Solving high dimensional FBSDE with deep signature techniques with application to nonlinear options pricing

Hui Sun; Feng Bao

doi:10.3934/dcdsb.2024121

Article Contents

2025, Volume 30, Issue 4: 1083-1105. Doi: 10.3934/dcdsb.2024121

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Solving high dimensional FBSDE with deep signature techniques with application to nonlinear options pricing

Hui Sun^1, , and
Feng Bao^{1, 2,}

1.
Citibank, Markets and Global Function, Wilmington DE 19801, USA
2.
Department of Mathematics, Florida State University, FL 32306, USA

^*Corresponding author: Hui Sun
^*Corresponding author: Hui Sun

Disclaimer: the content of the paper is of the first author's own research interest and does not represent any of the corporate opinion

Received: January 2024

Revised: July 2024

Early access: September 18, 2024

Published online: September 18, 2024

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Abstract

We report two methods for solving FBSDEs of path-dependent types in high-dimensions. Inspired by the work from [31], [32], and [20], we propose deep learning frameworks for solving such problems using path signatures as underlying features. Our two methods (forward/backward) demonstrate comparable/better accuracy and efficiency compared to the state of the art [14], [13], and [18]. More importantly, leveraging the techniques developed in [5], we are able to solve the problem of high dimension (100 and above), which is a limitation in [14] and [13]. We also provide convergence proofs for both methods with the proof of the the forward method following similar lines to [14], and the backward methods inspired by [16] in the Markovian case.

Keywords:

Forward-backward stochastic differential equations,
signature,
Dimension reduction,
option pricing.

Mathematics Subject Classification: Primary: 65C30, 60H35; Secondary: 65M75.

Citation:

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Figure 1. Predicted $ Y_0 $ value versus the number of interations trained

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Figure 2. Predicted $ Y_0 $ value versus the number of interations trained

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Table 1. Example 1 results comparison

	Exact	Method in [14]	Forward method 1	Backward method 2
Result $ Y_0 $	0.5828	0.579	0.581	0.578
Error	–	0.6 %	0.3 %	0.8%

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Table 2. Example 2 result comparison, $ d = 20 $

$ d=20 $	Exact	Method in [14]	Forward method 1	Backward method 2
Result $ Y_0 $	6.66	6.6	6.58	6.71
Error	–	1 %	1.2 %	0.75%
$ d=100 $
Result $ Y_0 $	33.33	–	33.15	33.50
Error	–	NA	0.538 %	0.534%

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Table 3. Example 3 result comparison $ d = 1 $

d=1	European	Method in [13]	Method in [18]	Backward
Result $ Y_0 $	4.732	4.963	5.113	5.03
Confidence Interval	–	[4.896, 5.03]	[5.009, 5.217]	[4.97, 5.10]
d=5
Result $ Y_0 $	3.078	3.190	3.335	3.11
Confidence Interval	–	[3.115, 3.266]	[3.207, 3.462]	[3.08, 3.155]
d=10
Result $ Y_0 $	2.701	2.914	3.142	2.76
Confidence Interval	–	[2.844, 2.983]	[2.975, 3.309]	[2.734, 2.813]
d=20
Result $ Y_0 $	2.51	3.093	3.095	2.61
Confidence Interval	–	[3.017, 3.168]	[2.883, 3.3308]	[2.587, 2.627]

