Valoración no lineal de derivados financieros en mercados con iliquidez mediante extensiones del teorema de Feynman-Kac y algoritmos de aprendizaje automático

ilustraciones, diagramas, tablas

Autores:
Moreno Trujillo, John Freddy
Tipo de recurso:
Doctoral thesis
Fecha de publicación:
2024
Institución:
Universidad Nacional de Colombia
Repositorio:
Universidad Nacional de Colombia
Idioma:
spa
OAI Identifier:
oai:repositorio.unal.edu.co:unal/86877
Acceso en línea:
https://repositorio.unal.edu.co/handle/unal/86877
https://repositorio.unal.edu.co/
Palabra clave:
330 - Economía::332 - Economía financiera
510 - Matemáticas::519 - Probabilidades y matemáticas aplicadas
000 - Ciencias de la computación, información y obras generales::006 - Métodos especiales de computación
DERIVADOS FINANCIEROS
OPCIONES (FINANZAS)
TEORIA DEL APRENDIZAJE COMPUTACIONAL
APRENDIZAJE SUPERVISADO (APRENDIZAJE AUTOMATICO)
ECUACIONES DIFERENCIALES NO LINEALES
REDES NEURALES (COMPUTADORES)
Derivative securities
Options (Finance)
Computational learning theory
Supervised learning (Machine learning)
Differential equations, nonlinear
Neural networks (Computer science)
Mercado ilíquido
Valoración de derivados
Ecuaciones diferenciales parciales no lineales
Representación de Feynman-Kac
Redes neuronales artificiales
Illiquid market
Derivative valuation
Nonlinear partial differential equations
Feynman-Kac representation
Artificial neural networks
Rights
openAccess
License
Atribución-NoComercial-SinDerivadas 4.0 Internacional
id UNACIONAL2_4cb43827e83391db4fecbb20420d6513
oai_identifier_str oai:repositorio.unal.edu.co:unal/86877
network_acronym_str UNACIONAL2
network_name_str Universidad Nacional de Colombia
repository_id_str
dc.title.spa.fl_str_mv Valoración no lineal de derivados financieros en mercados con iliquidez mediante extensiones del teorema de Feynman-Kac y algoritmos de aprendizaje automático
dc.title.translated.eng.fl_str_mv Nonlinear valuation of financial derivatives in illiquid markets using extensions of the Feynman-Kac theorem and machine learning algorithms
title Valoración no lineal de derivados financieros en mercados con iliquidez mediante extensiones del teorema de Feynman-Kac y algoritmos de aprendizaje automático
spellingShingle Valoración no lineal de derivados financieros en mercados con iliquidez mediante extensiones del teorema de Feynman-Kac y algoritmos de aprendizaje automático
330 - Economía::332 - Economía financiera
510 - Matemáticas::519 - Probabilidades y matemáticas aplicadas
000 - Ciencias de la computación, información y obras generales::006 - Métodos especiales de computación
DERIVADOS FINANCIEROS
OPCIONES (FINANZAS)
TEORIA DEL APRENDIZAJE COMPUTACIONAL
APRENDIZAJE SUPERVISADO (APRENDIZAJE AUTOMATICO)
ECUACIONES DIFERENCIALES NO LINEALES
REDES NEURALES (COMPUTADORES)
Derivative securities
Options (Finance)
Computational learning theory
Supervised learning (Machine learning)
Differential equations, nonlinear
Neural networks (Computer science)
Mercado ilíquido
Valoración de derivados
Ecuaciones diferenciales parciales no lineales
Representación de Feynman-Kac
Redes neuronales artificiales
Illiquid market
Derivative valuation
Nonlinear partial differential equations
Feynman-Kac representation
Artificial neural networks
title_short Valoración no lineal de derivados financieros en mercados con iliquidez mediante extensiones del teorema de Feynman-Kac y algoritmos de aprendizaje automático
title_full Valoración no lineal de derivados financieros en mercados con iliquidez mediante extensiones del teorema de Feynman-Kac y algoritmos de aprendizaje automático
title_fullStr Valoración no lineal de derivados financieros en mercados con iliquidez mediante extensiones del teorema de Feynman-Kac y algoritmos de aprendizaje automático
title_full_unstemmed Valoración no lineal de derivados financieros en mercados con iliquidez mediante extensiones del teorema de Feynman-Kac y algoritmos de aprendizaje automático
title_sort Valoración no lineal de derivados financieros en mercados con iliquidez mediante extensiones del teorema de Feynman-Kac y algoritmos de aprendizaje automático
dc.creator.fl_str_mv Moreno Trujillo, John Freddy
dc.contributor.advisor.spa.fl_str_mv Hoyos Gómez, Nancy Milena
dc.contributor.author.spa.fl_str_mv Moreno Trujillo, John Freddy
dc.contributor.orcid.spa.fl_str_mv Moreno Trujillo, John Freddy [0000-0002-2772-6931]
dc.contributor.cvlac.spa.fl_str_mv Moreno Trujillo, John Freddy [https://scienti.minciencias.gov.co/cvlac/visualizador/generarCurriculoCv.do?cod_rh=0001028324]
dc.contributor.googlescholar.spa.fl_str_mv Moreno Trujillo, John Freddy [https://scholar.google.com/citations?user=j7aRNrAAAAAJ&hl=es]
dc.subject.ddc.spa.fl_str_mv 330 - Economía::332 - Economía financiera
510 - Matemáticas::519 - Probabilidades y matemáticas aplicadas
000 - Ciencias de la computación, información y obras generales::006 - Métodos especiales de computación
topic 330 - Economía::332 - Economía financiera
510 - Matemáticas::519 - Probabilidades y matemáticas aplicadas
000 - Ciencias de la computación, información y obras generales::006 - Métodos especiales de computación
DERIVADOS FINANCIEROS
OPCIONES (FINANZAS)
TEORIA DEL APRENDIZAJE COMPUTACIONAL
APRENDIZAJE SUPERVISADO (APRENDIZAJE AUTOMATICO)
ECUACIONES DIFERENCIALES NO LINEALES
