publication . Article . Preprint . 2018

Computation of Optimal Transport and Related Hedging Problems via Penalization and Neural Networks

Eckstein, Stephan; Kupper, Michael;
Open Access
  • Published: 23 Feb 2018 Journal: Applied Mathematics & Optimization (issn: 0095-4616, eissn: 1432-0606, Copyright policy)
  • Publisher: Springer Science and Business Media LLC
Abstract
This paper presents a widely applicable approach to solving (multi-marginal, martingale) optimal transport and related problems via neural networks. The core idea is to penalize the optimization problem in its dual formulation and reduce it to a finite dimensional one which corresponds to optimizing a neural network with smooth objective function. We present numerical examples from optimal transport, martingale optimal transport, portfolio optimization under uncertainty and generative adversarial networks that showcase the generality and effectiveness of the approach.
Subjects
free text keywords: Control and Optimization, Applied Mathematics, Numerical analysis, Duality (optimization), Optimization problem, Knightian uncertainty, Portfolio optimization, Mathematical optimization, Computation, Martingale (probability theory), Artificial neural network, Mathematics, Mathematics - Optimization and Control, Quantitative Finance - Mathematical Finance, Statistics - Machine Learning
40 references, page 1 of 3

[1] M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V. Vanhoucke, V. Vasudevan, F. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng. TensorFlow: Large-scale machine learning on heterogeneous systems, 2015. Software available from tensorflow.org.

[2] A. Alfonsi, J. Corbetta, and B. Jourdain. Sampling of probability measures in the convex order and approximation of martingale optimal transport problems. 2017. [OpenAIRE]

[3] M. Arjovsky, S. Chintala, and L. Bottou. Wasserstein GAN. arXiv preprint arXiv:1701.07875, 2017.

[4] P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath. Coherent measures of risk. Mathematical Finance, 9(3):203-228, 1999. [OpenAIRE]

[5] D. Bartl, P. Cheridito, M. Kupper, and L. Tangpi. Duality for increasing convex functionals with countably many marginal constraints. Banach Journal of Mathematical Analysis, 11(1):72-89, 2017.

[6] D. Bartl, S. Drapeau, and L. Tangpi. Computational aspects of robust optimized certainty equivalents. arXiv preprint arXiv:1706.10186, 2017.

[7] D. Bartl, M. Kupper, T. Lux, and A. Papapantoleon. Sharpness of improved Fréchet-Hoeffding bounds: an optimal transport approach. arXiv preprint arXiv:1709.00641, 2017. [OpenAIRE]

[8] M. Beiglböck, P. Henry-Labordère, and F. Penkner. Model-independent bounds for option prices: A mass transport approach. Finance and Stochastics, 17(3):477-501, 2013.

[9] A. Ben-Tal and M. Teboulle. An old-new concept of convex risk measures: The optimized certainty equivalent. Mathematical Finance, 17(3):449-476, 2007.

[10] J.-D. Benamou, G. Carlier, M. Cuturi, L. Nenna, and G. Peyré. Iterative Bregman projections for regularized transportation problems. SIAM Journal on Scientific Computing, 37(2):A1111- A1138, 2015.

[11] C. Bernard, X. Jiang, and R. Wang. Risk aggregation with dependence uncertainty. Insurance: Mathematics and Economics, 54:93-108, 2014.

[12] C. Bernard, L. Rüschendorf, S. Vanduffel, and J. Yao. How robust is the value-at-risk of credit risk portfolios? The European Journal of Finance, 23(6):507-534, 2017.

[13] H. Bühler, L. Gonon, J. Teichmann, and B. Wood. arXiv:1802.03042, 2018.

[14] G. Carlier, V. Duval, G. Peyré, and B. Schmitzer. Convergence of entropic schemes for optimal transport and gradient flows. SIAM Journal on Mathematical Analysis, 49(2):1385-1418, 2017.

