1932

Abstract

We survey the active interface of statistical learning methods and quantitative finance models. Our focus is on the use of statistical surrogates, also known as functional approximators, for learning input–output relationships relevant for financial tasks. Given the disparate terminology used among statisticians and financial mathematicians, we begin by reviewing the main ingredients of surrogate construction and the motivating financial tasks. We then summarize the major surrogate types, including (deep) neural networks, Gaussian processes, gradient boosting machines, smoothing splines, and Chebyshev polynomials. The second half of the article dives deeper into the major applications of statistical learning in finance, covering () parametric option pricing, () learning the implied/local volatility surface, () learning option sensitivities, () American option pricing, and () model calibration. We also briefly detail statistical learning for stochastic control and reinforcement learning, two areas of research exploding in popularity in quantitative finance.

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2023-03-09
2024-10-09
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Literature Cited

  1. Ackerer D, Tagasovska N, Vatter T. 2020. Deep smoothing of the implied volatility surface. Adv. Neural Inform. Proc. Syst. 33:11552–63
    [Google Scholar]
  2. Agarwal A, Juneja S. 2015. Nearest neighbor based estimation technique for pricing Bermudan options. Int. Game Theory Rev. 17:11540002
    [Google Scholar]
  3. Aid R, Campi L, Langrené N, Pham H. 2014. A probabilistic numerical method for optimal multiple switching problems in high dimension. SIAM J. Financ. Math. 5:1191–231
    [Google Scholar]
  4. Bachouch A, Huré C, Langrené N, Pham H. 2022. Deep neural networks algorithms for stochastic control problems on finite horizon: numerical applications. Methodol. Comput. Appl. Probab. 24:1143–78
    [Google Scholar]
  5. Balata A, Ludkovski M, Maheshwari A, Palczewski J. 2021. Statistical learning for probability-constrained stochastic optimal control. Eur. J. Oper. Res. 290:2640–56
    [Google Scholar]
  6. Balata A, Palczewski J. 2017. Regress-later Monte Carlo for optimal inventory control with applications in energy. arXiv:1703.06461 [math.OC]
  7. Balata A, Palczewski J. 2018. Regress-later Monte Carlo for optimal control of Markov processes. arXiv:1712.09705 [math.OC]
  8. Baldacci B, Manziuk I, Mastrolia T, Rosenbaum M. 2019. Market making and incentives design in the presence of a dark pool: a deep reinforcement learning approach. arXiv:1912.01129 [q-fin.MF]
  9. Bayer C, Horvath B, Muguruza A, Stemper B, Tomas M. 2019. On deep calibration of (rough) stochastic volatility models. arXiv:1908.08806 [q-fin.MF]
  10. Becker S, Cheridito P, Jentzen A. 2019. Deep optimal stopping. J. Mach. Learn. Res. 20:2712–36
    [Google Scholar]
  11. Becker S, Cheridito P, Jentzen A. 2020. Pricing and hedging American-style options with deep learning. J. Risk Financ. Manag. 13:7158
    [Google Scholar]
  12. Belomestny D. 2011a. On the rates of convergence of simulation-based optimization algorithms for optimal stopping problems. Ann. Appl. Probab. 21:1215–39
    [Google Scholar]
  13. Belomestny D. 2011b. Pricing Bermudan options by nonparametric regression: optimal rates of convergence for lower estimates. Finance Stochast. 15:4655–83
    [Google Scholar]
  14. Belomestny D, Kolodko A, Schoenmakers J. 2010a. Regression methods for stochastic control problems and their convergence analysis. SIAM J. Control Optim. 48:53562–88
