Deep signature FBSDE algorithm

Qi Feng; Man Luo; Zhaoyu Zhang

doi:10.3934/naco.2022028

Article Contents

2023, Volume 13, Issue 3&4: 500-522. Doi: 10.3934/naco.2022028

This issue Previous Article From mean field games to Navier-Stokes equations Next Article Static Markowitz mean-variance portfolio selection model with long-term bonds

Deep signature FBSDE algorithm

1.
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, United States
2.
Guotai Junan, Road 655, Xinzha Rd, Bohua Plaza, Jing'An, Shanghai, China, 200071
3.
Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, United States

^*Corresponding author: Qi Feng
^*Corresponding author: Qi Feng

Received: November 2021

Revised: August 2022

Early access: September 2022

Published: September 2023

This work is dedicated to Professor Jin Ma's 65th birthday

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Abstract

We propose a deep signature/log-signature FBSDE algorithm to solve forward-backward stochastic differential equations (FBSDEs) with state and path dependent features. By incorporating the deep signature/log-signature transformation into the recurrent neural network (RNN) model, our algorithm shortens the training time, improves the accuracy, and extends the time horizon comparing to methods in the existing literature. Moreover, our algorithms can be applied to a wide range of applications such as state and path dependent option pricing involving high-frequency data, model ambiguity, and stochastic games, which are linked to parabolic partial differential equations (PDEs), and path-dependent PDEs (PPDEs). Lastly, we also derive the convergence analysis of the deep signature/log-signature FBSDE algorithm.

Keywords:

Forward-backward stochastic differential equations,
signature,
recurrent neural network,
option pricing.

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

Citation:

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Figure 1. Convergence on lookback option prices ($ T = 1 $) via different methods

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Figure 2. Option pricing errors across different methods and time steps

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Figure 3. Computation times over different methods and time steps

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Figure 4. High frequency long duration lookback option pricing example ($ T = 10 $)

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Figure 5. High frequency long duration lookback option pricing example ($ T = 10 $ zoomed plot)

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Figure 6. A better bound for bid and ask prices

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Figure 7. Zoomed plots for bid prices

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Figure 8. Zoomed plots for ask prices

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Figure 9. Nonlinear Example

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Table 1. Best ask price for GBM European call option

Simple NN $n=100$	Sig-LSTM $\tilde{n}=5$, $n=100$	Sig-LSTM $\tilde{n}=5$, $n=500$	Sig-LSTM $\tilde{n}=5$, $n=1000$	Sig-LSTM $\tilde{n}=5$, $n=5000$
25.526	25.48	25.46	25.46	25.45

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Table 2. Bid ask prices for European call under Heston model

MARS Bid $n=25$	MARS Bid $n=100$	Vanilla-LSTM Bid $n=100$	Vanilla-LSTM Bid $n=200$	Sig-LSTM Bid $\tilde{n}=5$, $n=100$	Sig-LSTM Bid $\tilde{n}=5$, $n=500$	Sig-LSTM Bid $\tilde{n}=5$, $n=5000$
9.74	9.62	9.59	9.58	9.53	9.527	9.50
MARS Ask $n=25$	MARS Ask $n=100$	Vanilla-LSTM Ask $n=100$	Vanilla-LSTM Ask $n=200$	Sig-LSTM Ask $\tilde{n}=5$, $n=100$	Sig-LSTM Ask $\tilde{n}=5$, $n=500$	Sig-LSTM Ask $\tilde{n}=5$, $n=5000$
12.16	12.25	12.17	12.41	12.48	12.52	12.57

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Table 3. $ Y_0$ in the Nonlinear Example

	$\tilde{n}=5$	$\tilde{n}=20$	$\tilde{n}=50$	$\tilde{n}=100$
$n=100$	0.986	0.9979	0.9988	–
$n=1000$	0.987	0.9982	0.9991	0.9997

