Recovering the local volatility in Black–Scholes model by numerical differentiation

Abstract

In this article, a numerical method for recovering the local volatility in Black–Scholes model is proposed based on the Dupire formula in which the numerical derivatives are used. By Tikhonov regularization, a new numerical differentiation method in two-dimensional (2-D) case is presented. The convergent analysis and numerical examples are also given. It shows that our method is efficient and stable.

Keywords:

• Black–Scholes model
• Local volatility
• Dupire formula
• Numerical differentiation
• Tikhonov regularization

Keywords:

• AMS Subject Classifications: 65M32
• 65D25

Acknowledgments

This research is partly supported by NSF of China (No. 10271032). The authors would like to thank Prof. T. Wei (Lanzhou University, China) and Mr. Y. B. Wang (Fudan University, China) for sending us their papers [15,16] and the useful comments and suggestions.