    [Google Scholar]
  15. Belomestny D, Milstein GN, Schoenmakers J. 2010b. Sensitivities for Bermudan options by regression methods. Decis. Econ. Finance 33:2117–38
    [Google Scholar]
  16. Belomestny D, Schoenmakers J. 2018. Advanced Simulation-Based Methods for Optimal Stopping and Control: With Applications in Finance London: Palgrave Macmillan
    [Google Scholar]
  17. Belomestny D, Schoenmakers J, Spokoiny V, Tavyrikov Y. 2018. Optimal stopping via deeply boosted backward regression. arXiv:1808.02341 [math.NA]
  18. Benth FE, Detering N, Lavagnini S. 2021. Accuracy of deep learning in calibrating HJM forward curves. Digit. Finance 3:3209–48
    [Google Scholar]
  19. Binois M, Gramacy RB, Ludkovski M. 2018. Practical heteroskedastic Gaussian process modeling for large simulation experiments. J. Comput. Graph. Stat. 27:4808–21
    [Google Scholar]
  20. Bouchard B, Warin X 2011. Monte-Carlo valorisation of American options: facts and new algorithms to improve existing methods. Numerical Methods in Finance R Carmona, PD Moral, P Hu, N Oudjane 215–55. Heidelberg, Ger: Springer
    [Google Scholar]
  21. Broadie M, Cao M. 2008. Improved lower and upper bound algorithms for pricing American options by simulation. Quant. Finance 8:8845–61
    [Google Scholar]
  22. Buehler H, Gonon L, Teichmann J, Wood B. 2019. Deep hedging. Quant. Finance 19:81271–91
    [Google Scholar]
  23. Cao J, Chen J, Hull J, Poulos Z. 2021. Deep hedging of derivatives using reinforcement learning. J. Financ. Data Sci. 3:110–27
    [Google Scholar]
  24. Capponi A, Lehalle CA, eds. 2022. Machine Learning in Financial Markets: A Guide to Contemporary Practice Cambridge, UK: Cambridge Univ. Press. In press
    [Google Scholar]
  25. Capriotti L, Giles M. 2012. Adjoint Greeks made easy. Risk 25:992–98
    [Google Scholar]
  26. Capriotti L, Jiang Y, Macrina A. 2017. AAD and least-square Monte Carlo: fast Bermudan-style options and XVA Greeks. Algorithmic Finance 6:1–235–49
    [Google Scholar]
  27. Carmona R, Laurière M. 2021. Deep learning for mean field games and mean field control with applications to finance. arXiv:2107.04568 [math.OC]
  28. Carmona R, Ludkovski M. 2010. Valuation of energy storage: an optimal switching approach. Quant. Finance 10:4359–74
    [Google Scholar]
  29. Carmona R, Touzi N. 2008. Optimal multiple stopping and valuation of swing options. Math. Finance 18:2239–68
    [Google Scholar]
  30. Charpentier A, Elie R, Remlinger C. 2021. Reinforcement learning in economics and finance. Comput. Econ. In press. https://doi.org/10.1007/s10614-021-10119-4
    [Google Scholar]
  31. Chataigner M. 2021. Some contributions of machine learning to quantitative finance: volatility, nowcasting, CVA compression. PhD Thesis, Univ. Paris-Saclay Paris:
    [Google Scholar]
  32. Chataigner M, Cousin A, Crépey S, Dixon M, Gueye D. 2021. Beyond surrogate modeling: learning the local volatility via shape constraints. SIAM J. Financ. Math. 12:3SC58–69
    [Google Scholar]
  33. Chataigner M, Crépey S, Dixon M. 2020. Deep local volatility. Risks 8:382
    [Google Scholar]
  34. Chen T, Ludkovski M. 2021. A machine learning approach to adaptive robust utility maximization and hedging. SIAM J. Financ. Math. 12:31226–56
    [Google Scholar]
  35. Chen V, Ruppert D, Shoemaker C. 1999. Applying experimental design and regression splines to high-dimensional continuous-state stochastic dynamic programming. Oper. Res. 47:138–53
    [Google Scholar]
  36. Cheridito P, Gersey B. 2021. Computation of conditional expectations with guarantees. arXiv:2112.01804 [stat.CO]
