Calibration of the temporally varying volatility and interest rate functions

Abstract

In this study, we develop a calibration method of the temporally varying volatility and interest rate functions using the Black–Scholes (BS) partial differential equation and the observed market option prices with different strikes and expiries. The proposed method uses the piecewise linear interpolations between data points which are defined at the middle points of maturity dates. When we construct the volatility and interest rate, we use the exponential function so that the interpolated values are always positive. Numerical experiments with synthetic and real market data demonstrate the superior performance of the proposed method.

Keywords:

• Black–Scholes equation
• volatility
• interest rate
• calibration
• option price

2010 Mathematics Subject Classification:

• finance

Acknowledgments

The corresponding author (J.S. Kim) was supported by the Brain Korea 21 FOUR through the National Research Foundation of Korea funded by the Ministry of Education of Korea. The authors thank the reviewers for their constructive and helpful comments on the revision of this article.

Disclosure statement

No potential conflict of interest was reported by the author(s).