Reconstruction of the local volatility function using the Black–Scholes model

Sangkwon Kim, Hyunsoo Han, Hanbyeol Jang, Darae Jeong, Chaeyoung Lee, Wonjin Lee, Junseok Kim

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)


In this paper, we propose a robust and accurate numerical algorithm to reconstruct a local volatility function using the Black–Scholes (BS) partial differential equation (PDE). Using the BS PDE and given market data, option prices at strike prices and expiry times, a time-dependent local volatility function is computed. The proposed algorithm consists of the following steps: (1) The time-dependent volatility function is computed using a recently developed method; (2) A Monte Carlo simulation technique is used to find the effective region which has a strong influence on option prices; and we partition the effective domain into several sub-regions and define a local volatility function based on the time-dependent volatility function on the sub-regions; and (3) Finally, we calibrate the local volatility function using the fully implicit finite difference method and the conjugate gradient optimization algorithm. We demonstrate the robustness and accuracy of the proposed local volatility reconstruction algorithm using manufactured volatility surface and real market data.

Original languageEnglish
Article number101341
JournalJournal of Computational Science
Publication statusPublished - 2021 Apr

Bibliographical note

Funding Information:
The authors greatly appreciate the reviewers for their constructive comments and suggestions, which have significantly improved the quality of this paper. The corresponding author (J.S. Kim) was supported by the National Research Foundation (NRF), Korea, under project BK21 FOUR.

Publisher Copyright:
© 2021 Elsevier B.V.


  • Black–Scholes equation
  • Finite difference method
  • Local volatility
  • Monte Carlo simulation

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science
  • Modelling and Simulation


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