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Solving high-dimensional optimal stopping problems using deep learning

Published online by Cambridge University Press:  27 April 2021

RiskLab, Department of Mathematics, ETH Zürich, 8092Zürich, Switzerland;
RiskLab, Department of Mathematics, ETH Zürich, 8092Zürich, Switzerland;
Seminar for Applied Mathematics, Department of Mathematics, ETH Zürich, 8092Zürich, Switzerland and Faculty of Mathematics and Computer Science, University of Münster, 48149Münster, Germany email:
Seminar for Applied Mathematics, Department of Mathematics, ETH Zürich, 8092Zürich, Switzerland and D ONE Solutions AG, 8003Zürich, Switzerland email:
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Nowadays many financial derivatives, such as American or Bermudan options, are of early exercise type. Often the pricing of early exercise options gives rise to high-dimensional optimal stopping problems, since the dimension corresponds to the number of underlying assets. High-dimensional optimal stopping problems are, however, notoriously difficult to solve due to the well-known curse of dimensionality. In this work, we propose an algorithm for solving such problems, which is based on deep learning and computes, in the context of early exercise option pricing, both approximations of an optimal exercise strategy and the price of the considered option. The proposed algorithm can also be applied to optimal stopping problems that arise in other areas where the underlying stochastic process can be efficiently simulated. We present numerical results for a large number of example problems, which include the pricing of many high-dimensional American and Bermudan options, such as Bermudan max-call options in up to 5000 dimensions. Most of the obtained results are compared to reference values computed by exploiting the specific problem design or, where available, to reference values from the literature. These numerical results suggest that the proposed algorithm is highly effective in the case of many underlyings, in terms of both accuracy and speed.

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AitSahlia, F. & Carr, P. (1997) American options: a comparison of numerical methods. In: Rogers, L. C. G. and Talay, D. (editors) Numerical Methods in Finance. Publications of the Newton Institute, Vol. 13, Cambridge University Press, Cambridge, pp. 6787.CrossRefGoogle Scholar
Andersen, L. (2000) A simple approach to the pricing of Bermudan swaptions in the multi-factor LIBOR market model. J. Comput. Finance 3(2), 532.CrossRefGoogle Scholar
Andersen, L. & Broadie, M. (2004) Primal-dual simulation algorithm for pricing multidimensional American options. Manage. Sci. 50(9), 12221234.CrossRefGoogle Scholar
Bally, V. & Pagès, G. (2003) Error analysis of the optimal quantization algorithm for obstacle problems. Stochastic Process. Appl. 106(1), 140.CrossRefGoogle Scholar
Barraquand, J. & Martineau, D. (1995) Numerical valuation of high dimensional multivariate American securities. J. Financial. Quant. Anal. 30(3), 383405.CrossRefGoogle Scholar
Barron, A. R. (1993) Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inform. Theory 39(3), 930945.CrossRefGoogle Scholar
Bayer, C., Häppölä, J. & Tempone, R. (2019) Implied stopping rules for American basket options from Markovian projection. Quant. Finance 19(3), 371390.CrossRefGoogle Scholar
Bayer, C., Tempone, R. & Wolfers, S. (2018) Pricing American options by exercise rate optimization. ArXiv e-prints, 17 p. arXiv:1809.07300.Google Scholar
Becker, S., Cheridito, P. & Jentzen, A. (2019) Deep optimal stopping. J. Mach. Learn. Res. 20, Paper No. 74, 125.Google Scholar
