{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:20:36Z","timestamp":1760059236627,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,3]],"date-time":"2025-06-03T00:00:00Z","timestamp":1748908800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"CIFRE grant from Natixis"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>The market risk measurement of a trading portfolio in banks, specifically the practical implementation of the value-at-risk (VaR) and expected shortfall (ES) models, involves intensive recalls of the pricing engine. Machine learning algorithms may offer a solution to this challenge. In this study, we investigate the application of the Gaussian process (GP) regression and multi-fidelity modeling technique as approximation for the pricing engine. More precisely, multi-fidelity modeling combines models of different fidelity levels, defined as the degree of detail and precision offered by a predictive model or simulation, to achieve rapid yet precise prediction. We use the regression models to predict the prices of mono- and multi-asset equity option portfolios. In our numerical experiments, conducted with data limitation, we observe that both the standard GP model and multi-fidelity GP model outperform both the traditional approaches used in banks and the well-known neural network model in term of pricing accuracy as well as risk calculation efficiency.<\/jats:p>","DOI":"10.3390\/computation13060134","type":"journal-article","created":{"date-parts":[[2025,6,3]],"date-time":"2025-06-03T03:57:18Z","timestamp":1748923038000},"page":"134","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Performance-Enhancing Market Risk Calculation Through Gaussian Process Regression and Multi-Fidelity Modeling"],"prefix":"10.3390","volume":"13","author":[{"given":"N.","family":"Lehdili","sequence":"first","affiliation":[{"name":"Market and Counterparty Risk Modeling (MCRM), Enterprise Risk Management Department (ERM), Natixis CIB, 75013 Paris, France"}]},{"given":"P.","family":"Oswald","sequence":"additional","affiliation":[{"name":"Market and Counterparty Risk Modeling (MCRM), Enterprise Risk Management Department (ERM), Natixis CIB, 75013 Paris, France"}]},{"given":"H. D.","family":"Nguyen","sequence":"additional","affiliation":[{"name":"Laboratoire de Probabilit\u00e9s, Statistique et Mod\u00e9lisation, Universit\u00e9 Paris Cite\u00e9, 75006 Paris, France"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1023\/A:1009779322802","article-title":"Non-linear value-at-risk","volume":"2","author":"Schaefer","year":"1999","journal-title":"Rev. Financ."},{"key":"ref_2","unstructured":"Dowd, K. (2003). An Introduction to Market Risk Measurement, John Wiley & Sons."},{"key":"ref_3","unstructured":"Dowd, K. (2007). Measuring Market Risk, John Wiley & Sons."},{"key":"ref_4","unstructured":"Basel Committee on Banking Supervision (2025, May 12). Minimum Capital Requirements for Market Risk. Available online: https:\/\/www.bis.org\/bcbs\/publ\/d457.pdf."},{"key":"ref_5","unstructured":"Ferguson, R., and Green, A. (2018). Deeply learning derivatives. arXiv."},{"key":"ref_6","unstructured":"Financial Stability Board (2025, May 12). Artificial Intelligence and Machine Learning in Financial Services: Market Developments and Financial Stability Implications. Available online: https:\/\/www.fsb.org\/wp-content\/uploads\/P011117.pdf."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Liu, S., Oosterlee, C.W., and Bohte, S.M. (2019). Pricing options and computing implied volatilities using neural networks. Risks, 7.","DOI":"10.3390\/risks7010016"},{"key":"ref_8","unstructured":"Basel Committee on Banking Supervision (2025, May 12). Fundamental Review of the Trading Book: A Revised Market Risk Framework. Available online: https:\/\/www.bis.org\/publ\/bcbs265.pdf."},{"key":"ref_9","first-page":"1","article-title":"Gaussian process regression for derivative portfolio modeling and application to CVA computations","volume":"24","author":"Dixon","year":"2019","journal-title":"J. Comput. Financ."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1635","DOI":"10.1080\/14697688.2018.1495335","article-title":"Machine learning for quantitative finance: Fast derivative pricing, hedging and fitting","volume":"18","author":"Madan","year":"2018","journal-title":"Quant. Financ."},{"key":"ref_11","first-page":"7576","article-title":"Gpytorch: Blackbox matrix-matrix Gaussian process inference with GPU acceleration","volume":"31","author":"Gardner","year":"2018","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_12","unstructured":"Le Gratiet, L. (2013). Multi-Fidelity Gaussian Process Regression for Computer Experiments. [Ph.D. Thesis, Universit\u00e9 Paris-Diderot-Paris VII]. Available online: https:\/\/theses.hal.science\/tel-00866770\/PDF\/manuscrit.pdf."},{"key":"ref_13","unstructured":"Lehdili, N., Oswald, P., and Gueneau, H. (2025, May 12). Market Risk Assessment of a Trading Book Using Statistical and Machine Learning. Available online: https:\/\/www.researchgate.net\/publication\/337059465_Market_Risk_Assessment_of_a_trading_book_using_Statistical_and_Machine_Learning?channel=doi&linkId=5dc2cf8da6fdcc21280babf0&showFulltext=true."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"374","DOI":"10.69554\/CYHX1007","article-title":"Machine learning in risk measurement: Gaussian process regression for value-at-risk and expected shortfall","volume":"12","author":"Wilkens","year":"2019","journal-title":"J. Risk Manag. Financ. Institutions"},{"key":"ref_15","first-page":"1","article-title":"Neural networks for option pricing and hedging: A literature review","volume":"24","author":"Ruf","year":"2019","journal-title":"J. Comput. Financ."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"109020","DOI":"10.1016\/j.jcp.2019.109020","article-title":"A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse pde problems","volume":"401","author":"Meng","year":"2020","journal-title":"J. Comput. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1547","DOI":"10.1090\/mcom\/3514","article-title":"Deep backward schemes for high-dimensional nonlinear PDEs","volume":"89","author":"Pham","year":"2020","journal-title":"Math. Comput."