REDES NEURALES (COMPUTADORES)
Derivative securities
Options (Finance)
Computational learning theory
Supervised learning (Machine learning)
Differential equations, nonlinear
Neural networks (Computer science)
Mercado ilíquido
Valoración de derivados
Ecuaciones diferenciales parciales no lineales
Representación de Feynman-Kac
Redes neuronales artificiales
Illiquid market
Derivative valuation
Nonlinear partial differential equations
Feynman-Kac representation
Artificial neural networks
dc.subject.lemb.spa.fl_str_mv DERIVADOS FINANCIEROS
OPCIONES (FINANZAS)
TEORIA DEL APRENDIZAJE COMPUTACIONAL
APRENDIZAJE SUPERVISADO (APRENDIZAJE AUTOMATICO)
ECUACIONES DIFERENCIALES NO LINEALES
REDES NEURALES (COMPUTADORES)
dc.subject.lemb.eng.fl_str_mv Derivative securities
Options (Finance)
Computational learning theory
Supervised learning (Machine learning)
Differential equations, nonlinear
Neural networks (Computer science)
dc.subject.proposal.spa.fl_str_mv Mercado ilíquido
Valoración de derivados
Ecuaciones diferenciales parciales no lineales
Representación de Feynman-Kac
Redes neuronales artificiales
dc.subject.proposal.eng.fl_str_mv Illiquid market
Derivative valuation
Nonlinear partial differential equations
Feynman-Kac representation
Artificial neural networks
description ilustraciones, diagramas, tablas
publishDate 2024
dc.date.accessioned.none.fl_str_mv 2024-09-30T18:56:11Z
dc.date.available.none.fl_str_mv 2024-09-30T18:56:11Z
dc.date.issued.none.fl_str_mv 2024
dc.type.spa.fl_str_mv Trabajo de grado - Doctorado
dc.type.driver.spa.fl_str_mv info:eu-repo/semantics/doctoralThesis
dc.type.version.spa.fl_str_mv info:eu-repo/semantics/acceptedVersion
dc.type.coar.spa.fl_str_mv http://purl.org/coar/resource_type/c_db06
dc.type.content.spa.fl_str_mv Text
dc.type.redcol.spa.fl_str_mv http://purl.org/redcol/resource_type/TD
format http://purl.org/coar/resource_type/c_db06
status_str acceptedVersion
dc.identifier.uri.none.fl_str_mv https://repositorio.unal.edu.co/handle/unal/86877
dc.identifier.instname.spa.fl_str_mv Universidad Nacional de Colombia
dc.identifier.reponame.spa.fl_str_mv Repositorio Institucional Universidad Nacional de Colombia
dc.identifier.repourl.spa.fl_str_mv https://repositorio.unal.edu.co/
url https://repositorio.unal.edu.co/handle/unal/86877
https://repositorio.unal.edu.co/
identifier_str_mv Universidad Nacional de Colombia
Repositorio Institucional Universidad Nacional de Colombia
dc.language.iso.spa.fl_str_mv spa
language spa
dc.relation.references.spa.fl_str_mv Acharya, V. V. and Pedersen, L. H. (2005). Asset pricing with liquidity risk. Journal of financial Economics, 77(2):375–410.
Albrecher, H., Guillaume, F., and Schoutens, W. (2013). Implied liquidity: model sensitivity. Journal of Empirical Finance, 23:48–67.
Amihud, Y. (2002). Illiquidity and stock returns: Cross-section and time-series effects. Journal of Financial Markets, 5(1):31–56.
Amihud, Y. (2019). Illiquidity and stock returns: A revisit. Critical Finance Review, (8):203–221.
Amihud, Y. and Mendelson, H. (1986). Asset pricing and the bid-ask spread. Journal of Financial Economics, 17(2):223–249.
Arenas, A. J., González-Parra, G., and Caraballo, B. M. (2013). A nonstandard finite difference scheme for a nonlinear black–scholes equation. Mathematical and Computer Modelling, 57(7-8):1663–1670.
Avellaneda, M., Levy, A., and Parás, A. (1995). Pricing and hedging derivative securities in markets with uncertain volatilities. Applied Mathematical Finance, 2(2):73–88.
Bachouch, A., Huré, C., Langrené, N., and Pham, H. (2021). Deep neural networks algorithms for stochastic control problems on finite horizon: numerical applications. Methodology and Computing in Applied Probability, pages 1–36.
Back, K. (1993). Asymmetric information and options. The Review of Financial Studies, 6(3):435–472.
Beck, C., Becker, S., Grohs, P., Jaafari, N., and Jentzen, A. (2021). Solving the kolmogorov pde by means of deep learning. Journal of Scientific Computing, 88:1–28.
Beck, C., Hutzenthaler, M., Jentzen, A., and Kuckuck, B. (2020). An overview on deep learning-based approximation methods for partial differential equations. arXiv preprint arXiv:2012.12348.
Beck, C., Weinan, E., and Jentzen, A. (2019). Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations. Journal of Nonlinear Science, 29(4):1563–1619.
Bergman, Y. Z. (1995). Option pricing with differential interest rates. The Review of Financial Studies, 8(2):475–500.
Berkman, H. and Koch, P. D. (2008). Noise trading and the price formation process. Journal of Empirical Finance, 15(2):232–250.
Bertsimas, D. and Lo, A. W. (1998). Optimal control of execution costs. Journal of financial markets, 1(1):1–50.
Bismut, J.-M. (1973). Conjugate convex functions in optimal stochastic control. Journal of Mathematical Analysis and Applications, 44(2):384–404.
Björk, T. (2009). Arbitrage theory in continuous time. Oxford university press.
Black, F. and Scholes, M. (1973a). The pricing of options and corporate liabilities. Journal of political Economy, 81(3):637–654.
Black, F. and Scholes, M. (1973b). The pricing of options and corporate liabilities. Journal of political economy, 81(3):637–654.
Bouchard, B. (2015). Lecture notes on BSDEs Main existence and stability results. PhD thesis, CEREMADE-CEntre de REcherches en MAthématiques de la DEcision.
