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

Wanchaloem Wunkaew, Yuqing Liu, Kirill V. Golubnichiy

TL;DR

This work addresses the ill-posed problem of extracting implied volatility from option prices by inverting a Binomial lattice model rather than relying solely on the Black-Scholes PDE. It combines the Newton-Raphson root finder with Automatic Differentiation to efficiently compute $f'(σ)$ for $f(σ)=C_{\text{Binomial}}(σ)-C_{market}$ across a 10-layer lattice, with $r=0$ in generated data and an initial guess $σ_0=0.2$. Empirical results show that Automatic Differentiation reduces the number of iterations required to converge compared to a Secant approach on synthetic data, while real-market data exhibit lower convergence rates (~75%) due to model–data gaps and possible multiple roots. The methodology offers a practical, differentiable pathway to implied volatility estimation in discrete models and can be extended to American options and other asset-pricing frameworks, enabling more robust volatility-based trading signals.

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.

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

TL;DR

This work addresses the ill-posed problem of extracting implied volatility from option prices by inverting a Binomial lattice model rather than relying solely on the Black-Scholes PDE. It combines the Newton-Raphson root finder with Automatic Differentiation to efficiently compute for across a 10-layer lattice, with in generated data and an initial guess . Empirical results show that Automatic Differentiation reduces the number of iterations required to converge compared to a Secant approach on synthetic data, while real-market data exhibit lower convergence rates (~75%) due to model–data gaps and possible multiple roots. The methodology offers a practical, differentiable pathway to implied volatility estimation in discrete models and can be extended to American options and other asset-pricing frameworks, enabling more robust volatility-based trading signals.

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.
Paper Structure (17 sections, 13 equations, 5 figures, 2 tables)

This paper contains 17 sections, 13 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical Model
  3. Binomial Model
  4. Numerical Methods
  5. Newton-Raphson method
  6. Automatic Differentiation
  7. Methodology
  8. Calculating implied volatility inversely via binomial model
  9. Testing
  10. Testing the Numerical Method
  11. Testing with Real-World Data
  12. Numerical Findings
  13. Generated Data
  14. Stock options selected from American markets from 2017- 2018
  15. Stock options selected from American markets from 2020- 2021
  16. ...and 2 more sections

Figures (5)

  • Figure 1: Simulated stock price based on Geometric Brownian motion.
  • Figure 2: An example of simulated option prices generated from the stock prices above and the Binomial model with four different volatilities, a strike price of 150, and a fixed maturity of 180 days.
  • Figure 3: (left) The scatter plot of implied volatilities calculated from the binomial model and their corresponding actual values from the dataset (Generated Data); (right) The implied volatilities distributions (Generated Data).
  • Figure 4: (left) The scatter plot of implied volatilities calculated from the binomial model and their corresponding actual values from the dataset (The 2017-2018 US market); (right) The implied volatilities distributions (The 2017-2018 US market).
  • Figure 5: (left) The scatter plot of implied volatilities calculated from the binomial model and their corresponding actual values from the dataset (The 2020-2021 US market); (right) The implied volatilities distributions (The 2020-2021 US market).