Reduced models for sparse grid discretizations of the multi-asset Black-Scholes equation
This work presents reduced models for pricing basket options with the Black-Scholes and the Heston model. Basket options lead to multi-dimensional partial differential equations (PDEs) that quickly become computationally infeasible to discretize on full tensor grids. We therefore rely on sparse grid...
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| Format: | Article |
| Language: | English |
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Springer US
2017
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| Online Access: | http://hdl.handle.net/1721.1/107122 https://orcid.org/0000-0002-5045-046X |
| _version_ | 1826197900479168512 |
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| author | Gómez, Pablo Bungartz, Hans-Joachim Peherstorfer, Benjamin |
| author2 | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics |
| author_facet | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Gómez, Pablo Bungartz, Hans-Joachim Peherstorfer, Benjamin |
| author_sort | Gómez, Pablo |
| collection | MIT |
| description | This work presents reduced models for pricing basket options with the Black-Scholes and the Heston model. Basket options lead to multi-dimensional partial differential equations (PDEs) that quickly become computationally infeasible to discretize on full tensor grids. We therefore rely on sparse grid discretizations of the PDEs, which allow us to cope with the curse of dimensionality to some extent. We then derive reduced models with proper orthogonal decomposition. Our numerical results with the Black-Scholes model show that sufficiently accurate results are achieved while gaining speedups between 80 and 160 compared to the high-fidelity sparse grid model for 2-, 3-, and 4-asset options. For the Heston model, results are presented for a single-asset option that leads to a two-dimensional pricing problem, where we achieve significant speedups with our model reduction approach based on high-fidelity sparse grid models. |
| first_indexed | 2024-09-23T10:55:21Z |
| format | Article |
| id | mit-1721.1/107122 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T10:55:21Z |
| publishDate | 2017 |
| publisher | Springer US |
| record_format | dspace |
| spelling | mit-1721.1/1071222022-09-27T15:58:14Z Reduced models for sparse grid discretizations of the multi-asset Black-Scholes equation Gómez, Pablo Bungartz, Hans-Joachim Peherstorfer, Benjamin Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Peherstorfer, Benjamin This work presents reduced models for pricing basket options with the Black-Scholes and the Heston model. Basket options lead to multi-dimensional partial differential equations (PDEs) that quickly become computationally infeasible to discretize on full tensor grids. We therefore rely on sparse grid discretizations of the PDEs, which allow us to cope with the curse of dimensionality to some extent. We then derive reduced models with proper orthogonal decomposition. Our numerical results with the Black-Scholes model show that sufficiently accurate results are achieved while gaining speedups between 80 and 160 compared to the high-fidelity sparse grid model for 2-, 3-, and 4-asset options. For the Heston model, results are presented for a single-asset option that leads to a two-dimensional pricing problem, where we achieve significant speedups with our model reduction approach based on high-fidelity sparse grid models. 2017-02-23T17:44:01Z 2017-02-23T17:44:01Z 2015-05 2014-01 2016-05-23T12:17:08Z Article http://purl.org/eprint/type/JournalArticle 1019-7168 1572-9044 http://hdl.handle.net/1721.1/107122 Peherstorfer, Benjamin, Pablo Gómez, and Hans-Joachim Bungartz. “Reduced Models for Sparse Grid Discretizations of the Multi-Asset Black-Scholes Equation.” Adv Comput Math 41, no. 5 (May 20, 2015): 1365–1389. https://orcid.org/0000-0002-5045-046X en http://dx.doi.org/10.1007/s10444-015-9421-4 Advances in Computational Mathematics Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer Science+Business Media New York application/pdf Springer US Springer US |
| spellingShingle | Gómez, Pablo Bungartz, Hans-Joachim Peherstorfer, Benjamin Reduced models for sparse grid discretizations of the multi-asset Black-Scholes equation |
| title | Reduced models for sparse grid discretizations of the multi-asset Black-Scholes equation |
| title_full | Reduced models for sparse grid discretizations of the multi-asset Black-Scholes equation |
| title_fullStr | Reduced models for sparse grid discretizations of the multi-asset Black-Scholes equation |
| title_full_unstemmed | Reduced models for sparse grid discretizations of the multi-asset Black-Scholes equation |
| title_short | Reduced models for sparse grid discretizations of the multi-asset Black-Scholes equation |
| title_sort | reduced models for sparse grid discretizations of the multi asset black scholes equation |
| url | http://hdl.handle.net/1721.1/107122 https://orcid.org/0000-0002-5045-046X |
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