Using the Newton-Raphson method with Automatic Differentiation to numerically solve the implied volatility of stock options via the Binomial model

In the paper by Michael Klibanov et al. [7], a novel method is proposed to calculate the implied volatility of European stock options as a solution to the ill-posed inverse problem for the Black-Scholes equation. Additionally, the paper introduces a trading strategy based on the difference between the implied volatility of the option and the volatility of the underlying stock. Besides the Black-Scholes equation, the Binomial model is another method used to price European options, and the implied volatility can also be calculated using this model. In this paper, we apply the Newton-Raphson method in conjunction with Automatic Differentiation to numerically approximate the implied volatility of a stock option using the Binomial/Lattice model [4]. We provide an explanation of the mathematical model, the methodology, and the results from our tests using simulated data from the Geometric Brownian Motion model and the Binomial model itself, as well as data from the U.S. market from 2020 to 2022.