  37. Clément E, Lamberton D, Protter P. 2002. An analysis of a least squares regression algorithm for American option pricing. Finance Stochast. 6:449–71
    [Google Scholar]
  38. Cong F, Oosterlee CW. 2016. Multi-period mean–variance portfolio optimization based on Monte-Carlo simulation. J. Econ. Dyn. Control 64:23–38
    [Google Scholar]
  39. Cousin A, Maatouk H, Rullière D. 2016. Kriging of financial term-structures. Eur. J. Oper. Res. 255:2631–48
    [Google Scholar]
  40. Crépey S, Dixon MF. 2020. Gaussian process regression for derivative portfolio modeling and application to credit valuation adjustment computations. J. Comput. Finance 24:147–81
    [Google Scholar]
  41. Culkin R, Das SR. 2017. Machine learning in finance: the case of deep learning for option pricing. J. Invest. Manag. 15:492–100
    [Google Scholar]
  42. Davis J, Devos L, Reyners S, Schoutens W. 2020. Gradient boosting for quantitative finance. J. Comput. Finance 24:41–40
    [Google Scholar]
  43. De Spiegeleer J, Madan DB, Reyners S, Schoutens W. 2018. Machine learning for quantitative finance: fast derivative pricing, hedging and fitting. Quant. Finance 18:101635–43
    [Google Scholar]
  44. Deisenroth MP, Rasmussen CE, Peters J. 2009. Gaussian process dynamic programming. Neurocomputing 72:71508–24
    [Google Scholar]
  45. Deschatre T, Mikael J 2020. Deep combinatorial optimisation for optimal stopping time problems: application to swing options pricing. arXiv:2001.11247 [q-fin.CP]
  46. Dixon MF, Halperin I, Bilokon P. 2020. Machine Learning in Finance Cham, Switz: Springer
    [Google Scholar]
  47. Dugas C, Bengio Y, Bélisle F, Nadeau C, Garcia R. 2000. Incorporating second-order functional knowledge for better option pricing. Adv. Neural Inform. Proc. Syst. 13:472–78
    [Google Scholar]
  48. Dugas C, Bengio Y, Bélisle F, Nadeau C, Garcia R. 2009. Incorporating functional knowledge in neural networks. J. Mach. Learn. Res. 10:1239–62
    [Google Scholar]
  49. Duvenaud D. 2014. Automatic model construction with Gaussian processes PhD Thesis, Univ. Cambridge Cambridge, UK:
    [Google Scholar]
  50. Egloff D. 2005. Monte Carlo algorithms for optimal stopping and statistical learning. Ann. Appl. Probab. 15:21396–432
    [Google Scholar]
  51. Egloff D, Kohler M, Todorovic N. 2007. A dynamic look-ahead Monte Carlo algorithm for pricing Bermudan options. Ann. Appl. Probability 17:41138–71
    [Google Scholar]
  52. Elie R, Perolat J, Laurière M, Geist M, Pietquin O. 2020. On the convergence of model free learning in mean field games. Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 347143–50. Palo Alto, CA: AAAI
    [Google Scholar]
  53. Fecamp S, Mikael J, Warin X 2020. Deep learning for discrete-time hedging in incomplete markets. J. Comput. Finance 25:251–85
    [Google Scholar]
  54. Fengler MR. 2009. Arbitrage-free smoothing of the implied volatility surface. Quant. Finance 9:4417–28
    [Google Scholar]
  55. Ferguson R, Green AD. 2018. Deeply learning derivatives SSRN Work. Pap. 3244821
    [Google Scholar]
  56. Fromkorth A, Kohler M. 2011. Analysis of least squares regression estimates in case of additional errors in the variables. J. Stat. Plan. Inference 141:1172–88
    [Google Scholar]
  57. Fu H, Jin X, Pan G, Yang Y 2012. Estimating multiple option Greeks simultaneously using random parameter regression. J. Comput. Finance 16:285–118
    [Google Scholar]
  58. Garcia R, Gençay R. 2000. Pricing and hedging derivative securities with neural networks and a homogeneity hint. J. Econom. 94:1–293–115
    [Google Scholar]