[15] P. Cheridito, M. Kupper, and L. Tangpi. Representation of increasing convex functionals with countably additive measures. arXiv preprint arXiv:1502.05763, 2015. [OpenAIRE]

40 references, page 1 of 3
Abstract
This paper presents a widely applicable approach to solving (multi-marginal, martingale) optimal transport and related problems via neural networks. The core idea is to penalize the optimization problem in its dual formulation and reduce it to a finite dimensional one which corresponds to optimizing a neural network with smooth objective function. We present numerical examples from optimal transport, martingale optimal transport, portfolio optimization under uncertainty and generative adversarial networks that showcase the generality and effectiveness of the approach.
Subjects
free text keywords: Control and Optimization, Applied Mathematics, Numerical analysis, Duality (optimization), Optimization problem, Knightian uncertainty, Portfolio optimization, Mathematical optimization, Computation, Martingale (probability theory), Artificial neural network, Mathematics, Mathematics - Optimization and Control, Quantitative Finance - Mathematical Finance, Statistics - Machine Learning
40 references, page 1 of 3

[1] M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V. Vanhoucke, V. Vasudevan, F. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng. TensorFlow: Large-scale machine learning on heterogeneous systems, 2015. Software available from tensorflow.org.

[2] A. Alfonsi, J. Corbetta, and B. Jourdain. Sampling of probability measures in the convex order and approximation of martingale optimal transport problems. 2017. [OpenAIRE]

[3] M. Arjovsky, S. Chintala, and L. Bottou. Wasserstein GAN. arXiv preprint arXiv:1701.07875, 2017.

[4] P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath. Coherent measures of risk. Mathematical Finance, 9(3):203-228, 1999. [OpenAIRE]

[5] D. Bartl, P. Cheridito, M. Kupper, and L. Tangpi. Duality for increasing convex functionals with countably many marginal constraints. Banach Journal of Mathematical Analysis, 11(1):72-89, 2017.

[6] D. Bartl, S. Drapeau, and L. Tangpi. Computational aspects of robust optimized certainty equivalents. arXiv preprint arXiv:1706.10186, 2017.

[7] D. Bartl, M. Kupper, T. Lux, and A. Papapantoleon. Sharpness of improved Fréchet-Hoeffding bounds: an optimal transport approach. arXiv preprint arXiv:1709.00641, 2017. [OpenAIRE]

[8] M. Beiglböck, P. Henry-Labordère, and F. Penkner. Model-independent bounds for option prices: A mass transport approach. Finance and Stochastics, 17(3):477-501, 2013.

[9] A. Ben-Tal and M. Teboulle. An old-new concept of convex risk measures: The optimized certainty equivalent. Mathematical Finance, 17(3):449-476, 2007.

[10] J.-D. Benamou, G. Carlier, M. Cuturi, L. Nenna, and G. Peyré. Iterative Bregman projections for regularized transportation problems. SIAM Journal on Scientific Computing, 37(2):A1111- A1138, 2015.

[11] C. Bernard, X. Jiang, and R. Wang. Risk aggregation with dependence uncertainty. Insurance: Mathematics and Economics, 54:93-108, 2014.

[12] C. Bernard, L. Rüschendorf, S. Vanduffel, and J. Yao. How robust is the value-at-risk of credit risk portfolios? The European Journal of Finance, 23(6):507-534, 2017.

[13] H. Bühler, L. Gonon, J. Teichmann, and B. Wood. arXiv:1802.03042, 2018.

[14] G. Carlier, V. Duval, G. Peyré, and B. Schmitzer. Convergence of entropic schemes for optimal transport and gradient flows. SIAM Journal on Mathematical Analysis, 49(2):1385-1418, 2017.

[15] P. Cheridito, M. Kupper, and L. Tangpi. Representation of increasing convex functionals with countably additive measures. arXiv preprint arXiv:1502.05763, 2015. [OpenAIRE]

40 references, page 1 of 3
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