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.

在本文中，我们引入了标准化调用函数的概念，并通过双曲正切得到了布莱克和斯科尔斯调用函数的新近似公式。与文献中提出的其他解决方案不同，该公式是可逆的；因此，它对于定价和风险管理以及从报价期权中提取隐含波动率非常有用。后者特别重要，因为它表明标的风险并且它是期权价格的主要组成部分。这就是交易台所关注的。此外，我们估计了建议解决方案的近似误差，并将计算隐含方法的结果与文献中最常用的方法进行比较，并讨论了这种方法的挑战。