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

Yun Liu; Lifeng Guo; Xijuan Liu; Yun Liu; Lifeng Guo; Xijuan Liu

doi:10.3934/math.20261002

AIMS Mathematics

2026, Volume 11, Issue 8: 24928-24957. doi: 10.3934/math.20261002

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Research article

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

1.
College of Information Engineering, Tarim University, Alar, Xinjiang 843300, China
2.
Key Laboratory of Tarim Oasis Agriculture (Tarim University) Ministry of Education, Alar, Xinjiang 843300, China

Received: 07 April 2026 Revised: 27 July 2026 Accepted: 06 August 2026 Published: 13 August 2026
MSC : 35R30, 49J20, 91G20

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.
- option pricing,
- implied volatility,
- parabolic coefficient inverse problem,
- optimal control,
- existence,
- Lipschitz stability
Citation: Yun Liu, Lifeng Guo, Xijuan Liu. A regularized semi-discrete optimal control framework for recovering local volatility surfaces[J]. AIMS Mathematics, 2026, 11(8): 24928-24957. doi: 10.3934/math.20261002

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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.

References

[1]	F. Black, M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ, 81 (1973), 637–654. https://doi.org/10.1086/260062 doi: 10.1086/260062
[2]	L. S. Jiang, C. Xu, X. Ren, S. Li, Mathematical model and case analysis of financial derivatives pricing, Higher Educ. Press, 2008.
[3]	V. Isakov, Recovery of time dependent volatility coefficient by linearization, Evol. Equ. Control. The, 3 (2014), 119–134. https://doi.org/10.3934/eect.2014.3.119 doi: 10.3934/eect.2014.3.119
[4]	Q. Chen, Recovery of local volatility for financial assets with mean-reverting price processes, Math. Control. Relat. F., 8 (2018), 625–635. https://doi.org/10.3934/mcrf.2018026 doi: 10.3934/mcrf.2018026
[5]	Z. C. Deng, Y. C. Hon, V. Isakov, Recovery of time-dependent volatility in option pricing model, Inverse Probl., 32 (2016), 115010. https://doi.org/10.1088/0266-5611/32/11/115010 doi: 10.1088/0266-5611/32/11/115010
[6]	B. Dupire, Pricing with a smile, Risk, 7 (1994), 18–20.
[7]	R. Cont, J. da Fonseca, Dynamics of implied volatility surfaces, Quant. Financ, 2 (2002), 45–60. https://doi.org/10.1088/1469-7688/2/1/304 doi: 10.1088/1469-7688/2/1/304
[8]	S. Crpey, Calibration of the local volatility in a generalized Black-Scholes model using Tikhonov regularization, SIAM. J. Math. Anal, 34 (2003), 1183–1206. https://doi.org/10.1137/S0036141001400202 doi: 10.1137/S0036141001400202
[9]	L. Yang, L. Yin, Z. Deng, Drift coefficient inversion problem of Kolmogorov-type equation, AIMS Math., 6 (2021), 3432–3454. https://doi.org/10.3934/math.2021205 doi: 10.3934/math.2021205
[10]	B. Nabubie, S. Wang, Numerical techniques for determining implied volatility in option pricing, J. Comput. Appl. Math., 422 (2023), 114913. https://doi.org/10.1016/j.cam.2022.114913 doi: 10.1016/j.cam.2022.114913
[11]	X. Y. Jiang, X. Xu, On implied volatility recovery of a time-fractional Black-Scholes equation for double barrier options, Appl. Anal., 100 (2021), 3145–3160. https://doi.org/10.1080/00036811.2020.1712369 doi: 10.1080/00036811.2020.1712369
[12]	I. Bouchouev, V. Isakov, The inverse problem of option pricing, Inverse Probl., 13 (1997), L11–L17. https://doi.org/10.1088/0266-5611/13/5/001 doi: 10.1088/0266-5611/13/5/001
[13]	I. Bouchouev, V. Isakov, Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets, Inverse Probl., 15 (1999), 95–116. https://doi.org/10.1088/0266-5611/15/3/201 doi: 10.1088/0266-5611/15/3/201
[14]	I. Bouchouev, V. Isakov, N. Valdivia, Recovery of volatility coefficient by linearization, Quant. Financ., 2 (2002), 257–263. https://doi.org/10.1088/1469-7688/2/4/302 doi: 10.1088/1469-7688/2/4/302
[15]	L. S. Jiang, Y. S. Tao, Identifying the volatility of underlying assets from option prices, Inverse Probl., 17 (2001), 137. https://doi.org/10.1088/0266-5611/17/1/311 doi: 10.1088/0266-5611/17/1/311
[16]	L. Jiang, Q. Chen, L. Wang, J. Zhang, A new well-posed algorithm to recover implied local volatility, Quant. Financ., 3 (2003), 451–457. https://doi.org/10.1088/1469-7688/3/6/304 doi: 10.1088/1469-7688/3/6/304
[17]	Z. C. Deng, J. N. Yu, L. Yang, Optimization method for an evolutional type inverse heat conduction problem, J. Phys. A-Math. Theor., 41 (2008), 035201. https://doi.org/10.1088/1751-8113/41/3/035201 doi: 10.1088/1751-8113/41/3/035201
[18]	L. S. Jiang, B. J. Bian, The regularized implied local volatility equations-a new model to recover the volatility of underlying asset from observed market option price, Discrete. Cont. Dyn.-B, 17 (2012), 2017–2046. https://doi.org/10.3934/dcdsb.2012.17.2017 doi: 10.3934/dcdsb.2012.17.2017
[19]	H. Egger, H. W. Engl, Tikhonov regularization applied to the inverse problem of option pricing: Convergence analysis and rates, Inverse Probl. 21 (2005), 1027–1045. https://doi.org/10.1088/0266-5611/21/3/014 doi: 10.1088/0266-5611/21/3/014
[20]	A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, 1964.
[21]	Y. Yimamu, Z. C. Deng, C. N. Sam, Y. C. Hon, Total variation regularization analysis for inverse volatility option pricing problem, Int. J. Comput. Math., 101 (2024). https://doi.org/10.1080/00207160.2024.2345660 doi: 10.1080/00207160.2024.2345660
[22]	W. Gong, Y. Sun, L. Yang, An inverse problem for calibrating the volatility in time fractional jump-diffusion option pricing model, J. Comput. Appl. Math., 489 (2027), 117884. https://doi.org/10.1016/j.cam.2026.117884 doi: 10.1016/j.cam.2026.117884
[23]	W. Gong, Z. Xu, Reconstruction of local volatility surface from american options, J. Inverse Ill-Posed Probl., 31 (2023), 91–102. https://doi.org/10.1515/jiip-2019-0085 doi: 10.1515/jiip-2019-0085
[24]	L. C. Evans, Partial Differential Equations, American Mathematical Society, 2022.
[25]	V. Isakov, Inverse Problems for Partial Differential Equations, Springer, 2017.
[26]	A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems, WILEY, 1977.
[27]	H. W. Engl, R. Ramlau, Regularization of Inverse Problems, Springer Berlin Heidelberg, 2015. https://doi.org/10.1007/978-3-540-70529-1_52
[28]	J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, 1971. https://doi.org/10.1007/978-3-642-65024-6

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