  59. Gaß M, Glau K, Mahlstedt M, Mair M. 2018. Chebyshev interpolation for parametric option pricing. Finance Stochast. 22:3701–31
    [Google Scholar]
  60. Gatheral J. 2011. The Volatility Surface: A Practitioner's Guide. Hoboken, NJ: John Wiley & Sons
    [Google Scholar]
  61. Gençay R, Qi M. 2001. Pricing and hedging derivative securities with neural networks: Bayesian regularization, early stopping, and bagging. IEEE Trans. Neural Netw. 12:4726–34
    [Google Scholar]
  62. Germain M, Pham H, Warin X. 2021. Neural networks-based algorithms for stochastic control and PDEs in finance. arXiv:2101. [math.OC]
  63. Gevret H, Langrené N, Lelong J, Warin X, Maheshwari A. 2018. STochastic OPTimization library in C++ Res. Rep., EDF Lab. Paris:
    [Google Scholar]
  64. Giurca A, Borovkova S. 2021. Delta hedging of derivatives using deep reinforcement learning SSRN Work. Pap. 3847272
    [Google Scholar]
  65. Glasserman P. 2004. Monte Carlo Methods in Financial Engineering. New York: Springer
    [Google Scholar]
  66. Glasserman P, Yu B. 2004a. Number of paths versus number of basis functions in American option pricing. Ann. Appl. Probab. 14:42090–119
    [Google Scholar]
  67. Glasserman P, Yu B 2004b. Simulation for American options: regression now or regression later?. Monte Carlo and Quasi-Monte Carlo Methods 2002 H Niederreiter 213–26. Berlin: Springer
    [Google Scholar]
  68. Glau K, Herold P, Madan DB, Pötz C. 2019a. The Chebyshev method for the implied volatility. J. Comput. Finance 23:31–31
    [Google Scholar]
  69. Glau K, Kressner D, Statti F. 2020. Low-rank tensor approximation for Chebyshev interpolation in parametric option pricing. SIAM J. Financ. Math. 11:3897–927
    [Google Scholar]
  70. Glau K, Mahlstedt M. 2019. Improved error bound for multivariate Chebyshev polynomial interpolation. Int. J. Comput. Math. 96:112302–14
    [Google Scholar]
  71. Glau K, Mahlstedt M, Pötz C. 2019b. A new approach for American option pricing: the dynamic Chebyshev method. SIAM J. Sci. Comput. 41:1B153–80
    [Google Scholar]
  72. Gonon L, Schwab C. 2021. Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models. Finance Stochast. 25:4615–57
    [Google Scholar]
  73. Goudenège L, Molent A, Zanette A. 2019. Variance reduction applied to machine learning for pricing Bermudan/American options in high dimension. arXiv:1903.11275 [q-fin.CP]
  74. Goudenège L, Molent A, Zanette A. 2020. Machine learning for pricing American options in high-dimensional Markovian and non-Markovian models. Quant. Finance 20:4573–91
    [Google Scholar]
  75. Gramacy RB. 2020. Surrogates: Gaussian Process Modeling, Design, and Optimization for the Applied Sciences Boca Raton, FL: CRC
    [Google Scholar]
  76. Gramacy RB, Ludkovski M. 2015. Sequential design for optimal stopping problems. SIAM J. Financ. Math. 6:1748–75
    [Google Scholar]
  77. Guéant O, Manziuk I. 2019. Deep reinforcement learning for market making in corporate bonds: beating the curse of dimensionality. Appl. Math. Finance 26:5387–452
    [Google Scholar]
  78. Györfi L, Kohler M, Krzyzak A, Walk H. 2002. A Distribution-Free Theory of Nonparametric Regression, Vol. 1 New York: Springer
    [Google Scholar]
  79. Halperin I. 2020. QLBS: Q-learner in the Black-Scholes (-Merton) worlds. J. Deriv. 28:199–122
    [Google Scholar]
  80. Hambly B, Xu R, Yang H. 2021. Recent advances in reinforcement learning in finance. arXiv:2112.04553 [q-fin.MF]
  81. Han J, E W 2016. Deep learning approximation for stochastic control problems. arXiv:1611.07422 [cs.LG]
