{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T23:46:32Z","timestamp":1740181592665,"version":"3.37.3"},"reference-count":22,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2024,7,3]],"date-time":"2024-07-03T00:00:00Z","timestamp":1719964800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,7,3]],"date-time":"2024-07-03T00:00:00Z","timestamp":1719964800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"External University of Colombia"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["SN COMPUT. SCI."],"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The classic financial market model proposed by Black and Scholes allows solving, under a series of simplifying assumptions, the problem of valuing financial derivatives. However, these assumptions deviate from the reality of global markets, particularly assuming perfect liquidity. In this work, a financial market model is proposed in which the illiquidity of the risky asset is stochastic, described by a mean-reversion process. The dynamics of the asset price in this context are established alongside the valuation PDE for derivatives. Additionally, as an approximation method to the solution of this PDE, the implementation of the extension of the Feynman\u2013Kac representation theorem to the nonlinear case using deep neural networks is proposed.<\/jats:p>","DOI":"10.1007\/s42979-024-02931-0","type":"journal-article","created":{"date-parts":[[2024,7,3]],"date-time":"2024-07-03T09:01:41Z","timestamp":1719997301000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Neural Networks and the Nonlinear Feynman\u2013Kac Theorem Applied to Financial Options Pricing"],"prefix":"10.1007","volume":"5","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2772-6931","authenticated-orcid":false,"given":"John F.","family":"Moreno T.","sequence":"first","affiliation":[]},{"given":"Carlos","family":"Zapata","sequence":"additional","affiliation":[]},{"given":"Daniel","family":"Aragon","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,7,3]]},"reference":[{"issue":"3","key":"2931_CR1","doi-asserted-by":"publisher","first-page":"637","DOI":"10.1086\/260062","volume":"81","author":"F Black","year":"1973","unstructured":"Black F, Scholes M. The pricing of options and corporate liabilities. J Polit Econ. 1973;81(3):637\u201354.","journal-title":"J Polit Econ"},{"issue":"2","key":"2931_CR2","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1108\/SEF-10-2018-0319","volume":"37","author":"A Tripathi","year":"2019","unstructured":"Tripathi A, Dixit A, et al. Liquidity of financial markets: a review. Stud Econ Fin. 2019;37(2):201\u201327.","journal-title":"Stud Econ Fin"},{"issue":"2","key":"2931_CR3","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1016\/0304-405X(86)90065-6","volume":"17","author":"Y Amihud","year":"1986","unstructured":"Amihud Y, Mendelson H. Asset pricing and the bid-ask spread. J Fin Econ. 1986;17(2):223\u201349.","journal-title":"J Fin Econ"},{"issue":"1","key":"2931_CR4","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1016\/S1386-4181(01)00024-6","volume":"5","author":"Y Amihud","year":"2002","unstructured":"Amihud Y. Illiquidity and stock returns: cross-section and time-series effects. J Fin Markets. 2002;5(1):31\u201356.","journal-title":"J Fin Markets"},{"key":"2931_CR5","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1561\/104.00000073","volume":"8","author":"Y Amihud","year":"2019","unstructured":"Amihud Y. Illiquidity and stock returns: a revisit. Crit Fin Rev. 2019;8:203\u201321.","journal-title":"Crit Fin Rev"},{"issue":"1","key":"2931_CR6","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1016\/S0304-405X(99)00057-4","volume":"56","author":"T Chordia","year":"2000","unstructured":"Chordia T, Roll R, Subrahmanyam A. Commonality in liquidity. J Fin Econ. 2000;56(1):3\u201328.","journal-title":"J Fin Econ"},{"issue":"2","key":"2931_CR7","doi-asserted-by":"publisher","first-page":"501","DOI":"10.1111\/0022-1082.00335","volume":"56","author":"T Chordia","year":"2001","unstructured":"Chordia T, Roll R, Subrahmanyam A. Market liquidity and trading activity. J Fin. 2001;56(2):501\u201330.","journal-title":"J Fin"},{"key":"2931_CR8","doi-asserted-by":"publisher","first-page":"1007","DOI":"10.1007\/978-0-387-77117-5_64","volume-title":"Handbook of quantitative finance and risk management","author":"U Cetin","year":"2010","unstructured":"Cetin U, Jarrow RA, Protter P. Liquidity risk and arbitrage pricing theory. In: Handbook of quantitative finance and risk management. Berlin: Springer; 2010. p. 1007\u201324."