Total variation regularization analysis for inverse volatility option pricing problem

Abstract

We investigate in this paper an inverse problem of recovering the space-dependent volatility in option pricing. To enhance precision across the domain, we transform the original problem into an inverse source problem of a bounded degenerate parabolic equation, utilizing linearization and variable substitution. Unlike classical methods, we apply a total variation regularization combined with a novel generalized finite integration technique. This approach accommodates volatility jumps in an overnight rate scenario. Leveraging an optimal control framework, we demonstrate that the inverse problem can be reformulated as an optimal control problem whose existence, necessary conditions, local uniqueness and stability for the minimizer of control functional are obtained. For numerical verification, we derive the Euler equation and design a discretization algorithm with generalized finite integration technique. Numerical examples showcase the robustness of our approach, highlighting advantages in accuracy and effectiveness over other strategies.

Keywords:

• Inverse volatility problem
• degenerate parabolic equation
• total variation regularization
• generalized finite integration method
• discretization algorithm

(2000) AMS Subject Classifications:

• 35R30
• 49J20

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partially supported by a grant from the Research Grant Council of the Hong Kong Special Administrative Region (GRF CityU No. 11305321) and grants from NNSF of China (Nos. 61663018, 11461039, 11961042), Foundation of A Hundred Youth Talents Training Program of Lanzhou Jiaotong University (2011028) and NSF of Gansu Province of China (No. 18JR3RA122).