  82. Han J, Hu R. 2020. Deep fictitious play for finding Markovian Nash equilibrium in multi-agent games. Proc. Mach. Learn. Res. 107:221–45
    [Google Scholar]
  83. Haugh M, Kogan L. 2004. Pricing American options: a duality approach. Oper. Res. 52:2258–70
    [Google Scholar]
  84. Hornik K, Stinchcombe M, White H. 1990. Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks. Neural Netw. 3:5551–60
    [Google Scholar]
  85. Horvath B, Muguruza A, Tomas M 2021. Deep learning volatility: a deep neural network perspective on pricing and calibration in (rough) volatility models. Quant. Finance 21:111–27
    [Google Scholar]
  86. Hu R. 2019. Deep fictitious play for stochastic differential games. arXiv:1903.09376 [math.OC]
  87. Hu W, Zastawniak T. 2020. Pricing high-dimensional American options by kernel ridge regression. Quant. Finance 20:5851–65
    [Google Scholar]
  88. Huh J. 2019. Pricing options with exponential Lévy neural network. Expert Syst. Appl. 127:128–40
    [Google Scholar]
  89. Huré C, Pham H, Bachouch A, Langrené N. 2021. Deep neural networks algorithms for stochastic control problems on finite horizon: convergence analysis. SIAM J. Numer. Anal. 59:1525–57
    [Google Scholar]
  90. Hutchinson JM, Lo AW, Poggio T. 1994. A nonparametric approach to pricing and hedging derivative securities via learning networks. J. Finance 49:3851–89
    [Google Scholar]
  91. Itkin A. 2019. Deep learning calibration of option pricing models: some pitfalls and solutions. arXiv:1906.03507 [q-fin.CP]
  92. Jain S, Leitao Á, Oosterlee CW. 2019. Rolling adjoints: fast Greeks along Monte Carlo scenarios for early-exercise options. J. Comput. Sci. 33:95–112
    [Google Scholar]
  93. Jain S, Oosterlee CW. 2015. The stochastic grid bundling method: efficient pricing of Bermudan options and their Greeks. Appl. Math. Comput. 269:412–31
    [Google Scholar]
  94. James G, Witten D, Hastie T, Tibshirani R. 2013. An Introduction to Statistical Learning. New York: Springer
    [Google Scholar]
  95. Ke G, Meng Q, Finley T, Wang T, Chen W et al. 2017. LightGBM: A highly efficient gradient boosting decision tree. Adv. Neural Inform. Proc. Syst. 30:3146–54
    [Google Scholar]
  96. Kharroubi I, Langrené N, Pham H. 2014. A numerical algorithm for fully nonlinear HJB equations: an approach by control randomization. Monte Carlo Methods Appl. 20:2145–65
    [Google Scholar]
  97. Kirkby J, Deng S. 2019. Swing option pricing by dynamic programming with B-spline density projection. Int. J. Theor. Appl. Finance 22:081950038
    [Google Scholar]
  98. Kohler M. 2008. A regression-based smoothing spline Monte Carlo algorithm for pricing American options in discrete time. Adv. Stat. Anal. 92:2153–78
    [Google Scholar]
  99. Kohler M 2010. A review on regression-based Monte Carlo methods for pricing American options. Recent Developments in Applied Probability and Statistics L Devroye, B Karasözen, M Kohler, R Kornpp 37–58. Heidelberg, Ger: Springer
    [Google Scholar]
  100. Kohler M, Krzyżak A. 2012. Pricing of American options in discrete time using least squares estimates with complexity penalties. J. Stat. Plan. Inference 142:82289–307
    [Google Scholar]
  101. Kohler M, Krzyżak A, Todorovic N. 2010. Pricing of high-dimensional American options by neural networks. Math. Finance 20:3383–410
    [Google Scholar]
  102. Kolm PN, Ritter G. 2019. Dynamic replication and hedging: a reinforcement learning approach. J. Financ. Data Sci. 1:1159–71
    [Google Scholar]
  103. Laurière M 2021. Numerical methods for mean field games and mean field type control. Mean Field Games F Delarue 221–82. Providence, RI: Am. Math. Soc.
