Efficiency of Some Predictor–Corrector Methods with Fourth-Order Compact Scheme for a System of Free Boundary Options
The trade-off between numerical accuracy and computational cost is always an important factor to consider when pricing options numerically, due to the inherent irregularity and existence of non-linearity in many models. In this work, we first present fast and accurate (1,2) and (2,2) predictor–corre...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-08-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/12/8/762 |
| _version_ | 1797585561128534016 |
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| author | Chinonso Nwankwo Weizhong Dai |
| author_facet | Chinonso Nwankwo Weizhong Dai |
| author_sort | Chinonso Nwankwo |
| collection | DOAJ |
| description | The trade-off between numerical accuracy and computational cost is always an important factor to consider when pricing options numerically, due to the inherent irregularity and existence of non-linearity in many models. In this work, we first present fast and accurate (1,2) and (2,2) predictor–corrector methods with a fourth-order compact finite difference scheme for pricing coupled system of the non-linear free boundary option pricing problem consisting of the option value and delta sensitivity. To predict the optimal exercise boundary, we set up a high-order boundary scheme, which is strategically derived using a combination of the fourth-order Robin boundary scheme and the fourth-order compact finite difference scheme near boundary. Furthermore, we implement a three-step high-order correction scheme for computing interior values of the option value and delta sensitivity. The discrete matrix system of this correction scheme has a tri-diagonal structure and strictly diagonal dominance. This nice feature allows for the implementation of the Thomas algorithm, thereby enabling fast computation. The optimal exercise boundary value is also corrected in each of the three correction steps with the derived Robin boundary scheme. Our implementations are fast on both coarse and very refined grids and provide highly accurate numerical approximations. Moreover, we recover a reasonable convergence rate. Further extensions to high-order predictor two-step corrector schemes are elaborated. |
| first_indexed | 2024-03-11T00:07:53Z |
| format | Article |
| id | doaj.art-ce917358f42842a38062402d24fe875e |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-11T00:07:53Z |
| publishDate | 2023-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-ce917358f42842a38062402d24fe875e2023-11-19T00:14:52ZengMDPI AGAxioms2075-16802023-08-0112876210.3390/axioms12080762Efficiency of Some Predictor–Corrector Methods with Fourth-Order Compact Scheme for a System of Free Boundary OptionsChinonso Nwankwo0Weizhong Dai1Department of Mathematics and Statistics, University of Calgary, Calgary, AB T2N 1N4, CanadaMathematics and Statistics, Louisiana Tech University, Ruston, LA 71272, USAThe trade-off between numerical accuracy and computational cost is always an important factor to consider when pricing options numerically, due to the inherent irregularity and existence of non-linearity in many models. In this work, we first present fast and accurate (1,2) and (2,2) predictor–corrector methods with a fourth-order compact finite difference scheme for pricing coupled system of the non-linear free boundary option pricing problem consisting of the option value and delta sensitivity. To predict the optimal exercise boundary, we set up a high-order boundary scheme, which is strategically derived using a combination of the fourth-order Robin boundary scheme and the fourth-order compact finite difference scheme near boundary. Furthermore, we implement a three-step high-order correction scheme for computing interior values of the option value and delta sensitivity. The discrete matrix system of this correction scheme has a tri-diagonal structure and strictly diagonal dominance. This nice feature allows for the implementation of the Thomas algorithm, thereby enabling fast computation. The optimal exercise boundary value is also corrected in each of the three correction steps with the derived Robin boundary scheme. Our implementations are fast on both coarse and very refined grids and provide highly accurate numerical approximations. Moreover, we recover a reasonable convergence rate. Further extensions to high-order predictor two-step corrector schemes are elaborated.https://www.mdpi.com/2075-1680/12/8/762American optionsoptimal exercise boundaryfourth-order compact finite difference schemepredictor–corrector methods |
| spellingShingle | Chinonso Nwankwo Weizhong Dai Efficiency of Some Predictor–Corrector Methods with Fourth-Order Compact Scheme for a System of Free Boundary Options Axioms American options optimal exercise boundary fourth-order compact finite difference scheme predictor–corrector methods |
| title | Efficiency of Some Predictor–Corrector Methods with Fourth-Order Compact Scheme for a System of Free Boundary Options |
| title_full | Efficiency of Some Predictor–Corrector Methods with Fourth-Order Compact Scheme for a System of Free Boundary Options |
| title_fullStr | Efficiency of Some Predictor–Corrector Methods with Fourth-Order Compact Scheme for a System of Free Boundary Options |
| title_full_unstemmed | Efficiency of Some Predictor–Corrector Methods with Fourth-Order Compact Scheme for a System of Free Boundary Options |
| title_short | Efficiency of Some Predictor–Corrector Methods with Fourth-Order Compact Scheme for a System of Free Boundary Options |
| title_sort | efficiency of some predictor corrector methods with fourth order compact scheme for a system of free boundary options |
| topic | American options optimal exercise boundary fourth-order compact finite difference scheme predictor–corrector methods |
| url | https://www.mdpi.com/2075-1680/12/8/762 |
| work_keys_str_mv | AT chinonsonwankwo efficiencyofsomepredictorcorrectormethodswithfourthordercompactschemeforasystemoffreeboundaryoptions AT weizhongdai efficiencyofsomepredictorcorrectormethodswithfourthordercompactschemeforasystemoffreeboundaryoptions |