},{"key":"ref_18","unstructured":"Gouden\u00e8ge, L., Molent, A., and Zanette, A. (2021). Variance reduction applied to machine learning for pricing Bermudan\/American options in high dimension. Oleg Kudryavtsev, Antonino Zanette. Applications of L\u00e9vy Processes, Nova Science Publishers."},{"key":"ref_19","unstructured":"Fern\u00e1ndez-Godino, M.G., Park, C., Kim, N.-H., and Haftka, R.T. (2016). Review of multi-fidelity models. arXiv."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"106339","DOI":"10.1016\/j.ast.2020.106339","article-title":"Overview of gaussian process based multi-fidelity techniques with variable relationship between fidelities, application to aerospace systems","volume":"107","author":"Brevault","year":"2020","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Roncalli, T. (2020). Handbook of Financial Risk Management, Chapman and Hall\/CRC.","DOI":"10.1201\/9781315144597"},{"key":"ref_22","first-page":"1","article-title":"Monte carlo methods for value-at-risk and conditional value-at-risk: A review","volume":"24","author":"Hong","year":"2014","journal-title":"ACM Trans. Model. Comput. Simul. (TOMACS)"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Cr\u00e9pey, S. (2013). Financial Modeling, A Backward Stochastic Differential Equations Perspective, Springer.","DOI":"10.1007\/978-3-642-37113-4"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1093\/rfs\/14.1.113","article-title":"Valuing American options by simulation: A simple least-squares approach","volume":"14","author":"Longstaff","year":"2001","journal-title":"Rev. Financ. Stud."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1515\/fcds-2018-0011","article-title":"Supervised machine learning with control variates for american option pricing","volume":"43","author":"Mu","year":"2018","journal-title":"Found. Comput. Decis. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Ruiz, I., and Zeron, M. (2021). Machine Learning for Risk Calculations: A Practitioner\u2019s View, John Wiley & Sons.","DOI":"10.1002\/9781119791416"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1088\/1469-7688\/4\/1\/001","article-title":"Approximated moment-matching dynamics for basket-options pricing","volume":"4","author":"Brigo","year":"2003","journal-title":"Quant. Financ."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Rasmussen, C.E., and Williams, C.K. (2006). Gaussian Processes for Machine learning, The MIT Press.","DOI":"10.7551\/mitpress\/3206.001.0001"},{"key":"ref_29","unstructured":"Murphy, K.P. (2012). Machine Learning: A Probabilistic Perspective, The MIT Press."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1077","DOI":"10.1287\/opre.2015.1419","article-title":"Risk estimation via regression","volume":"63","author":"Broadie","year":"2015","journal-title":"Oper. Res."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"274","DOI":"10.1111\/mafi.12368","article-title":"Pathwise CVA regressions with oversimulated defaults","volume":"33","author":"Saadeddine","year":"2023","journal-title":"Math. Financ."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1093\/biomet\/87.1.1","article-title":"Predicting the output from a complex computer code when fast approximations are available","volume":"87","author":"Kennedy","year":"2000","journal-title":"Biometrika"},{"key":"ref_33","first-page":"3251","article-title":"Multi-fidelity optimization via surrogate modelling","volume":"463","author":"Forrester","year":"2007","journal-title":"R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_34","first-page":"567","article-title":"Variational learning of inducing variables in sparse Gaussian processes","volume":"5","author":"Titsias","year":"2009","journal-title":"Proc. Mach. Learn. Res."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1111\/j.1540-6261.1987.tb02569.x","article-title":"Efficient analytic approximation of american option values","volume":"42","author":"Whaley","year":"1987","journal-title":"J. Financ."},{"key":"ref_36","unstructured":"Cr\u00e9pey, S., Li, B., Nguyen, H.D., and Saadeddine, B. (2024). CVA sensitivities, hedging and risk. arXiv."},{"key":"ref_37","unstructured":"Kingma, D.P., and Ba, J. (2015, January 7\u20139). Adam: A method for stochastic optimization. Proceedings of the 3rd International Conference on Learning Representations, San Diego, CA, USA."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"68","DOI":"10.25080\/gerudo-f2bc6f59-009","article-title":"Emukit: A python toolkit for decision making under uncertainty","volume":"22","author":"Paleyes","year":"2023","journal-title":"Proc. Python Sci. Conf."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Nocedal, J., and Wright, S.J. (1999). Numerical Optimization, Springer.","DOI":"10.1007\/b98874"},{"key":"ref_40","first-page":"1407","article-title":"Product kernel interpolation for scalable Gaussian processes","volume":"84","author":"Gardner","year":"2018","journal-title":"Int. Conf. Artif. Intell. And Stat."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"859","DOI":"10.1080\/01621459.2017.1285773","article-title":"Variational inference: A review for statisticians","volume":"112","author":"Blei","year":"2017","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_42","first-page":"1303","article-title":"Stochastic variational inference","volume":"14","author":"Hoffman","year":"2013","journal-title":"J. Mach. Learn. Res."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1038\/nature14539","article-title":"Deep learning","volume":"521","author":"LeCun","year":"2015","journal-title":"Nature"}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/6\/134\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:46:22Z","timestamp":1760031982000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/6\/134"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,3]]},"references-count":43,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["computation13060134"],"URL":"https:\/\/doi.org\/10.3390\/computation13060134","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2025,6,3]]}}}