    [Google Scholar]
  104. Leal L, Laurière M, Lehalle CA. 2020. Learning a functional control for high-frequency finance. arXiv:2006.09611 [math.OC]
  105. Lemieux C. 2009. Monte Carlo and Quasi-Monte Carlo Sampling New York: Springer
    [Google Scholar]
  106. Liu S, Borovykh A, Grzelak LA, Oosterlee CW. 2019. A neural network-based framework for financial model calibration. J. Math. Ind. 9:19
    [Google Scholar]
  107. Longstaff FA, Schwartz ES. 2001. Valuing American options by simulation: a simple least-squares approach. Rev. Financ. Stud. 14:1113–47
    [Google Scholar]
  108. Ludkovski M. 2018. Kriging metamodels and experimental design for Bermudan option pricing. J. Comput. Finance 22:137–77
    [Google Scholar]
  109. Ludkovski M. 2020. mlOSP: Towards a unified implementation of regression Monte Carlo algorithms. arXiv:2012.00729 [q-fin.CP]
  110. Ludkovski M. 2021. mlOSP: Regression Monte Carlo algorithms for optimal stopping. R package version 1.0. https://github.com/mludkov/mlOSP
    [Google Scholar]
  111. Ludkovski M. 2022. Regression Monte Carlo for impulse control. Math. Action 11:73–90
    [Google Scholar]
  112. Ludkovski M, Maheshwari A. 2020. Simulation methods for stochastic storage problems: a statistical learning perspective. Energy Syst. 11:2377–415
    [Google Scholar]
  113. Ludkovski M, Saporito Y. 2022. KrigHedge: Gaussian process surrogates for Delta hedging. Appl. Math. Finance 28:4330–60
    [Google Scholar]
  114. Maran A, Pallavicini A, Scoleri S. 2021. Chebyshev Greeks: Smoothing Gamma without bias. SSRN Work. Pap. 3872744
  115. Mazières D, Boogert A. 2013. A radial basis function approach to gas storage valuation. J. Energy Mark. 6:219–50
    [Google Scholar]
  116. Meinshausen N, Hambly BM. 2004. Monte Carlo methods for the valuation of multiple-exercise options. Math. Finance 14:4557–83
    [Google Scholar]
  117. Nadarajah S, Margot F, Secomandi N. 2017. Comparison of least squares Monte Carlo methods with applications to energy real options. Eur. J. Oper. Res. 256:1196–204
    [Google Scholar]
  118. Olivares P, Alvarez A. 2016. Pricing basket options by polynomial approximations. J. Appl. Math. 2016:9747394
    [Google Scholar]
  119. Rasmussen CE, Williams CKI. 2006. Gaussian Processes for Machine Learning Cambridge, MA: MIT Press
    [Google Scholar]
  120. Reppen AM, Soner HM, Tissot-Daguette V. 2022. Neural optimal stopping boundary. arXiv:2205.04595 [q-fin.PR]
  121. Risk J, Ludkovski M. 2018. Sequential design and spatial modeling for portfolio tail risk measurement. SIAM J. Financ. Math. 9:41137–74
    [Google Scholar]
  122. Roustant O, Ginsbourger D, Deville Y. 2012. DiceKriging, DiceOptim: two R packages for the analysis of computer experiments by Kriging-based metamodeling and optimization. J. Stat. Softw. 51:11–55
    [Google Scholar]
  123. Ruf J, Wang W. 2020. Neural networks for option pricing and hedging: a literature review. J. Comput. Finance 24:11–46
    [Google Scholar]
  124. Ruf J, Wang W. 2022. Hedging with linear regressions and neural networks. J. Bus. Econ. Stat. 40:41442–54
    [Google Scholar]
  125. Ruppert D. 2004. Statistics and Finance: An Introduction New York: Springer
    [Google Scholar]
  126. Sirignano J, Spiliopoulos K. 2018. DGM: a deep learning algorithm for solving partial differential equations. J. Comput. Phys. 375:1339–64
    [Google Scholar]
  127. Tompaidis S, Yang C. 2013. Pricing American-style options by Monte Carlo simulation: alternatives to ordinary least squares. J. Comput. Finance 18:1121–43
    [Google Scholar]
  128. Tsitsiklis JN, van Roy B. 2001. Regression methods for pricing complex American-style options. IEEE Trans. Neural Netw. 12:4694–703
    [Google Scholar]
  129. Whalley AE, Wilmott P. 1997. An asymptotic analysis of an optimal hedging model for option pricing with transaction costs. Math. Finance 7:3307–24
    [Google Scholar]
  130. Yang Y, Zheng Y, Hospedales T 2017. Gated neural networks for option pricing: Rationality by design. Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence S Singh, S Markovitch 52–58. Palo Alto: AAAI Press
    [Google Scholar]
  131. Zanger DZ. 2018. Convergence of a least-squares Monte Carlo algorithm for American option pricing with dependent sample data. Math. Finance 28:1447–79
    [Google Scholar]
  132. Zhang R, Langrené N, Tian Y, Zhu Z, Klebaner F, Hamza K. 2019. Dynamic portfolio optimization with liquidity cost and market impact: a simulation-and-regression approach. Quant. Finance 19:3519–32
    [Google Scholar]
  133. Zheng Y, Yang Y, Chen B 2021. Incorporating prior financial domain knowledge into neural networks for implied volatility surface prediction. Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining3968–75. New York: Assoc. Comput. Mach.
    [Google Scholar]
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