Reconstructing the Local Volatility Surface from Market Option Prices

Soobin Kwak, Youngjin Hwang, Yongho Choi, Jian Wang, Sangkwon Kim and Junseok Kim
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Soobin Kwak: Department of Mathematics, Korea University, Seoul 02841, Korea
Youngjin Hwang: Department of Mathematics, Korea University, Seoul 02841, Korea
Yongho Choi: Department of Computer & Information Engineering (Information Security), Daegu University, Gyeongsan-si 38453, Korea
Jian Wang: School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
Sangkwon Kim: Department of Mathematics, Korea University, Seoul 02841, Korea
Junseok Kim: Department of Mathematics, Korea University, Seoul 02841, Korea

Mathematics, 2022, vol. 10, issue 14, 1-12

Abstract: We present an efficient and accurate computational algorithm for reconstructing a local volatility surface from given market option prices. The local volatility surface is dependent on the values of both the time and underlying asset. We use the generalized Black–Scholes (BS) equation and finite difference method (FDM) to numerically solve the generalized BS equation. We reconstruct the local volatility function, which provides the best fit between the theoretical and market option prices by minimizing a cost function that is a quadratic representation of the difference between the two option prices. This is an inverse problem in which we want to calculate a local volatility function consistent with the observed market prices. To achieve robust computation, we place the sample points of the unknown volatility function in the middle of the expiration dates. We perform various numerical experiments to confirm the simplicity, robustness, and accuracy of the proposed method in reconstructing the local volatility function.

Keywords: local volatility function; Black–Scholes equations; option pricing; finite difference method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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