Brunetti, C. and Caldarera, A. (2004). Asset prices and asset correlations in illiquid markets. Available at SSRN 625184.
Caldarera, A., Brunetti, C., et al. (2005). Asset prices and asset correlations in illiquid markets. In 2005 Meeting Papers, number 288. Society for Economic Dynamics.
Cetin, U., Jarrow, R., Protter, P., and Warachka, M. (2006). Pricing options in an extended black scholes economy with illiquidity: Theory and empirical evidence. The Review of Financial Studies, 19(2):493–529.
Cetin, U., Jarrow, R. A., and Protter, P. (2010). Liquidity risk and arbitrage pricing theory. Springer.
Cheridito, P., Soner, H. M., Touzi, N., and Victoir, N. (2007). Second-order backward stochastic differential equations and fully nonlinear parabolic pdes. Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, 60(7):1081–1110.
Chordia, T., Roll, R., and Subrahmanyam, A. (2000). Commonality in liquidity. Journal of Financial Economics, 56(1):3–28.
Chordia, T., Roll, R., and Subrahmanyam, A. (2001). Market liquidity and trading activity. The Journal of Finance, 56(2):501–530.
Company, R., Jódar, L., and Pintos, J.-R. (2010a). Numerical analysis and computing for option pricing models in illiquid markets. Mathematical and computer modelling, 52(7- 8):1066–1073.
Company, R., Jódar, L., Ponsoda, E., and Ballester, C. (2010b). Numerical analysis and simulation of option pricing problems modeling illiquid markets. Computers & Mathematics with Applications, 59(8):2964–2975.
Corcuera, J. M., Guillaume, F., Madan, D. B., and Schoutens, W. (2012). Implied liquidity: towards stochastic liquidity modelling and liquidity trading. International Journal of Portfolio Analysis and Management, 1(1):80–91.
Cybenko, G. (1989). Approximation by superpositions of a sigmoidal function. Mathematics of control, signals and systems, 2(4):303–314.
Duffie, D. (2010). Dynamic asset pricing theory. Princeton University Press.
El Karoui, N. and Mazliak, L. (1997). Backward stochastic differential equations, volume 364. CRC Press.
Esser, A. (2004). Pricing in (in) complete markets: Structural analysis and applications. Springer Science & Business Media.
Fama, E. F. (1965). The behavior of stock-market prices. The journal of Business, 38(1):34–105.
Feng, S.-P., Hung, M.-W., and Wang, Y.-H. (2014). Option pricing with stochastic liquidity risk: Theory and evidence. Journal of Financial Markets, 18:77–95.
Forsyth, P. A. and Vetzal, K. R. (2012). Numerical methods for nonlinear pdes in finance. In Handbook of Computational Finance, pages 503–528. Springer.
Frey, R. (1998). Perfect option hedging for a large trader. Finance and Stochastics, 2(2):115–141.
Frey, R. (2000). Market illiquidity as a source of model risk in dynamic hedging. Model Risk, pages 125–138.
Frey, R. and Patie, P. (2002). Risk management for derivatives in illiquid markets: A simulation study. Advances in finance and stochastics, pages 137–159.
Frey, R. and Stremme, A. (1997). Market volatility and feedback effects from dynamic hedging. Mathematical finance, 7(4):351–374.
Geiss, S. and Ylinen, J. (2021). Decoupling on the Wiener Space, related Besov Spaces, and applications to BSDEs, volume 272. American Mathematical Society.
Gennotte, G. and Leland, H. (1990). Market liquidity, hedging, and crashes. The American Economic Review, pages 999–1021.
Germain, M., Pham, H., and Warin, X. (2021). Neural networks-based algorithms for stochastic control and pdes in finance. arXiv preprint arXiv:2101.08068.
Glover, K. J., Duck, P. W., and Newton, D. P. (2010). On nonlinear models of markets with finite liquidity: some cautionary notes. SIAM Journal on Applied Mathematics, 70(8):3252–3271.
Gökay, S., Roch, A. F., and Soner, H. M. (2011). Liquidity models in continuous and discrete time. Springer.
Gregory, J. (2015). The xVA Challenge: counterparty credit risk, funding, collateral and capital. John Wiley & Sons.
Guéant, O. (2016). The Financial Mathematics of Market Liquidity: From optimal execution to market making, volume 33. CRC Press.
Guo, J. andWang, W. (2015). On the numerical solution of nonlinear option pricing equation in illiquid markets. Computers & Mathematics with Applications, 69(2):117–133.
Gurney, K. (2018). An introduction to neural networks. CRC press
Guyon, J. and Henry-Labordere, P. (2013). Nonlinear option pricing. CRC Press.
Han, J., Jentzen, A., and E, W. (2018). Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences, 115(34):8505– 8510.
Heider, P. (2010). Numerical methods for non-linear black–scholes equations. Applied Mathematical Finance, 17(1):59–81.
Henry-Labordere, P. (2012). Counterparty risk valuation: A marked branching diffusion approach. arXiv preprint arXiv:1203.2369.
Henry-Labordere, P., Tan, X., and Touzi, N. (2014). A numerical algorithm for a class of bsdes via the branching process. Stochastic Processes and their Applications, 124(2):1112–1140.
Hornik, K., Stinchcombe, M., and White, H. (1989). Multilayer feedforward networks are universal approximators. Neural networks, 2(5):359–366.
Hornik, K., Stinchcombe, M., and White, H. (1990). Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks. Neural networks, 3(5):551–560.
Huberman, G. and Stanzl, W. (2004). Arbitrage-free price update and price-impact functions. Econometrica, 72(4):1247–1275.
Huré, C., Pham, H., and Warin, X. (2020). Deep backward schemes for high-dimensional nonlinear pdes. Mathematics of Computation, 89(324):1547–1579.
Jarrow, R. A. (1994). Derivative security markets, market manipulation, and option pricing theory. Journal of financial and quantitative analysis, 29(2):241–261.
Karatzas, I. and Shreve, S. (2014). Brownian motion and stochastic calculus, volume 113. springer.
Karatzas, I. and Shreve, S. E. (1998). Methods of mathematical finance, volume 39. Springer.
