An inverse volatility problem in a degenerate parabolic equation in a bounded domain

Yilihamujiang Yimamu; Zui-Cha Deng; Liu Yang; Yilihamujiang Yimamu; Zui-Cha Deng; Liu Yang

doi:10.3934/math.20221056

AIMS Mathematics

2022, Volume 7, Issue 10: 19237-19266. doi: 10.3934/math.20221056

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Research article

An inverse volatility problem in a degenerate parabolic equation in a bounded domain

1.
Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, People's Republic of China
2.
Institute of Engineering Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, People's Republic of China

Received: 08 May 2022 Accepted: 20 July 2022 Published: 30 August 2022
MSC : 5R30, 49J20

In this paper, an inverse problem of determining the space-dependent volatility from the observed market prices of options with different strikes is studied. Being different from other inverse volatility problem with classical parabolic equations, we apply the linearization method and introduce some variable substitutions to convert the original problem into an inverse source problem in a degenerate parabolic equation in 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. After the necessary conditions are deduced, the uniqueness and stability of the minimizer are proved. Then, the Landweber iterative method is used to obtain a stable numerical solution of the inverse problem and some numerical experiments are also performed. The numerical results show that the algorithm which we proposed is robust and the unknown coefficient is recovered quite well.
- inverse volatility problem,
- Linearization and variable substitutions,
- degenerate parabolic equation,
- optimal control,
- Landweber iteration
Citation: Yilihamujiang Yimamu, Zui-Cha Deng, Liu Yang. An inverse volatility problem in a degenerate parabolic equation in a bounded domain[J]. AIMS Mathematics, 2022, 7(10): 19237-19266. doi: 10.3934/math.20221056

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Abstract

In this paper, an inverse problem of determining the space-dependent volatility from the observed market prices of options with different strikes is studied. Being different from other inverse volatility problem with classical parabolic equations, we apply the linearization method and introduce some variable substitutions to convert the original problem into an inverse source problem in a degenerate parabolic equation in 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. After the necessary conditions are deduced, the uniqueness and stability of the minimizer are proved. Then, the Landweber iterative method is used to obtain a stable numerical solution of the inverse problem and some numerical experiments are also performed. The numerical results show that the algorithm which we proposed is robust and the unknown coefficient is recovered quite well.

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