Reconstruction of the Time-Dependent Volatility Function Using the Black–Scholes Model
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...
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| Main Authors: | , , , , , , , |
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| Format: | Article |
| Language: | English |
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Wiley
2018-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2018/3093708 |
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| _version_ | 1832554030561230848 |
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| author | Yuzi Jin Jian Wang Sangkwon Kim Youngjin Heo Changwoo Yoo Youngrock Kim Junseok Kim Darae Jeong |
| author_facet | Yuzi Jin Jian Wang Sangkwon Kim Youngjin Heo Changwoo Yoo Youngrock Kim Junseok Kim Darae Jeong |
| author_sort | Yuzi Jin |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-a0d77ef4de1941efada4771eae67080b |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-a0d77ef4de1941efada4771eae67080b2025-02-03T05:52:32ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2018-01-01201810.1155/2018/30937083093708Reconstruction of the Time-Dependent Volatility Function Using the Black–Scholes ModelYuzi Jin0Jian Wang1Sangkwon Kim2Youngjin Heo3Changwoo Yoo4Youngrock Kim5Junseok Kim6Darae Jeong7Department of Mathematics, Jilin Institute of Chemical Technology, Jilin 132022, ChinaDepartment of Mathematics, Korea University, Seoul 02841, Republic of KoreaDepartment of Mathematics, Korea University, Seoul 02841, Republic of KoreaDepartment of Mathematics, Korea University, Seoul 02841, Republic of KoreaDepartment of Financial Engineering, Korea University, Seoul 02841, Republic of KoreaMajor in Mathematics Education, Hankuk University of Foreign Studies, Seoul 02450, Republic of KoreaDepartment of Mathematics, Korea University, Seoul 02841, Republic of KoreaDepartment of Mathematics, Kangwon National University, Gangwon-do 24341, Republic of KoreaWe 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.http://dx.doi.org/10.1155/2018/3093708 |
| spellingShingle | Yuzi Jin Jian Wang Sangkwon Kim Youngjin Heo Changwoo Yoo Youngrock Kim Junseok Kim Darae Jeong Reconstruction of the Time-Dependent Volatility Function Using the Black–Scholes Model Discrete Dynamics in Nature and Society |
| title | Reconstruction of the Time-Dependent Volatility Function Using the Black–Scholes Model |
| title_full | Reconstruction of the Time-Dependent Volatility Function Using the Black–Scholes Model |
| title_fullStr | Reconstruction of the Time-Dependent Volatility Function Using the Black–Scholes Model |
| title_full_unstemmed | Reconstruction of the Time-Dependent Volatility Function Using the Black–Scholes Model |
| title_short | Reconstruction of the Time-Dependent Volatility Function Using the Black–Scholes Model |
| title_sort | reconstruction of the time dependent volatility function using the black scholes model |
| url | http://dx.doi.org/10.1155/2018/3093708 |
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