Ku, H. and Zhang, H. (2018). Option pricing for a large trader with price impact and liquidity costs. Journal of Mathematical Analysis and Applications, 459(1):32–52.
Kyle, A. S. (1985). Continuous auctions and insider trading. Econometrica: Journal of the Econometric Society, pages 1315–1335.
Lagaris, I. E., Likas, A., and Fotiadis, D. I. (1998). Artificial neural networks for solving ordinary and partial differential equations. IEEE transactions on neural networks, 9(5):987–1000.
Leippold, M. and Schärer, S. (2017). Discrete-time option pricing with stochastic liquidity. Journal of Banking & Finance, 75:1–16.
Leland, H. E. (1985). Option pricing and replication with transactions costs. The journal of finance, 40(5):1283–1301.
Li, Z., Zhang, W.-G., and Liu, Y.-J. (2018a). Analytical valuation for geometric asian options in illiquid markets. Physica A: Statistical Mechanics and its Applications, 507:175–191.
Li, Z., Zhang, W.-G., and Liu, Y.-J. (2018b). European quanto option pricing in presence of liquidity risk. The North American Journal of Economics and Finance, 45:230–244.
Liesenfeld, R. and Richard, J.-F. (2003). Univariate and multivariate stochastic volatility models: estimation and diagnostics. Journal of empirical finance, 10(4):505–531.
Liu, H. and Yong, J. (2005). Option pricing with an illiquid underlying asset market. Journal of Economic Dynamics and Control, 29(12):2125–2156.
Liu, W. (2006). A liquidity-augmented capital asset pricing model. Journal of financial Economics, 82(3):631–671.
Lo, A. W., Mamaysky, H., and Wang, J. (2004). Asset prices and trading volume under fixed transactions costs. Journal of Political Economy, 112(5):1054–1090.
Loeper, G. (2018). Option pricing with linear market impact and nonlinear black–scholes equations. The Annals of Applied Probability, 28(5):2664–2726.
Longstaff, F. A., Mithal, S., and Neis, E. (2005). Corporate yield spreads: Default risk or liquidity? new evidence from the credit default swap market. The journal of finance, 60(5):2213–2253.
Ludkovski, M. and Shen, Q. (2013). European option pricing with liquidity shocks. International Journal of Theoretical and Applied Finance, 16(07):1350043.
Ma, J., Morel, J.-M., and Yong, J. (1999). Forward-backward stochastic differential equations and their applications. Number 1702. Springer Science & Business Media.
Madan, C. and Cherny, A. (2010). Illiquid markets as a counterparty: An introduction to conic finance, robert h. Smith School Research Paper No. RHS, pages 06–115.
Mandelbrot, B. B. (1997). The variation of certain speculative prices. Springer.
McKean, H. P. (1975). Application of brownian motion to the equation of kolmogorov-petrovskii-piskunov. Communications on pure and applied mathematics, 28(3):323–331.
Merton, R. C. (1973). Theory of rational option pricing. The Bell Journal of economics and management science, pages 141–183.
Merton, R. C. and Samuelson, P. A. (1992). Continuous-time finance.
Merton, R. C. and Samuelson, P. A. (1990). Continuous-time finance.
Mönch, B. (2006). Strategic trading in illiquid markets, volume 553. Springer Science & Business Media.
Nielsen, M. A. (2015). Neural networks and deep learning, volume 25. Determination pressSan Francisco, CA, USA.
Oksendal, B. (2013). Stochastic differential equations: an introduction with applications. Springer Science & Business Media.
Pardoux, E. and Peng, S. (1990). Adapted solution of a backward stochastic differential equation. Systems & control letters, 14(1):55–61.
Pardoux, E. and Tang, S. (1999). Forward-backward stochastic differential equations and quasilinear parabolic pdes. Probability theory and related fields, 114:123–150.
Pástor, L. and Stambaugh, R. F. (2003). Liquidity risk and expected stock returns. Journal of Political Economy, 111(3):642–685.
Pham, H., Warin, X., and Germain, M. (2021). Neural networks-based backward scheme for fully nonlinear pdes. SN Partial Differential Equations and Applications, 2(1):16.
Raissi, M. (2024). Forward–backward stochastic neural networks: deep learning of highdimensional partial differential equations. In Peter Carr Gedenkschrift: Research Advances in Mathematical Finance, pages 637–655. World Scientific.
Raissi, M. and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357:125–141.
Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics, 378:686–707.
Ronnie Sircar, K. and Papanicolaou, G. (1998). General black-scholes models accounting for increased market volatility from hedging strategies. Applied Mathematical Finance, 5(1):45–82.
Shreve, S. E. (2004). Stochastic calculus for finance II: Continuous-time models, volume 11. Springer Science & Business Media.
Skorokhod, A. V. (1964). Branching diffusion processes. Theory of Probability & Its Applications, 9(3):445–449.
Subramanian, A. and Jarrow, R. A. (2001). The liquidity discount. Mathematical Finance, 11(4):447–474.
Tadmor, E. (2012). A review of numerical methods for nonlinear partial differential equations. Bulletin of the American Mathematical Society, 49(4):507–554.
Taylor, S. J. (1994). Modeling stochastic volatility: A review and comparative study. Mathematical finance, 4(2):183–204.
Tripathi, A., Dixit, A., et al. (2019). Liquidity of financial markets: a review. Studies in Economics and Finance.
Trujillo, J. F. M. (2015). Modelos estocásticos en finanzas. U. Externado de Colombia.
Vayanos, D. (2001). Strategic trading in a dynamic noisy market. The Journal of Finance, 56(1):131–171.
Vayanos, D. (2004). Flight to quality, flight to liquidity, and the pricing of risk. Technical report, National bureau of economic research.
Venegas-Martínez, F. (2008). Riesgos financieros y económicos, productos derivados y decisiones económicas bajo incertidumbre, volume 1. Escuela Superior de Economía, Instituto Politécnico Nacional
Warin, X. (2018). Nesting monte carlo for high-dimensional non-linear pdes. Monte Carlo Methods and Applications, 24(4):225–247.
