Recovery of Implied Volatility in a Spatial-Fractional Black–Scholes Equation Under a Finite Moment Log Stable Model
In this paper, we study direct and inverse problems for a spatial-fractional Black–Scholes equation with space-dependent volatility. For the direct problem, we provide CN-WSGD (Crank–Nicholson and the weighted and shifted Grünwald difference) scheme to solve the initial boundary value problem. The l...
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MDPI AG
2025-08-01
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| author | Xiaoying Jiang Chunmei Shi Yujie Wei |
| author_facet | Xiaoying Jiang Chunmei Shi Yujie Wei |
| author_sort | Xiaoying Jiang |
| collection | DOAJ |
| description | In this paper, we study direct and inverse problems for a spatial-fractional Black–Scholes equation with space-dependent volatility. For the direct problem, we provide CN-WSGD (Crank–Nicholson and the weighted and shifted Grünwald difference) scheme to solve the initial boundary value problem. The latter aims to recover the implied volatility via observable option prices. Using a linearization technique, we rigorously derive a mathematical formulation of the inverse problem in terms of a Fredholm integral equation of the first kind. Based on an integral equation, an efficient numerical reconstruction algorithm is proposed to recover the coefficient. Numerical results for both problems are provided to illustrate the validity and effectiveness of proposed methods. |
| format | Article |
| id | doaj-art-b9b96057c0a14c8d97f398182c8b71c8 |
| institution | DOAJ |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-b9b96057c0a14c8d97f398182c8b71c82025-08-20T03:02:49ZengMDPI AGMathematics2227-73902025-08-011315248010.3390/math13152480Recovery of Implied Volatility in a Spatial-Fractional Black–Scholes Equation Under a Finite Moment Log Stable ModelXiaoying Jiang0Chunmei Shi1Yujie Wei2School of Mathematics and Computer Science, Zhejiang A&F University, Hangzhou 311300, ChinaSchool of Mathematics and Computer Science, Zhejiang A&F University, Hangzhou 311300, ChinaSchool of Mathematics Sciences, Zhejiang University, Hangzhou 310027, ChinaIn this paper, we study direct and inverse problems for a spatial-fractional Black–Scholes equation with space-dependent volatility. For the direct problem, we provide CN-WSGD (Crank–Nicholson and the weighted and shifted Grünwald difference) scheme to solve the initial boundary value problem. The latter aims to recover the implied volatility via observable option prices. Using a linearization technique, we rigorously derive a mathematical formulation of the inverse problem in terms of a Fredholm integral equation of the first kind. Based on an integral equation, an efficient numerical reconstruction algorithm is proposed to recover the coefficient. Numerical results for both problems are provided to illustrate the validity and effectiveness of proposed methods.https://www.mdpi.com/2227-7390/13/15/2480spatial-fractional Black–Scholes equationCN-WSGD schemelinearization techniquereconstruction algorithm |
| spellingShingle | Xiaoying Jiang Chunmei Shi Yujie Wei Recovery of Implied Volatility in a Spatial-Fractional Black–Scholes Equation Under a Finite Moment Log Stable Model Mathematics spatial-fractional Black–Scholes equation CN-WSGD scheme linearization technique reconstruction algorithm |
| title | Recovery of Implied Volatility in a Spatial-Fractional Black–Scholes Equation Under a Finite Moment Log Stable Model |
| title_full | Recovery of Implied Volatility in a Spatial-Fractional Black–Scholes Equation Under a Finite Moment Log Stable Model |
| title_fullStr | Recovery of Implied Volatility in a Spatial-Fractional Black–Scholes Equation Under a Finite Moment Log Stable Model |
| title_full_unstemmed | Recovery of Implied Volatility in a Spatial-Fractional Black–Scholes Equation Under a Finite Moment Log Stable Model |
| title_short | Recovery of Implied Volatility in a Spatial-Fractional Black–Scholes Equation Under a Finite Moment Log Stable Model |
| title_sort | recovery of implied volatility in a spatial fractional black scholes equation under a finite moment log stable model |
| topic | spatial-fractional Black–Scholes equation CN-WSGD scheme linearization technique reconstruction algorithm |
| url | https://www.mdpi.com/2227-7390/13/15/2480 |
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