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A regularized semi-discrete optimal control framework for recovering local volatility surfaces

Year: 2026
Type: article
Abstract: This paper uses standard time-stepping optimal control to construct a supplementary regularized method for recovering $\sigma(S, t)$ in the Black-Scholes model. With Dupire transformation, the volatility calibration problem becomes a coefficient identification problem of non-divergence parabolic equations. Combining Tikhonov regularization with a semi-discrete time stepping to break the two-dimensional inversion problem into a series of one-dimensional subproblems at each time layer, it reduces the inherent ill-posedness of the direct Dupire inversion. We derive forward-adjoint optimality conditions and uniform a priori bounds and prove that each discrete subproblem admits minimizers with existence, local uniqueness, and Lipschitz stability. Our main theoretical result shows that semi-discrete minimizers converge to the weak solution of the continuous regularized implied local volatility equation (RILVE) system when the time step approaches zero. Numerical simulations on synthetic flat, smile, and skew volatility data show that this layered regularized method can produce accurate volatility results and resist observational noise. This paper provides full theoretical deductions and numerical tests to support and improve the existing RILVE volatility inversion model.
DOI: 10.3934/math.20261002
Institution: Tarim University
Language: English
Open Access status: gold
Estimated APC paid: $1,200
OpenAlex ID: W7202344265
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Cites: 23
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Related to: 10