Watanabe, S. (1965). On the branching process for brownian particles with an absorbing boundary. Journal of Mathematics of Kyoto University, 4(2):385–398.
Weinan, E., Hutzenthaler, M., Jentzen, A., and Kruse, T. (2017). Linear scaling algorithms for solving high-dimensional nonlinear parabolic differential equations. SAM Research Report, 2017.
Wilmott, P., Hoggard, T., and Whalley, A. E. (1994). Hedging option portfolios in the presence of transaction costs. Advances in Futures and Options Research, 7.
Wilmott, P. and Schönbucher, P. J. (2000). The feedback effect of hedging in illiquid markets. SIAM Journal on Applied Mathematics, 61(1):232–272.
Zhang, Y., Ding, S., and Duygun, M. (2019). Derivatives pricing with liquidity risk. Journal of Futures Markets, 39(11):1471–1485.
dc.rights.coar.fl_str_mv http://purl.org/coar/access_right/c_abf2
dc.rights.license.spa.fl_str_mv Atribución-NoComercial-SinDerivadas 4.0 Internacional
dc.rights.uri.spa.fl_str_mv http://creativecommons.org/licenses/by-nc-nd/4.0/
dc.rights.accessrights.spa.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv Atribución-NoComercial-SinDerivadas 4.0 Internacional
http://creativecommons.org/licenses/by-nc-nd/4.0/
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.format.extent.spa.fl_str_mv x, 121 páginas
dc.format.mimetype.spa.fl_str_mv application/pdf
dc.publisher.spa.fl_str_mv Universidad Nacional de Colombia
dc.publisher.program.spa.fl_str_mv Bogotá - Ciencias Económicas - Doctorado en Ciencias Económicas
dc.publisher.faculty.spa.fl_str_mv Facultad de Ciencias Económicas
dc.publisher.place.spa.fl_str_mv Bogotá, Colombia
dc.publisher.branch.spa.fl_str_mv Universidad Nacional de Colombia - Sede Bogotá
institution Universidad Nacional de Colombia
bitstream.url.fl_str_mv https://repositorio.unal.edu.co/bitstream/unal/86877/1/license.txt
https://repositorio.unal.edu.co/bitstream/unal/86877/2/80059098.2024.pdf
https://repositorio.unal.edu.co/bitstream/unal/86877/3/80059098.2024.pdf.jpg
bitstream.checksum.fl_str_mv eb34b1cf90b7e1103fc9dfd26be24b4a
c50d4a86bfab6b110e50d8903474ed09
0b562b23bcd06e4f663a9bc377529abe
bitstream.checksumAlgorithm.fl_str_mv MD5
MD5
MD5
repository.name.fl_str_mv Repositorio Institucional Universidad Nacional de Colombia
repository.mail.fl_str_mv repositorio_nal@unal.edu.co
_version_ 1812169656436260864
spelling Atribución-NoComercial-SinDerivadas 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Hoyos Gómez, Nancy Milena12999640324d4aba16dd48e0d34cd393Moreno Trujillo, John Freddy579eacbd7db32a284077f5586cef0d15Moreno Trujillo, John Freddy [0000-0002-2772-6931]Moreno Trujillo, John Freddy [https://scienti.minciencias.gov.co/cvlac/visualizador/generarCurriculoCv.do?cod_rh=0001028324]Moreno Trujillo, John Freddy [https://scholar.google.com/citations?user=j7aRNrAAAAAJ&hl=es]2024-09-30T18:56:11Z2024-09-30T18:56:11Z2024https://repositorio.unal.edu.co/handle/unal/86877Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/ilustraciones, diagramas, tablasEn esta investigación se proponen y desarrollan cuatro modelos de mercados financieros ilíquidos, en los cuales se caracteriza la dinámica del precio de los activos riesgosos y la relación emergente entre dicha dinámica y las estrategias de negociación de los agentes. Además, se deducen las correspondientes ecuaciones diferenciales parciales para la valoración de activos contingentes. Específicamente, se presentan: 1. Un modelo de mercado con un factor de iliquidez proporcional al precio del activo; 2. Un modelo en el que la iliquidez es función del precio del activo; 3. Un modelo que incluye iliquidez proporcional, con la presencia de agentes ruidosos (noise traders); 4. Un modelo en el cual la iliquidez es estocástica y está descrita mediante un proceso de reversión a la media de tipo raíz cuadrada. Las ecuaciones de valoración obtenidas son extensiones no lineales de la ecuación diferencial parcial de Black-Scholes, donde la no linealidad resulta del efecto de retroalimentación asociado a la iliquidez del mercado. También se propone, como alternativa para la aproximación a la solución de estas ecuaciones, la aplicación de la extensión del teorema de representación de Feynman-Kac a los casos semi-lineales y completamente no lineales, lo que da lugar a una representación discreta de la solución que puede implementarse de manera eficiente mediante el uso de redes neuronales artificiales (Texto tomado de la fuente).In this research, four models of illiquid financial markets are proposed and developed, characterizing the dynamics of risky asset prices and the emerging relationship between this dynamics and the trading strategies of agents. Additionally, the corresponding partial differential equations for the valuation of contingent assets are deduced. Specifically, the following models are presented: 1. A market model with a liquidity factor proportional to the asset price; 2. A model in which liquidity is a function of the asset price; 3. A model that includes proportional liquidity, with the presence of noise traders; 4. A model in which liquidity is stochastic and described by a mean-reverting square-root process. The resulting valuation equations are nonlinear extensions of the Black-Scholes partial differential equation, where the nonlinearity arises from the feedback effect associated with market illiquidity. As an alternative approach to solving these equations, the application of the extension of the Feynman-Kac representation theorem to semi-linear and fully nonlinear cases is proposed, leading to a discrete representation of the solution that can be efficiently implemented using artificial neural networks.DoctoradoDoctor en Ciencias EconómicasFinanzasx, 121 páginasapplication/pdfspaUniversidad Nacional de ColombiaBogotá - Ciencias Económicas - Doctorado en Ciencias