Using the Newton-Raphson Method with Automatic Differentiation to Numerically Solve Implied Volatility of Stock Option through Binomial Model

TL;DR

This paper introduces a method combining Newton-Raphson and Automatic Differentiation to efficiently compute implied volatility of stock options using the Binomial Model, validated with simulated and real market data.

Contribution

It presents a novel application of Newton-Raphson with Automatic Differentiation for implied volatility calculation within the Binomial Model, enhancing numerical accuracy and efficiency.

Findings

Effective approximation of implied volatility using the proposed method.

Validation with simulated data shows high accuracy.

Application to real market data demonstrates practical utility.

Abstract

In the paper written by Klibanov et al, it proposes a novel method to calculate implied volatility of a European stock options as a solution to ill-posed inverse problem for the Black-Scholes equation. In addition, it proposes a trading strategy based on the difference between implied volatility of the option and the volatility of the underlying stock. In addition to the Black-Scholes equation, Binomial model is another method used to price European options. And, the implied volatility can be also calculated through this model. In this paper, we apply the Newton-Raphson method together with Automatic Differention to numerically approximate the implied volatility of an arbitrary stock option through this model. We provide an explanation of the mathematical model and methods, the methodology, and the results from our test using the stimulated data from the Geometric Brownian Motion Model and the Binomial Model itself, and the data from the US market data from 2018 to 2021.