Implementing an Optimized Neural Network to Price American Option under Uncertain Volatility Model with Stochastic Bounds

Document Type : Research Article

Authors

1 Ibn Zohr University, Agadir, Morocco.

2 High School of Technologie, Ibn Zohr University, Dakhla, Morocco.

10.22054/jmmf.2026.90314.1261
Abstract
The purpose of this paper is to investigate the asymptotic behavior of the price of an American-style put option under an uncertain volatility model that takes into account the risk associated with market volatility. We propose a novel representation for the uncertain volatility model and examine its convergence. An approximation method for the pricing problem is developed by adding conditions to the Black-Scholes-Barenblatt equation and splitting it into two equations. To improve the accuracy of our results, we employ an optimized Neural Network approach that combines finite difference and radial basis function network.

Keywords


[1] L. Andersen, and J. Andreasen, Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing, Review of Derivatives Research, (2000) vol. 4 no. 3 pp. 231–262.
[2] M. Avellaneda, A. Levy, and A. Par´as, Pricing and hedging derivative securities in markets with uncertain volatilities, Applied Mathematical Finance, (1995) vol. 2 no. 2 pp. 73–88.
[3] G. Barone-Adesi, and R. E. Whaley, Efficient analytic approximation of American option values, The Journal of Finance, (1987) vol. 42 no. 2 pp. 301–320.
[4] F. Black, and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, (1973) vol. 81 no. 3 pp. 637–654.
[5] J. Doran, The influence of tracking error on volatility premium estimation, Journal of Risk, (2007) vol. 9 no. 3.
[6] Z. El Kharrazi, S. Saoud, and Z. Mahani, Pricing American Put Option Using RBF-NN: New Simulation of Black–Scholes, Moroccan Journal of Pure and Applied Analysis, (2022) vol. 8 no. 1 pp. 78–91.
[7] J.-P. Fouque, G. Papanicolaou, R. Sircar, and K. Sølna, Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives, Cambridge University Press, (2011).
[8] J.-P. Fouque, and B. Ren, Approximation for option prices under uncertain volatility, SIAM Journal on Financial Mathematics, (2014) vol. 5 no. 1 pp. 360–383.
[9] J.-P. Fouque, and N. Ning, Uncertain volatility models with stochastic bounds, SIAM Journal on Financial Mathematics, (2018) vol. 9 no. 4 pp. 1175–1207.
[10] M.-H. Giga, Y. Giga, and J. Saal, Nonlinear Partial Differential Equations: Asymptotic Behavior of Solutions and Self-Similar Solutions, Springer Science & Business Media, (2010).
[11] S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies, (1993) vol. 6 no. 2 pp. 327–343.
[12] J. Hull, and A. White, The pricing of options on assets with stochastic volatilities, The Journal of Finance, (1987) vol. 42 no. 2 pp. 281–300.
[13] S. Ikonen, and J. Toivanen, Operator splitting methods for pricing American options under stochastic volatility, Numerische Mathematik, (2009) vol. 113 no. 2 pp. 299–324.
[14] T. J. Lyons, Uncertain volatility and the risk-free synthesis of derivatives, Applied Mathematical Finance, (1995) vol. 2 no. 2 pp. 117–133.
[15] R. C. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, (1976) vol. 3 no. 1–2 pp. 125–144.
[16] F. Mehrdoust, I. Noorani, and A. Hamdi, Two-factor Heston model equipped with regimeswitching: American option pricing and model calibration by Levenberg–Marquardt optimization algorithm, Mathematics and Computers in Simulation, (2023) vol. 204 pp. 660–678.
[17] Z. Mezdoud, C. Hartmann, M. R. Remita, and O. Kebiri, α-Hypergeometric Uncertain Volatility Models and their Connection to 2BSDEs, arXiv preprint arXiv:2108.06965, (2021).
[18] C. Nwankwo, N. Umeorah, T. Ware, and W. Dai, Deep learning and American options via free boundary framework, arXiv preprint arXiv:2211.11803, (2022).
[19] P. Wilmott, Derivatives: The Theory and Practice of Financial Engineering, Wiley, (1998).
[20] J. Yong, and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer Science & Business Media, (1999) vol. 43.
[21] Q. Zhou, and X. Li, Vulnerable options pricing under uncertain volatility model, Journal of Inequalities and Applications, (2019) vol. 2019 no. 1 pp. 1–16.
[22] R. Zvan, P. A. Forsyth, and K. R. Vetzal, Penalty methods for American options with stochastic volatility, Journal of Computational and Applied Mathematics, (1998) vol. 91 no. 2 pp. 199–218.

Articles in Press, Accepted Manuscript
Available Online from 02 September 2026

  • Receive Date 08 February 2026
  • Revise Date 30 August 2026
  • Accept Date 31 August 2026