EconómicasFacultad de Ciencias EconómicasBogotá, ColombiaUniversidad Nacional de Colombia - Sede Bogotá330 - Economía::332 - Economía financiera510 - Matemáticas::519 - Probabilidades y matemáticas aplicadas000 - Ciencias de la computación, información y obras generales::006 - Métodos especiales de computaciónDERIVADOS FINANCIEROSOPCIONES (FINANZAS)TEORIA DEL APRENDIZAJE COMPUTACIONALAPRENDIZAJE SUPERVISADO (APRENDIZAJE AUTOMATICO)ECUACIONES DIFERENCIALES NO LINEALESREDES NEURALES (COMPUTADORES)Derivative securitiesOptions (Finance)Computational learning theorySupervised learning (Machine learning)Differential equations, nonlinearNeural networks (Computer science)Mercado ilíquidoValoración de derivadosEcuaciones diferenciales parciales no linealesRepresentación de Feynman-KacRedes neuronales artificialesIlliquid marketDerivative valuationNonlinear partial differential equationsFeynman-Kac representationArtificial neural networksValoración no lineal de derivados financieros en mercados con iliquidez mediante extensiones del teorema de Feynman-Kac y algoritmos de aprendizaje automáticoNonlinear valuation of financial derivatives in illiquid markets using extensions of the Feynman-Kac theorem and machine learning algorithmsTrabajo de grado - Doctoradoinfo:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_db06Texthttp://purl.org/redcol/resource_type/TDAcharya, V. V. and Pedersen, L. H. (2005). Asset pricing with liquidity risk. Journal of financial Economics, 77(2):375–410.Albrecher, H., Guillaume, F., and Schoutens, W. (2013). Implied liquidity: model sensitivity. Journal of Empirical Finance, 23:48–67.Amihud, Y. (2002). Illiquidity and stock returns: Cross-section and time-series effects. Journal of Financial Markets, 5(1):31–56.Amihud, Y. (2019). Illiquidity and stock returns: A revisit. Critical Finance Review, (8):203–221.Amihud, Y. and Mendelson, H. (1986). Asset pricing and the bid-ask spread. Journal of Financial Economics, 17(2):223–249.Arenas, A. J., González-Parra, G., and Caraballo, B. M. (2013). A nonstandard finite difference scheme for a nonlinear black–scholes equation. Mathematical and Computer Modelling, 57(7-8):1663–1670.Avellaneda, M., Levy, A., and Parás, A. (1995). Pricing and hedging derivative securities in markets with uncertain volatilities. Applied Mathematical Finance, 2(2):73–88.Bachouch, A., Huré, C., Langrené, N., and Pham, H. (2021). Deep neural networks algorithms for stochastic control problems on finite horizon: numerical applications. Methodology and Computing in Applied Probability, pages 1–36.Back, K. (1993). Asymmetric information and options. The Review of Financial Studies, 6(3):435–472.Beck, C., Becker, S., Grohs, P., Jaafari, N., and Jentzen, A. (2021). Solving the kolmogorov pde by means of deep learning. Journal of Scientific Computing, 88:1–28.Beck, C., Hutzenthaler, M., Jentzen, A., and Kuckuck, B. (2020). An overview on deep learning-based approximation methods for partial differential equations. arXiv preprint arXiv:2012.12348.Beck, C., Weinan, E., and Jentzen, A. (2019). Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations. Journal of Nonlinear Science, 29(4):1563–1619.Bergman, Y. Z. (1995). Option pricing with differential interest rates. The Review of Financial Studies, 8(2):475–500.Berkman, H. and Koch, P. D. (2008). Noise trading and the price formation process. Journal of Empirical Finance, 15(2):232–250.Bertsimas, D. and Lo, A. W. (1998). Optimal control of execution costs. Journal of financial markets, 1(1):1–50.Bismut, J.-M. (1973). Conjugate convex functions in optimal stochastic control. Journal of Mathematical Analysis and Applications, 44(2):384–404.Björk, T. (2009). Arbitrage theory in continuous time. Oxford university press.Black, F. and Scholes, M. (1973a). The pricing of options and corporate liabilities. Journal of political Economy, 81(3):637–654.Black, F. and Scholes, M. (1973b). The pricing of options and corporate liabilities. Journal of political economy, 81(3):637–654.Bouchard, B. (2015). Lecture notes on BSDEs Main existence and stability results. PhD thesis, CEREMADE-CEntre de REcherches en MAthématiques de la DEcision.Brunetti, C. and Caldarera, A. (2004). Asset prices and asset correlations in illiquid markets. Available at SSRN 625184.Caldarera, A., Brunetti, C., et al. (2005). Asset prices and asset correlations in illiquid markets. In 2005 Meeting Papers, number 288. Society for Economic Dynamics.Cetin, U., Jarrow, R., Protter, P., and Warachka, M. (2006). Pricing options in an extended black scholes economy with illiquidity: Theory and empirical evidence. The Review of Financial Studies, 19(2):493–529.Cetin, U., Jarrow, R. A., and Protter, P. (2010). Liquidity risk and arbitrage pricing theory. Springer.Cheridito, P., Soner, H. M., Touzi, N., and Victoir, N. (2007). Second-order backward stochastic differential equations and fully nonlinear parabolic pdes. Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, 60(7):1081–1110.Chordia, T., Roll, R., and Subrahmanyam, A. (2000). Commonality in liquidity. Journal of Financial Economics, 56(1):3–28.Chordia, T., Roll, R., and Subrahmanyam, A. (2001). Market liquidity and trading activity. The Journal of Finance, 56(2):501–530.Company, R., Jódar, L., and Pintos, J.