option formula