Becker, S., Cheridito, P. & Jentzen, A. (2020) Pricing and hedging American-style options with deep learning. J. Risk Financial Manag. 13(7), 158, 1–12.CrossRefGoogle Scholar
Bellman, R. (1957) Dynamic Programming, Princeton University Press, Princeton, NJ.Google ScholarPubMed
Belomestny, D. (2011a) On the rates of convergence of simulation-based optimization algorithms for optimal stopping problems. Ann. Appl. Probab. 21(1), 215239.CrossRefGoogle Scholar
Belomestny, D. (2011b) Pricing Bermudan options by nonparametric regression: optimal rates of convergence for lower estimates. Finance Stoch. 15(4), 655683.CrossRefGoogle Scholar
Belomestny, D. (2013) Solving optimal stopping problems via empirical dual optimization. Ann. Appl. Probab. 23(5), 19882019.CrossRefGoogle Scholar
Belomestny, D., Bender, C. & Schoenmakers, J. (2009) True upper bounds for Bermudan products via non-nested Monte Carlo. Math. Finance 19(1), 5371.CrossRefGoogle Scholar
Belomestny, D., Dickmann, F. & Nagapetyan, T. (2015) Pricing Bermudan options via multilevel approximation methods. SIAM J. Financial Math. 6(1), 448466.CrossRefGoogle Scholar
Belomestny, D., Ladkau, M. & Schoenmakers, J. (2015) Multilevel simulation based policy iteration for optimal stopping—convergence and complexity. SIAM/ASA J. Uncertain. Quantif. 3(1), 460483.CrossRefGoogle Scholar
Belomestny, D., Schoenmakers, J. & Dickmann, F. (2013) Multilevel dual approach for pricing American style derivatives. Finance Stoch. 17(4), 717742.CrossRefGoogle Scholar
Bender, C., Kolodko, A. & Schoenmakers, J. (2006) Policy iteration for American options: overview. Monte Carlo Methods Appl. 12(5–6), 347362.CrossRefGoogle Scholar
Bender, C., Kolodko, A. & Schoenmakers, J. (2008) Enhanced policy iteration for American options via scenario selection. Quant. Finance 8(2), 135146.CrossRefGoogle Scholar
Bender, C., Schweizer, N. & Zhuo, J. (2017) A primal-dual algorithm for BSDEs. Math. Finance 27(3), 866901.CrossRefGoogle Scholar
Berridge, S. J. & Schumacher, J. M. (2008) An irregular grid approach for pricing high-dimensional American options. J. Comput. Appl. Math. 222(1), 94111.CrossRefGoogle Scholar
Bouchard, B. & Touzi, N. (2004) Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stochastic Process. Appl. 111(2), 175206.Google Scholar
Broadie, M. & Cao, M. (2008) Improved lower and upper bound algorithms for pricing American options by simulation. Quant. Finance 8(8), 845861.CrossRefGoogle Scholar
Broadie, M. & Glasserman, P. (1997) Pricing American-style securities using simulation. J. Econom. Dyn. Control 21(8–9), 13231352.CrossRefGoogle Scholar
Broadie, M. & Glasserman, P. (2004) A stochastic mesh method for pricing high-dimensional American options. J. Comput. Finance 7(4), 3572.CrossRefGoogle Scholar
Broadie, M., Glasserman, P. & Ha, Z. (2000) Pricing American options by simulation using a stochastic mesh with optimized weights. In: Probabilistic Constrained Optimization. Nonconvex Optimization and Its Applications, Vol. 49, Kluwer Academic Publishers, Dordrecht, pp. 2644.CrossRefGoogle Scholar
Carriere, J. F. (1996) Valuation of the early-exercise price for options using simulations and nonparametric regression. Insurance Math. Econom. 19(1), 1930.CrossRefGoogle Scholar
Chen, N. & Glasserman, P. (2007) Additive and multiplicative duals for American option pricing. Finance Stoch. 11(2), 153179.CrossRefGoogle Scholar
Chen, Y. & Wan, J. W. L. (2019) Deep neural network framework based on backward stochastic differential equations for pricing and hedging American options in high dimensions. ArXiv e-prints, 35 p. arXiv:1909.11532.Google Scholar
