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

Yuzi Jin, Jian Wang, Sangkwon Kim, Youngjin Heo, Changwoo Yoo, Youngrock Kim, Junseok Kim, Darae Jeong

Research output: Contribution to journalArticle

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 languageEnglish
Article number3093708
JournalDiscrete Dynamics in Nature and Society
Volume2018
DOIs
Publication statusPublished - 2018 Jan 1

Fingerprint

Black-Scholes Model
Volatility
Cost functions
Cost Function
Steepest descent method
Black-Scholes Equation
Steepest Descent Method
Predictor-corrector
Robust Algorithm
Reconstruction Algorithm
Finite difference method
Piecewise Linear
Numerical Algorithms
Difference Method
Resolve
Finite Difference
Minimise
Numerical Results

ASJC Scopus subject areas

  • Modelling and Simulation

Cite this

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.

In: Discrete Dynamics in Nature and Society, Vol. 2018, 3093708, 01.01.2018.

Research output: Contribution to journalArticle

Jin, Yuzi ; Wang, Jian ; Kim, Sangkwon ; Heo, Youngjin ; Yoo, Changwoo ; Kim, Youngrock ; Kim, Junseok ; Jeong, Darae. / Reconstruction of the Time-Dependent Volatility Function Using the Black-Scholes Model. In: Discrete Dynamics in Nature and Society. 2018 ; Vol. 2018.
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