Citation: Yimamu, Y.; Deng, Z. Convergence of Inverse Volatility Problem Based on Degenerate Parabolic Equation. Mathematics 2022, 10, 2608. https://doi.org/10.3390/ math10152608 Academic Editor: Jürgen Frikel Received: 12 June 2022 Accepted: 22 July 2022 Published: 26 July 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). mathematics Article Convergence of Inverse Volatility Problem Based on Degenerate Parabolic Equation Yilihamujiang Yimamu and Zuicha Deng * Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China; lucifer − yili@163.com * Correspondence: zc − deng78@hotmail.com Abstract: Based on the theoretical framework of the Black–Scholes model, the convergence of the inverse volatility problem based on the degenerate parabolic equation is studied. Being different from other inverse volatility problems in classical parabolic equations, we introduce some variable substitutions to convert the original problem into an inverse principal coefficient problem in a degenerate parabolic equation on a bounded area, from which an unknown volatility can be recovered and deficiencies caused by artificial truncation can be solved. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established, and a rigorous mathematical proof is given for the convergence of the optimal solution. In the end, the gradient-type iteration method is applied to obtain the numerical solution of the inverse problem, and some numerical experiments are performed. Keywords: inverse volatility problem; degenerate parabolic equation; optimal control framework; existence; convergence; numerical experiments MSC: 35R25; 35R30; 65J20; 65M30 1. Introduction With the continuous development of science and technology, a large number of inverse problems [1–3] have appeared in the fields of weather forecasting, material nondestruc- tive testing, wave inverse scattering, biological imaging and option pricing. The inverse volatility problem has received much attention in recent years due to its significant impacts on the market value of options (see, e.g., [4–11]). In the real market, the volatility of the underlying asset price cannot be directly observed, but the market price of an option can be directly observed and can provide information about the volatility of the underlying asset price. The associated forward operators are based on solutions to the corresponding Black–Scholes/Dupire equations (see [12], and Section 2 of this paper). In this paper, we investigate the following problem: Problem P: In terms of a stock option without paying dividends, it is matter of public knowledge that V(s, t; K, T) for a call option satisfies the following Black–Scholes equation  ∂V ∂t + 1 2 σ 2 (s)s 2 ∂ 2 V ∂s 2 + sµ ∂V ∂s − rV = 0, (s, t) ∈ R + × (0, T), V(s, t)=(s − K) + = max(0, s − K), s ∈ R + . (1) Herein, the parameters s, K are the price of the underlying stock and the strike price. T, µ and r are, respectively, the date of expiry, the risk-neutral drift and the risk-free interest rate, which are assumed to be constants. The parameter σ(s) represents the volatility coefficient to be identified. Given the following additional condition: V(s ∗ , t ∗ , K, T)= V ∗ (K, T), K ∈ R + , (2) Mathematics 2022, 10, 2608. https://doi.org/10.3390/math10152608 https://www.mdpi.com/journal/mathematics