Challenges in approximating the Black and Scholes call formula with hyperbolic tangents

TL;DR

This paper proposes a new hyperbolic tangent-based approximation for the Black-Scholes call formula, aiming to improve pricing, risk management, and implied volatility extraction, while analyzing its accuracy and challenges.

Contribution

Introduces a novel hyperbolic tangent approximation for the Black-Scholes call function, enhancing computational methods for options pricing and implied volatility estimation.

Findings

The new approximation effectively estimates the call price.

Numerical analysis shows the approximation's error margins.

Comparison indicates challenges in accuracy and practical implementation.

Abstract

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. This formula 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. 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 literature we discuss the challenges of this approach.