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References

[1]	R. Archibald, F. Bao, Y. Cao and H. Sun, Numerical Analysis for Convergence of a Sample-Wise Backpropagation Method for Training Stochastic Neural Networks, SIAM Journal on Numerical Analysis, 62 (2024), 593-621. doi: 10.1137/22M1523765.
[2]	A. Bachouch, C. Huré, N. Langrené and H. Pham, Deep neural networks algorithms for stochastic control problems on finite horizon, part 2: Numerical applications, Methodol. Comput. Appl. Probab., 24 (2022), 143-178.
[3]	S. Becker, P. Cheridito and A. Jentzen, Deep optimal stopping, Journal of Machine Learning Research, 20 (2019), 1-25.
[4]	S. Becker, P. Cheridito, A. Jentzen and T. Welti, Solving high-dimensional optimal stopping problems using deep learning, European Journal of Applied Mathematics, 32 (2021), 470-514. doi: 10.1017/S0956792521000073.
[5]	P. Bonnier, P. Kidger, I. P. Arribas, C. Salvi and T. Lyons, Deep Signature Transforms, arXiv: 1905.08494
[6]	I. Chevyrev and A. Kormilitzin, A primer on the signature method in machine learning, arXiv: 1603.03788.
[7]	B. Dupire, Functional Itô calculus, Quantitative Finance, 19 (2019), 721-729. doi: 10.1080/14697688.2019.1575974.
[8]	I. Ekren, Viscosity solutions of obstacle problems for fully nonlinear path-dependent PDEs, Stochastic Processes and their Applications, 127 (2017), 3966-3996. doi: 10.1016/j.spa.2017.03.016.
[9]	I. Ekren, C. Keller, N. Touzi and J. Zhang, On viscosity solutions of path-dependent PDEs, Ann. Probab., 42 (2014), 204-236. doi: 10.1214/12-AOP788.
[10]	I. Ekren, N. Touzi and J. Zhang, Optimal stopping under nonlinear expectation, Stochastic Processes and their Applications, 124 (2014), 3277-3311. doi: 10.1016/j.spa.2014.04.006.
[11]	I. Ekren, N. Touzi and J. Zhang, Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part Ⅰ, Ann. Probab., 44 (2016), 1212-1253.
[12]	I. Ekren, N. Touzi and J. Zhang, Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part Ⅱ, Ann. Probab., 44 (2016), 2507-2553. doi: 10.1214/15-AOP1027.
[13]	Q. Feng, E. Bayraktar and Z. Zhang, Deep signature algorithm for multidimensional path-dependent options, SIAM Journal on Financial Mathematics, 15 (2024). doi: 10.1137/23M1571563.
[14]	Q. Feng, M. Luo and Z. Zhang, Deep Signature FBSDE algorithm, Numerical Algebra, Control and Optimization, 13 (2023), 500-522. doi: 10.3934/naco.2022028.
[15]	J.-P. Fouque and Z. Zhang, Deep learning methods for mean field control problems with delay, Frontiers in Applied Mathematics and Statistics, 6 (2020), 00011. doi: 10.3389/fams.2020.00011.
[16]	C. Gao, S. Gao, R. Hu and Z. Zhu, Convergence of the backward deep bsde method with applications to optimal stopping problems, SIAM J. Financial Math., 14 (2023), 1290-1303.
[17]	C. Huré, H. Pham, A. Bachouch and N. Langrené, Deep neural networks algorithms for stochastic control problems on finite horizon, part Ⅰ: Convergence analysis, SIAM J. Numer. Anal., 59 (2021), 525-557.
[18]	C. Huré, H. Pham and X. Warin, Deep Backward schemes for high-dimensional nonlinear PDEs, Mathematics of Computation, 89 (2020), 1547-1579. doi: 10.1090/mcom/3514.
[19]	A. J. Jacquier and M. Oumgari, Deep ppdes for rough local stochastic volatility, Available at SSRN 3400035, 2019.
[20]	S. Liao, T. Lyons, W. Yang and H. Ni, Learning stochastic differential equations using rnn with log signature features, preprint, 2019. arXiv: 1908.08286.
[21]	D. Levin, T. Lyons and H. Ni, Learning from the past, predicting the statistics for the future, learning an evolving system, preprint, 2013. arXiv: 1309.0260.
[22]	T. Lyons, Differential equations driven by rough signals (i): An extension of an inequality of lc young, Mathematical Research Letters, 1 (1994), 451-464. doi: 10.4310/MRL.1994.v1.n4.a5.
[23]	M. Musiela and M. Rutkowski, Martingale Methods in Financial Modeling, , Springer, 2005. doi: 10.1007/b137866.
[24]	S. Peng, Backward stochastic differential equation, nonlinear expectation and their applications, Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010.
[25]	S. Peng and F. Wang, Path-dependent PDE and nonlinear Feynman-Kac formula, Sci. China Math., 59 (2016), 19-36. doi: 10.1007/s11425-015-5086-1.
[26]	T. Pham and J. Zhang, Two person zero-sum game in weak formulation and path dependent Bellman-Isaacs equation, SIAM J. Control Optim., 52 (2014), 2090-2121. doi: 10.1137/120894907.
[27]	J. Ruan, Numerical methods for high-dimensional path-dependent pdes driven by stochastic volterra integral equations, University of Southern California Dissertations and Theses, Volume12/etd-RuanJie-8638, 2020.
[28]	Z. Ren, N. Touzi and J. Zhang, An overview on viscosity solutions of path-dependent PDEs, Stochastic Analysis and Applications 2014 - In Honor of Terry Lyons, Springer Proceedings in Mathematics and Statistics, 100 (2014), 397-453. doi: 10.1007/978-3-319-11292-3_15.
[29]	M. Sabate-Vidales, D. Siska and L. Szpruch, Solving path dependent pdes with lstm networks and path signatures, preprint, 2020, arXiv: 2011.10630.
[30]	Y. F. Saporito and Z. Zhang, PDGM: A neural network approach to solve path- dependent partial differential equations, preprint, 2020, arXiv: 2003.02035.
[31]	H. Wang, H. Chen, A. Sudjianto, R. Liu and Q. Shen, Deep Learning Based BSDE solver for LIBOR market model with application to bermudan swaption pricing and hedging, arXiv: 1807.06622
[32]	E. Weinan, J. Han and A. Jentzen, Deep learning-based numerical methods for high dimensional parabolic partial differential equations and backward stochastic differential equations, communications in Mathematics and Statistics, 5 (2017), 349-380. doi: 10.1007/s40304-017-0117-6.
[33]	Y. Yu, B. Hientzsch and N. Ganesan, Backward deep bsde methods and applications to nonlinear problems, Preprint, 2020, arXiv: 2006.07635. doi: 0.2139/ssrn.3626208.
[34]	J. Zhang, Backward stochastic differential equations, from linear to fully nonlinear theory, Probability Theory and Stochastic Modeling, Springer, 86.