},{"issue":"1","key":"2931_CR9","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1080\/135048698334727","volume":"5","author":"K Ronnie Sircar","year":"1998","unstructured":"Ronnie Sircar K, Papanicolaou G. General Black\u2013Scholes models accounting for increased market volatility from hedging strategies. Appl Math Fin. 1998;5(1):45\u201382.","journal-title":"Appl Math Fin"},{"issue":"1","key":"2931_CR10","doi-asserted-by":"publisher","first-page":"232","DOI":"10.1137\/S0036139996308534","volume":"61","author":"P Wilmott","year":"2000","unstructured":"Wilmott P, Sch\u00f6nbucher PJ. The feedback effect of hedging in illiquid markets. SIAM J Appl Math. 2000;61(1):232\u201372.","journal-title":"SIAM J Appl Math"},{"issue":"5","key":"2931_CR11","doi-asserted-by":"publisher","first-page":"1054","DOI":"10.1086\/422565","volume":"112","author":"AW Lo","year":"2004","unstructured":"Lo AW, Mamaysky H, Wang J. Asset prices and trading volume under fixed transactions costs. J Polit Econ. 2004;112(5):1054\u201390.","journal-title":"J Polit Econ"},{"key":"2931_CR12","doi-asserted-by":"crossref","unstructured":"Vayanos D. Flight to quality, flight to liquidity, and the pricing of risk. Technical report, National Bureau of Economic Research; 2004.","DOI":"10.3386\/w10327"},{"issue":"2","key":"2931_CR13","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1016\/j.jfineco.2004.06.007","volume":"77","author":"VV Acharya","year":"2005","unstructured":"Acharya VV, Pedersen LH. Asset pricing with liquidity risk. J Fin Econ. 2005;77(2):375\u2013410.","journal-title":"J Fin Econ"},{"key":"2931_CR14","doi-asserted-by":"publisher","first-page":"503","DOI":"10.1007\/978-3-642-17254-0_18","volume-title":"Handbook of computational finance","author":"PA Forsyth","year":"2012","unstructured":"Forsyth PA, Vetzal KR. Numerical methods for nonlinear PDES in finance. In: Handbook of computational finance. New York: Springer; 2012. p. 503\u201328."},{"issue":"4","key":"2931_CR15","doi-asserted-by":"publisher","first-page":"507","DOI":"10.1090\/S0273-0979-2012-01379-4","volume":"49","author":"E Tadmor","year":"2012","unstructured":"Tadmor E. A review of numerical methods for nonlinear partial differential equations. Bull Am Math Soc. 2012;49(4):507\u201354.","journal-title":"Bull Am Math Soc"},{"issue":"1","key":"2931_CR16","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1080\/13504860903075670","volume":"17","author":"P Heider","year":"2010","unstructured":"Heider P. Numerical methods for non-linear Black\u2013Scholes equations. Appl Math Fin. 2010;17(1):59\u201381.","journal-title":"Appl Math Fin"},{"key":"2931_CR17","doi-asserted-by":"crossref","unstructured":"Brunetti C, Caldarera A. Asset prices and asset correlations in illiquid markets. Available at SSRN 625184; 2004","DOI":"10.2139\/ssrn.625184"},{"key":"2931_CR18","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1016\/j.finmar.2013.05.002","volume":"18","author":"S-P Feng","year":"2014","unstructured":"Feng S-P, Hung M-W, Wang Y-H. Option pricing with stochastic liquidity risk: theory and evidence. J Fin Markets. 2014;18:77\u201395.","journal-title":"J Fin Markets"},{"key":"2931_CR19","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10915-021-01590-0","volume":"88","author":"C Beck","year":"2021","unstructured":"Beck C, Becker S, Grohs P, Jaafari N, Jentzen A. Solving the Kolmogorov PDE by means of deep learning. J Sci Comput. 2021;88:1\u201328.","journal-title":"J Sci Comput"},{"key":"2931_CR20","volume-title":"Backward stochastic differential equations","author":"N El Karoui","year":"1997","unstructured":"El Karoui N, Mazliak L. Backward stochastic differential equations, vol. 364. Boca Raton: CRC Press; 1997."},{"key":"2931_CR21","volume-title":"Forward\u2013backward stochastic differential equations and their applications","author":"J Ma","year":"1999","unstructured":"Ma J, Morel J-M, Yong J. Forward\u2013backward stochastic differential equations and their applications, vol. 1702. New York: Springer; 1999."},{"issue":"34","key":"2931_CR22","doi-asserted-by":"publisher","first-page":"8505","DOI":"10.1073\/pnas.1718942115","volume":"115","author":"J Han","year":"2018","unstructured":"Han J, Jentzen A, Ee W. Solving high-dimensional partial differential equations using deep learning. Proc Natl Acad Sci. 2018;115(34):8505\u201310.","journal-title":"Proc Natl Acad Sci"}],"container-title":["SN Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s42979-024-02931-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s42979-024-02931-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s42979-024-02931-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,3]],"date-time":"2024-07-03T09:11:22Z","timestamp":1719997882000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s42979-024-02931-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,3]]},"references-count":22,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,8]]}},"alternative-id":["2931"],"URL":"https:\/\/doi.org\/10.1007\/s42979-024-02931-0","relation":{},"ISSN":["2661-8907"],"issn-type":[{"type":"electronic","value":"2661-8907"}],"subject":[],"published":{"date-parts":[[2024,7,3]]},"assertion":[{"value":"11 December 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 April 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 July 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that there are no conflicts of interest regarding the publication of this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of Interest"}}],"article-number":"697"}}