-R. (2010a). Numerical analysis and computing for option pricing models in illiquid markets. Mathematical and computer modelling, 52(7- 8):1066–1073.Company, R., Jódar, L., Ponsoda, E., and Ballester, C. (2010b). Numerical analysis and simulation of option pricing problems modeling illiquid markets. Computers & Mathematics with Applications, 59(8):2964–2975.Corcuera, J. M., Guillaume, F., Madan, D. B., and Schoutens, W. (2012). Implied liquidity: towards stochastic liquidity modelling and liquidity trading. International Journal of Portfolio Analysis and Management, 1(1):80–91.Cybenko, G. (1989). Approximation by superpositions of a sigmoidal function. Mathematics of control, signals and systems, 2(4):303–314.Duffie, D. (2010). Dynamic asset pricing theory. Princeton University Press.El Karoui, N. and Mazliak, L. (1997). Backward stochastic differential equations, volume 364. CRC Press.Esser, A. (2004). Pricing in (in) complete markets: Structural analysis and applications. Springer Science & Business Media.Fama, E. F. (1965). The behavior of stock-market prices. The journal of Business, 38(1):34–105.Feng, S.-P., Hung, M.-W., and Wang, Y.-H. (2014). Option pricing with stochastic liquidity risk: Theory and evidence. Journal of Financial Markets, 18:77–95.Forsyth, P. A. and Vetzal, K. R. (2012). Numerical methods for nonlinear pdes in finance. In Handbook of Computational Finance, pages 503–528. Springer.Frey, R. (1998). Perfect option hedging for a large trader. Finance and Stochastics, 2(2):115–141.Frey, R. (2000). Market illiquidity as a source of model risk in dynamic hedging. Model Risk, pages 125–138.Frey, R. and Patie, P. (2002). Risk management for derivatives in illiquid markets: A simulation study. Advances in finance and stochastics, pages 137–159.Frey, R. and Stremme, A. (1997). Market volatility and feedback effects from dynamic hedging. Mathematical finance, 7(4):351–374.Geiss, S. and Ylinen, J. (2021). Decoupling on the Wiener Space, related Besov Spaces, and applications to BSDEs, volume 272. American Mathematical Society.Gennotte, G. and Leland, H. (1990). Market liquidity, hedging, and crashes. The American Economic Review, pages 999–1021.Germain, M., Pham, H., and Warin, X. (2021). Neural networks-based algorithms for stochastic control and pdes in finance. arXiv preprint arXiv:2101.08068.Glover, K. J., Duck, P. W., and Newton, D. P. (2010). On nonlinear models of markets with finite liquidity: some cautionary notes. SIAM Journal on Applied Mathematics, 70(8):3252–3271.Gökay, S., Roch, A. F., and Soner, H. M. (2011). Liquidity models in continuous and discrete time. Springer.Gregory, J. (2015). The xVA Challenge: counterparty credit risk, funding, collateral and capital. John Wiley & Sons.Guéant, O. (2016). The Financial Mathematics of Market Liquidity: From optimal execution to market making, volume 33. CRC Press.Guo, J. andWang, W. (2015). On the numerical solution of nonlinear option pricing equation in illiquid markets. Computers & Mathematics with Applications, 69(2):117–133.Gurney, K. (2018). An introduction to neural networks. CRC pressGuyon, J. and Henry-Labordere, P. (2013). Nonlinear option pricing. CRC Press.Han, J., Jentzen, A., and E, W. (2018). Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences, 115(34):8505– 8510.Heider, P. (2010). Numerical methods for non-linear black–scholes equations. Applied Mathematical Finance, 17(1):59–81.Henry-Labordere, P. (2012). Counterparty risk valuation: A marked branching diffusion approach. arXiv preprint arXiv:1203.2369.Henry-Labordere, P., Tan, X., and Touzi, N. (2014). A numerical algorithm for a class of bsdes via the branching process. Stochastic Processes and their Applications, 124(2):1112–1140.Hornik, K., Stinchcombe, M., and White, H. (1989). Multilayer feedforward networks are universal approximators. Neural networks, 2(5):359–366.Hornik, K., Stinchcombe, M., and White, H. (1990). Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks. Neural networks, 3(5):551–560.Huberman, G. and Stanzl, W. (2004). Arbitrage-free price update and price-impact functions. Econometrica, 72(4):1247–1275.Huré, C., Pham, H., and Warin, X. (2020). Deep backward schemes for high-dimensional nonlinear pdes. Mathematics of Computation, 89(324):1547–1579.Jarrow, R. A. (1994). Derivative security markets, market manipulation, and option pricing theory. Journal of financial and quantitative analysis, 29(2):241–261.Karatzas, I. and Shreve, S. (2014). Brownian motion and stochastic calculus, volume 113. springer.Karatzas, I. and Shreve, S. E. (1998). Methods of mathematical finance, volume 39. Springer.Ku, H. and Zhang, H. (2018). Option pricing for a large trader with price impact and liquidity costs. Journal of Mathematical Analysis and Applications, 459(1):32–52.Kyle, A. S. (1985). Continuous auctions and insider trading. Econometrica: Journal of the Econometric Society, pages 1315–1335.Lagaris, I. E., Likas, A., and Fotiadis, D. I. (1998). Artificial neural networks for solving ordinary and partial differential equations. IEEE transactions on neural networks, 9(5):987–1000.Leippold, M. and Schärer, S. (2017). Discrete-time option pricing with stochastic liquidity. Journal of Banking & Finance, 75:1–16.Leland, H. E. (1985). Option pricing and replication with transactions costs. The journal of finance, 40(5):1283–1301.Li, Z., Zhang, W.-G., and Liu, Y.-J. (2018a). Analytical valuation for geometric asian options in illiquid markets. Physica A: Statistical Mechanics and its Applications, 507:175–191.Li, Z., Zhang, W.-G., and Liu, Y.-J. (2018b). European quanto option pricing in presence of liquidity risk. The North American Journal of Economics and Finance, 45:230–244.Liesenfeld, R. and Richard, J.-F. (2003). Univariate and multivariate stochastic volatility models: estimation and diagnostics. Journal of empirical finance, 10(4):505–531.Liu, H. and Yong, J. (2005). Option pricing with an illiquid underlying asset market. Journal of Economic Dynamics and Control, 29(12):2125–2156.Liu, W. (2006). A liquidity-augmented capital asset pricing model. Journal of financial Economics, 82(3):631–671.Lo, A. W., Mamaysky, H., and Wang, J. (2004). Asset prices and trading volume under fixed transactions costs. Journal of Political Economy, 112(5):1054–1090.Loeper, G. (2018). Option pricing with linear market impact and nonlinear black–scholes equations. The Annals of Applied Probability, 28(5):2664–2726.Longstaff, F. A., Mithal, S., and Neis, E. (2005). Corporate yield spreads: Default risk or liquidity? new evidence from the credit default swap market. The journal of finance, 60(5):2213–2253.Ludkovski, M. and Shen, Q. (2013). European option pricing with liquidity shocks. International Journal of Theoretical and Applied Finance, 16(07):1350043.Ma, J., Morel, J.