Convergence of Inverse Volatility Problem Based on Degenerate Parabolic Equation
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...
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| Format: | Article |
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MDPI AG
2022-07-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/15/2608 |
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| author | Yilihamujiang Yimamu Zuicha Deng |
| author_facet | Yilihamujiang Yimamu Zuicha Deng |
| author_sort | Yilihamujiang Yimamu |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-09T05:13:05Z |
| format | Article |
| id | doaj.art-7934e82a3f614f909b7296ff6891c541 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T05:13:05Z |
| publishDate | 2022-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-7934e82a3f614f909b7296ff6891c5412023-12-03T12:47:27ZengMDPI AGMathematics2227-73902022-07-011015260810.3390/math10152608Convergence of Inverse Volatility Problem Based on Degenerate Parabolic EquationYilihamujiang Yimamu0Zuicha Deng1Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, ChinaDepartment of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, ChinaBased 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.https://www.mdpi.com/2227-7390/10/15/2608inverse volatility problemdegenerate parabolic equationoptimal control frameworkexistenceconvergencenumerical experiments |
| spellingShingle | Yilihamujiang Yimamu Zuicha Deng Convergence of Inverse Volatility Problem Based on Degenerate Parabolic Equation Mathematics inverse volatility problem degenerate parabolic equation optimal control framework existence convergence numerical experiments |
| title | Convergence of Inverse Volatility Problem Based on Degenerate Parabolic Equation |
| title_full | Convergence of Inverse Volatility Problem Based on Degenerate Parabolic Equation |
| title_fullStr | Convergence of Inverse Volatility Problem Based on Degenerate Parabolic Equation |
| title_full_unstemmed | Convergence of Inverse Volatility Problem Based on Degenerate Parabolic Equation |
| title_short | Convergence of Inverse Volatility Problem Based on Degenerate Parabolic Equation |
| title_sort | convergence of inverse volatility problem based on degenerate parabolic equation |
| topic | inverse volatility problem degenerate parabolic equation optimal control framework existence convergence numerical experiments |
| url | https://www.mdpi.com/2227-7390/10/15/2608 |
| work_keys_str_mv | AT yilihamujiangyimamu convergenceofinversevolatilityproblembasedondegenerateparabolicequation AT zuichadeng convergenceofinversevolatilityproblembasedondegenerateparabolicequation |