Christensen, S. (2014) A method for pricing American options using semi-infinite linear programming. Math. Finance 24(1), 156172.CrossRefGoogle Scholar
Company, R., Egorova, V., Jódar, L. & Soleymani, F. (2017) Computing stable numerical solutions for multidimensional American option pricing problems: a semi-discretization approach. ArXiv e-prints, 16 p. arXiv:1701.08545.Google Scholar
Cybenko, G. (1989) Approximation by superpositions of a sigmoidal function. Math. Control Signals Syst. 2(4), 303314.CrossRefGoogle Scholar
Da Prato, G. & Zabczyk, J. (1992) Stochastic Equations in Infinite Dimensions . Encyclopedia of Mathematics and its Applications, Vol. 44, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Davis, M. H. A. & Karatzas, I. (1994) A deterministic approach to optimal stopping. In: Probability, Statistics and Optimisation. Wiley Series in Probability and Mathematical Statistics, Wiley, Chichester, pp. 455466.Google Scholar
Desai, V. V., Farias, V. F. & Moallemi, C. C. (2012) Pathwise optimization for optimal stopping problems. Manage. Sci. 58(12), 22922308.CrossRefGoogle Scholar
E, W., Han, J. & Jentzen, A. (2017) Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Commun. Math. Stat. 5(4), 349380.Google Scholar
Egloff, D. (2005) Monte Carlo algorithms for optimal stopping and statistical learning. Ann. Appl. Probab. 15(2), 13961432.CrossRefGoogle Scholar
Egloff, D., Kohler, M. & Todorovic, N. (2007) A dynamic look-ahead Monte Carlo algorithm for pricing Bermudan options. Ann. Appl. Probab. 17(4), 11381171.CrossRefGoogle Scholar
Ferguson, T. S. (2021) Optimal Stopping and Applications: Chapter 1. Stopping Rule Problems. Mathematics Department, UCLA, available online at, last accessed on February 1, 2021.Google Scholar
Firth, N. P. (2005) High Dimensional American Options. PhD thesis, University of Oxford.Google Scholar
Fujii, M., Takahashi, A. & Takahashi, M. (2019) Asymptotic expansion as prior knowledge in deep learning method for high dimensional BSDEs. Asia-Pac. Financ. Markets 26, 391408.CrossRefGoogle Scholar
García, D. (2003) Convergence and biases of Monte Carlo estimates of American option prices using a parametric exercise rule. J. Econom. Dynam. Control 27(10), 18551879.CrossRefGoogle Scholar
Glasserman, P. (2004) Monte Carlo Methods in Financial Engineering. Applications of Mathematics (New York). Stochastic Modelling and Applied Probability, Vol. 53, Springer-Verlag, New York.Google Scholar
Glorot, X. & Bengio, Y. Understanding the difficulty of training deep feedforward neural networks. In: Y. W. Teh and M. Titterington (editors), Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics (2010-05). Proceedings of Machine Learning Research, PMLR, Vol. 9, pp. 249–256.Google Scholar
Gobet, E., Lemor, J.-P. & Warin, X. (2005) A regression-based Monte Carlo method to solve backward stochastic differential equations. Ann. Appl. Probab. 15(3), 21722202.CrossRefGoogle Scholar
Goldberg, D. A. & Chen, Y. (2018) Beating the curse of dimensionality in options pricing and optimal stopping. ArXiv e-prints, 62 p. arXiv:1807.02227.Google Scholar
Goudenège, L., Molent, A. & Zanette, A. (2019) Variance reduction applied to machine learning for pricing Bermudan/American options in high dimension. ArXiv e-prints, 25 p. arXiv:1903.11275.Google Scholar
Guyon, J. & Henry-Labordère, P. (2014) Nonlinear Option Pricing. Chapman & Hall/CRC Financial Mathematics Series, CRC Press, Boca Raton, FL.Google Scholar
Han, J., Jentzen, A. & E, W. (2018) Solving high-dimensional partial differential equations using deep learning. Proc. Natl. Acad. Sci. USA 115(34), 85058510.CrossRefGoogle ScholarPubMed
