### 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 Jan 1 |

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### ASJC Scopus subject areas

- Modelling and Simulation

### Cite this

*Discrete Dynamics in Nature and Society*,

*2018*, [3093708]. https://doi.org/10.1155/2018/3093708

**Reconstruction of the Time-Dependent Volatility Function Using the Black-Scholes Model.** / Jin, Yuzi; Wang, Jian; Kim, Sangkwon; Heo, Youngjin; Yoo, Changwoo; Kim, Youngrock; Kim, Junseok; Jeong, Darae.

Research output: Contribution to journal › Article

*Discrete Dynamics in Nature and Society*, vol. 2018, 3093708. https://doi.org/10.1155/2018/3093708

}

TY - JOUR

T1 - Reconstruction of the Time-Dependent Volatility Function Using the Black-Scholes Model

AU - Jin, Yuzi

AU - Wang, Jian

AU - Kim, Sangkwon

AU - Heo, Youngjin

AU - Yoo, Changwoo

AU - Kim, Youngrock

AU - Kim, Junseok

AU - Jeong, Darae

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85047598419&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85047598419&partnerID=8YFLogxK

U2 - 10.1155/2018/3093708

DO - 10.1155/2018/3093708

M3 - Article

VL - 2018

JO - Discrete Dynamics in Nature and Society

JF - Discrete Dynamics in Nature and Society

SN - 1026-0226

M1 - 3093708

ER -