## Abstract

We propose a simple and robust numerical algorithm to estimate a time-dependent volatility function from a set of market observations, using the Black-Scholes (BS) model. We employ a fully implicit finite difference method to solve the BS equation numerically. To define the time-dependent volatility function, we define a cost function that is the sum of the squared errors between the market values and the theoretical values obtained by the BS model using the time-dependent volatility function. To minimize the cost function, we employ the steepest descent method. However, in general, volatility functions for minimizing the cost function are nonunique. To resolve this problem, we propose a predictor-corrector technique. As the first step, we construct the volatility function as a constant. Then, in the next step, our algorithm follows the prediction step and correction step at half-backward time level. The constructed volatility function is continuous and piecewise linear with respect to the time variable. We demonstrate the ability of the proposed algorithm to reconstruct time-dependent volatility functions using manufactured volatility functions. We also present some numerical results for real market data using the proposed volatility function reconstruction algorithm.

Original language | English |
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Article number | 3093708 |

Journal | Discrete Dynamics in Nature and Society |

Volume | 2018 |

DOIs | |

Publication status | Published - 2018 |

### Bibliographical note

Funding Information:was supported by the Basic Science Research Program of the NRF (Korea) under Grant no. 2015R1D1A1A01059643/2.

Funding Information:

The first author (Yuzi Jin) was supported by the University-Level Major Social Science Project of Jilin Institute of Chemical Technology, no. 20, 2016. The author (Youngrock Kim)

Publisher Copyright:

© 2018 Yuzi Jin et al.

## ASJC Scopus subject areas

- Modelling and Simulation