Haugh, M. B. & Kogan, L. (2004) Pricing American options: a duality approach. Oper. Res. 52(2), 258270.CrossRefGoogle Scholar
Hornik, K., Stinchcombe, M. & White, H. (1989) Multilayer feedforward networks are universal approximators. Neural Networks 2(5), 359366.CrossRefGoogle Scholar
Ioffe, S. & Szegedy, C. (2015) Batch normalization: accelerating deep network training by reducing internal covariate shift. In: F. Bach and D. Blei (editors), Proceedings of the 32nd International Conference on Machine Learning (2015-07). Proceedings of Machine Learning Research, PMLR, Vol. 37, pp. 448–456.Google Scholar
Jain, S. & Oosterlee, C. W. (2012) Pricing high-dimensional Bermudan options using the stochastic grid method. Int. J. Comput. Math. 89(9), 11861211.CrossRefGoogle Scholar
Jamshidian, F. (2007) The duality of optimal exercise an d domineering claims: a Doob-Meyer decomposition approach to the Snell envelope. Stochastics 79(1–2), 2760.CrossRefGoogle Scholar
Jentzen, A., Salimova, D. & Welti, T. (2017) Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations. ArXiv e-prints, 60 p. arXiv:1710.07123. Published in J. Math. Anal. Appl. Google Scholar
Jiang, D. R. & Powell, W. B. (2015) An approximate dynamic programming algorithm for monotone value functions. Oper. Res. 63(6), 14891511.CrossRefGoogle Scholar
Kallsen, J. (2009) Option Pricing, Springer, Berlin, Heidelberg, pp. 599613.Google Scholar
Karatzas, I. & Shreve, S. E. (1991) Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics, Vol. 113, Springer-Verlag, New York.Google Scholar
Kingma, D. P. & Ba, J. (2014) Adam: a method for stochastic optimization. ArXiv e-prints, 15 p. arXiv:1412.6980. Published as a conference paper at ICLR 2015.Google Scholar
Klenke, A. (2014) Probability Theory. A Comprehensive Course, 2nd ed. Universitext, Springer, London.CrossRefGoogle Scholar
Kloeden, P. E. & Platen, E. (1992) Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York), Vol. 23, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Kohler, M. (2008) A regression-based smoothing spline Monte Carlo algorithm for pricing American options in discrete time. AStA Adv. Stat. Anal. 92(2), 153178.CrossRefGoogle Scholar
Kohler, M. (2010) A review on regression-based Monte Carlo methods for pricing American options. In: Recent Developments in Applied Probability and Statistics, Physica, Heidelberg, pp. 3758.CrossRefGoogle Scholar
Kohler, M. & Krzyżak, A. (2012) Pricing of American options in discrete time using least squares estimates with complexity penalties. J. Statist. Plann. Inference 142(8), 22892307.CrossRefGoogle Scholar
Kohler, M., Krzyżak, A. & Todorovic, N. (2010) Pricing of high-dimensional American options by neural networks. Math. Finance 20(3), 383410.CrossRefGoogle Scholar
Kohler, M., Krzyżak, A. & Walk, H. (2008) Upper bounds for Bermudan options on Markovian data using nonparametric regression and a reduced number of nested Monte Carlo steps. Statist. Decisions 26(4), 275288.Google Scholar
Kolodko, A. & Schoenmakers, J. (2006) Iterative construction of the optimal Bermudan stopping time. Finance Stoch. 10(1), 2749.CrossRefGoogle Scholar
Kulikov, A. V. & Gusyatnikov, P. P. (2016) Stopping times for fractional Brownian motion. In: Fonseca, R. J., Weber, G.-W. and Telhada, J. (editors), Computational Management Science: State of the Art 2014. Lecture Notes in Economics and Mathematical Systems, Vol. 682, Springer International Publishing, pp. 195200.CrossRefGoogle Scholar
Labart, C. & Lelong, J. (2011) A parallel algorithm for solving BSDEs - application to the pricing and hedging of American options. ArXiv e-prints, 25 p. arXiv:1102.4666. Published in Monte Carlo Methods Appl. Google Scholar
