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

Wanchaloem Wunkaew; Yuqing Liu; Kirill V. Golubnichiy

doi:10.3934/cac.2025012

Article Contents

September 2025, Volume 5, 43-52. Doi: 10.3934/cac.2025012

This volume Previous Article $ L^{\infty}(H_{{{\rm{div}}}}) \times W^{1,\infty}(L^2) $-norm superconvergence of mixed finite element schemes applied to multidimensional parabolic equations Next Article Pressure-robust Nitsche's method for Stokes problem with slip boundary conditions

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

1.
Department of Applied Mathematics and Statistics, Johns Hopkins University, MD 21218, USA
2.
Department of Mathematics, University of Washington, Seattle, WA 98195, USA
3.
Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA

^*Corresponding author: Kirill V. Golubnichiy
^*Corresponding author: Kirill V. Golubnichiy

Received: October 2024

Revised: February 2025

Published online: August 4, 2025

Abstract / Introduction Full Text(HTML) Figure(5) / Table(2) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 91G60, 60H15, 35R30.

Citation:

FullText(HTML)

Figure 1. Simulated stock price based on Geometric Brownian motion.

Download: Full-size image PowerPoint slide

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.

Download: Full-size image PowerPoint slide

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).

Download: Full-size image PowerPoint slide

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).

Download: Full-size image PowerPoint slide

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).

Download: Full-size image PowerPoint slide

Table 1. Generated Data (of total 1350 data).

	Secant	Autodiff
Number of Convergences	1322	1308
Average Number of Iterations	3.670	3.352

| Show Table

DownLoad: CSV

Table 2. Real Data (of total 1917 data).

	Secant	Autodiff
Number of Convergences	1431	1425
Average Number of Iterations	6.304	4.700

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	A. G. Baydin, B. A. Pearlmutter, A. A. Radul and J. M. Siskind, Automatic differentiation in machine learning: A survey, Journal of Machine Learning Research, 18 (2017), 1-43.
[2]	F. Black and M. Scholes, The pricing of options and corporate liabilities, World Scientific Reference on Contingent Claims Analysis in Corporate Finance: Foundations of CCA and Equity Valuation, 1 (2019), 3-21. doi: 10.1142/9789814759588_0001.
[3]	R. M. Corless and N. Fillion, A Graduate Introduction to Numerical Methods, Springer, New York, 2013.
[4]	J. C. Cox, S. A. Ross and M. Rubinstein, Option pricing: A simplified approach, Journal of financial Economics, 7 (1979), 229-263. doi: 10.1016/0304-405X(79)90015-1.
[5]	J. Hull, Options, Futures and Other Derivatives, 9$^{th}$ edition, Prentice Hall, Upper Saddle River 2014.
[6]	Y. S. Kim, S. Stoyanov, S. Rachev and F. Fabozzi, Multi-purpose binomial model: Fitting all moments to the underlying geometric Brownian motion, Economics Letters, 145 (2016), 225-229. doi: 10.1016/j.econlet.2016.05.035.
[7]	M. V. Klibanov, A. A. Shananin, K. V. Golubnichiy and S. M. Kravchenko, Forecasting Stock Options Prices via the Solution of an Ill-Posed Problem for the Black-Scholes Equation. arXiv: 2202.07174, Inverse Problems, 38 (2022).
[8]	A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, Š. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman and A. Scopatz, SymPy: Symbolic Computing in Python, PeerJ Computer Science, 3, e103, 2017.
[9]	L. B. Rall and G. F. Corliss, An introduction to automatic differentiation. computational differentiation: Techniques, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 89 (1996), 1-18.
[10]	S. M. Ross, An Elementary Introduction to Mathematical Finance, Cambridge University Press, 2011.