-M., and Yong, J. (1999). Forward-backward stochastic differential equations and their applications. Number 1702. Springer Science & Business Media.Madan, C. and Cherny, A. (2010). Illiquid markets as a counterparty: An introduction to conic finance, robert h. Smith School Research Paper No. RHS, pages 06–115.Mandelbrot, B. B. (1997). The variation of certain speculative prices. Springer.McKean, H. P. (1975). Application of brownian motion to the equation of kolmogorov-petrovskii-piskunov. Communications on pure and applied mathematics, 28(3):323–331.Merton, R. C. (1973). Theory of rational option pricing. The Bell Journal of economics and management science, pages 141–183.Merton, R. C. and Samuelson, P. A. (1992). Continuous-time finance.Merton, R. C. and Samuelson, P. A. (1990). Continuous-time finance.Mönch, B. (2006). Strategic trading in illiquid markets, volume 553. Springer Science & Business Media.Nielsen, M. A. (2015). Neural networks and deep learning, volume 25. Determination pressSan Francisco, CA, USA.Oksendal, B. (2013). Stochastic differential equations: an introduction with applications. Springer Science & Business Media.Pardoux, E. and Peng, S. (1990). Adapted solution of a backward stochastic differential equation. Systems & control letters, 14(1):55–61.Pardoux, E. and Tang, S. (1999). Forward-backward stochastic differential equations and quasilinear parabolic pdes. Probability theory and related fields, 114:123–150.Pástor, L. and Stambaugh, R. F. (2003). Liquidity risk and expected stock returns. Journal of Political Economy, 111(3):642–685.Pham, H., Warin, X., and Germain, M. (2021). Neural networks-based backward scheme for fully nonlinear pdes. SN Partial Differential Equations and Applications, 2(1):16.Raissi, M. (2024). Forward–backward stochastic neural networks: deep learning of highdimensional partial differential equations. In Peter Carr Gedenkschrift: Research Advances in Mathematical Finance, pages 637–655. World Scientific.Raissi, M. and Karniadakis, G. E. (2018). Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357:125–141.Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics, 378:686–707.Ronnie Sircar, K. and Papanicolaou, G. (1998). General black-scholes models accounting for increased market volatility from hedging strategies. Applied Mathematical Finance, 5(1):45–82.Shreve, S. E. (2004). Stochastic calculus for finance II: Continuous-time models, volume 11. Springer Science & Business Media.Skorokhod, A. V. (1964). Branching diffusion processes. Theory of Probability & Its Applications, 9(3):445–449.Subramanian, A. and Jarrow, R. A. (2001). The liquidity discount. Mathematical Finance, 11(4):447–474.Tadmor, E. (2012). A review of numerical methods for nonlinear partial differential equations. Bulletin of the American Mathematical Society, 49(4):507–554.Taylor, S. J. (1994). Modeling stochastic volatility: A review and comparative study. Mathematical finance, 4(2):183–204.Tripathi, A., Dixit, A., et al. (2019). Liquidity of financial markets: a review. Studies in Economics and Finance.Trujillo, J. F. M. (2015). Modelos estocásticos en finanzas. U. Externado de Colombia.Vayanos, D. (2001). Strategic trading in a dynamic noisy market. The Journal of Finance, 56(1):131–171.Vayanos, D. (2004). Flight to quality, flight to liquidity, and the pricing of risk. Technical report, National bureau of economic research.Venegas-Martínez, F. (2008). Riesgos financieros y económicos, productos derivados y decisiones económicas bajo incertidumbre, volume 1. Escuela Superior de Economía, Instituto Politécnico NacionalWarin, X. (2018). Nesting monte carlo for high-dimensional non-linear pdes. Monte Carlo Methods and Applications, 24(4):225–247.Watanabe, S. (1965). On the branching process for brownian particles with an absorbing boundary. Journal of Mathematics of Kyoto University, 4(2):385–398.Weinan, E., Hutzenthaler, M., Jentzen, A., and Kruse, T. (2017). Linear scaling algorithms for solving high-dimensional nonlinear parabolic differential equations. SAM Research Report, 2017.Wilmott, P., Hoggard, T., and Whalley, A. E. (1994). Hedging option portfolios in the presence of transaction costs. Advances in Futures and Options Research, 7.Wilmott, P. and Schönbucher, P. J. (2000). The feedback effect of hedging in illiquid markets. SIAM Journal on Applied Mathematics, 61(1):232–272.Zhang, Y., Ding, S., and Duygun, M. (2019). Derivatives pricing with liquidity risk. Journal of Futures Markets, 39(11):1471–1485.EstudiantesInvestigadoresMaestrosPúblico generalLICENSElicense.txtlicense.txttext/plain; charset=utf-85879https://repositorio.unal.edu.co/bitstream/unal/86877/1/license.txteb34b1cf90b7e1103fc9dfd26be24b4aMD51ORIGINAL80059098.2024.pdf80059098.2024.pdfTesis de Doctorado en Ciencias Económicasapplication/pdf2004063https://repositorio.unal.edu.co/bitstream/unal/86877/2/80059098.2024.pdfc50d4a86bfab6b110e50d8903474ed09MD52THUMBNAIL80059098.2024.pdf.jpg80059098.2024.pdf.jpgGenerated Thumbnailimage/jpeg5440https://repositorio.unal.edu.co/bitstream/unal/86877/3/80059098.2024.pdf.jpg0b562b23bcd06e4f663a9bc377529abeMD53unal/86877oai:repositorio.unal.edu.co:unal/868772024-09-30 23:18:00.083Repositorio Institucional Universidad Nacional de Colombiarepositorio_nal@unal.edu.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

Publicaciones similares