Lamberton, D. & Lapeyre, B. (2008) Introduction to Stochastic Calculus Applied to Finance, 2nd ed. Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
Lapeyre, B. & Lelong, J. (2019) Neural network regression for Bermudan option pricing. ArXiv e-prints, 23 p. arXiv:1907.06474.Google Scholar
Lelong, J. (2018) Dual pricing of American options by Wiener chaos expansion. SIAM J. Financial Math. 9(2), 493519.CrossRefGoogle Scholar
Lelong, J. (2019) Pricing path-dependent Bermudan options using Wiener chaos expansion: an embarrassingly parallel approach. ArXiv e-prints, 23 p. arXiv:1901.05672.Google Scholar
Longstaff, F. A. & Schwartz, E. S. (2001) Valuing American options by simulation: a simple least-squares approach. Rev. Financ. Stud. 14(1), 113147.CrossRefGoogle Scholar
Lord, R., Fang, F., Bervoets, F. & Oosterlee, C. W. (2008) A fast and accurate FFT-based method for pricing early-exercise options under Lévy processes. SIAM J. Sci. Comput. 30(4), 16781705.CrossRefGoogle Scholar
Maruyama, G. (1955) Continuous Markov processes and stochastic equations. Rend. Circ. Mat. Palermo (2) 4, 4890.CrossRefGoogle Scholar
Øksendal, B. (2003) Stochastic Differential Equations. An Introduction with Applications, 6th ed. Universitext, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Peskir, G. & Shiryaev, A. (2006) Optimal Stopping and Free-Boundary Problems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel.Google Scholar
Rogers, L. C. G. (2002) Monte Carlo valuation of American options. Math. Finance 12(3), 271286.CrossRefGoogle Scholar
Rogers, L. C. G. (2010) Dual valuation and hedging of Bermudan options. SIAM J. Financial Math. 1(1), 604608.CrossRefGoogle Scholar
Schoenmakers, J., Zhang, J. & Huang, J. (2013) Optimal dual martingales, their analysis, and application to new algorithms for Bermudan products. SIAM J. Financial Math. 4(1), 86116.CrossRefGoogle Scholar
Schweizer, M. (2002) On Bermudan options. In: Advances in Finance and Stochastics, Springer, Berlin, pp. 257270.CrossRefGoogle Scholar
Shreve, S. E. (2004) Stochastic Calculus for Finance II. Continuous-Time Models. Springer Finance, Springer-Verlag, New York.Google Scholar
Sirignano, J. & Spiliopoulos, K. (2017) Stochastic gradient descent in continuous time. SIAM J. Financial Math. 8(1), 933961.CrossRefGoogle Scholar
Sirignano, J. & Spiliopoulos, K. (2018) DGM: a deep learning algorithm for solving partial differential equations. J. Comput. Phys. 375, 13391364.CrossRefGoogle Scholar
Smirnov, M. Javascript options and implied volatility calculator. Columbia University. Last accessed on February 1, 2021.Google Scholar
Solan, E., Tsirelson, B. & Vieille, N. (2012) Random stopping times in stopping problems and stopping games. ArXiv e-prints, 21 p. arXiv:1211.5802.Google Scholar
Tilley, J. A. (1993) Valuing American options in a path simulation model. Trans. Soc. Actuaries 45, 83104.Google Scholar
Tsitsiklis, J. N. & Van Roy, B. (1999) Optimal stopping of Markov processes: Hilbert space theory, approximation algorithms, and an application to pricing high-dimensional financial derivatives. IEEE Trans. Automat. Control 44(10), 18401851.CrossRefGoogle Scholar
Tsitsiklis, J. N. & Van Roy, B. (2001) Regression methods for pricing complex American-style options. IEEE Trans. Neural Networks 12(4), 694703.CrossRefGoogle ScholarPubMed
Wang, H., Chen, H., Sudjianto, A., Liu, R. & Shen, Q. (2018) Deep learning-based BSDE solver for Libor market model with application to Bermudan swaption pricing and hedging. ArXiv e-prints, 36 p. arXiv:1807.06622.CrossRefGoogle Scholar