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References

[1]	I. P. Arribas, Derivatives pricing using signature payoffs, preprint, 2018, arXiv: 1809.09466.
[2]	C. Bayer, P. Friz and J. Gatheral, Pricing under rough volatility, Quantitative Finance, 16 (2016), 887-904. doi: 10.1080/14697688.2015.1099717.
[3]	D. Becherer and K. Kentia, Good deal hedging and valuation under combined uncertainty about drift and volatility, Probability, Uncertainty and Quantitative Risk, 2 (2017), Article Number: 13. doi: 10.1186/s41546-017-0024-5.
[4]	S. Biagini and M. Ç. Pnar, The robust merton problem of an ambiguity averse investor, Mathematics and Financial Economics, 11 (2017), 1-24. doi: 10.1007/s11579-016-0168-6.
[5]	T. R. Bielecki, I. Cialenco, I. Iyigunler and R. Rodriguez, Dynamic conic finance: Pricing and hedging in market models with transaction costs via dynamic coherent acceptability indices, International Journal of Theoretical and Applied Finance, 16 (2013), 1350002. doi: 10.1142/S0219024913500027.
[6]	P. Carr, H. Geman and D. B. Madan, Pricing and hedging in incomplete markets, Journal of Financial Economics, 62 (2001), 131-167.
[7]	A. Cherny and D. Madan, New measures for performance evaluation, The Review of Financial Studies, 22 (2009), 2571-2606.
[8]	I. Chevyrev and H. Oberhauser, Signature moments to characterize laws of stochastic process, Journal of Machine Learning Research, 23 (2022), 1-42.
[9]	J. H. Cochrane and J. Saa-Requejo, Beyond arbitrage: Good-deal asset price bounds in incomplete markets, Journal of Political Economy, 108 (2000), 79-119.
[10]	S. N. Cohen and M. Tegnér, European option pricing with stochastic volatility models under parameter uncertainty, in International Symposium on BSDEs, Springer, (2007), 123-167. doi: 10.1007/978-3-030-22285-7_5.
[11]	B. Dupire, Functional itô calculus, Quantitative Finance, 19 (2019), 721-729. doi: 10.1080/14697688.2019.1575974.
[12]	Q. Feng and J. Zhang, Cubature method for volterra sdes and rough volatility model, preprint, 2021, arXiv: 2110.12853.
[13]	J.-P. Fouque and Z. Zhang, Deep learning methods for mean field control problems with delay, Frontiers in Applied Mathematics and Statistics, 6, 2020, Available at: https://doi.org/10.3389/fams.2020.00011. doi: 10.3389/FAMS.2020.00011.
[14]	P. K. Friz and N. B. Victoir, Multidimensional Stochastic Processes as Rough Paths: Theory and Applications, volume 120, Cambridge University Press, 2010. doi: 10.1017/CBO9780511845079.
[15]	K.-I. Funahashi and Y. Nakamura, Approximation of dynamical systems by continuous time recurrent neural networks, Neural Networks, 6 (1993), 801-806.
[16]	L. Garlappi, R. Uppal and T. Wang, Portfolio selection with parameter and model uncertainty: A multi-prior approach, The Review of Financial Studies, 20 (2007), 41-81.
[17]	L. P. Hansen and R. Jagannathan, Implications of security market data for models of dynamic economies, Journal of Political Economy, 99 (1991), 225-262.
[18]	S. Hochreiter and J. Schmidhuber, Long short-term memory, Neural Computation, 9 (1997), 1735-1780.
[19]	A. J. Jacquier and M. Oumgari, Deep ppdes for rough local stochastic volatility, Available at SSRN 3400035, 2019.
[20]	P. Kidger, P. Bonnier, I. P. Arribas, C. Salvi and T. Lyons, Deep signature transforms, in Advances in Neural Information Processing Systems, (2019), 3105-3115.
[21]	F. J. Király and H. Oberhauser, Kernels for sequentially ordered data, Journal of Machine Learning Research, 20 (2019), Paper No. 31, 45 pp.
[22]	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
[23]	S. Liao, T. Lyons, W. Yang and H. Ni, Learning stochastic differential equations using rnn with log signature features, preprint, 2019, arXiv: 1908.08286.
[24]	M. Luo and J. Ma, On the Dynamic Frontiers of the Limit Order Books-A Principal Agent Problem View, Ph.D Thesis, University of Southern California, 2021. doi: 10.1007/978-3-030-98519-6_16.
[25]	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.
[26]	D. B. Madan and A. S. Cherny, Illiquid markets as a counterparty: An introduction to conic finance, Robert H. Smith School Research Paper No. RHS, (2010), 6-115. doi: 10.1142/S0219024910006157.
[27]	R. C. Merton, Theory of rational option pricing, The Bell Journal of Economics and Management Science, (1973), 141-183.
[28]	M. Min and T. Ichiba, Convolutional signature for sequential data, Digital Finance, (2022), 1-26.
[29]	M. Musiela and M. Rutkowski, Martingale Methods in Financial Modelling, Springer, 2005.
[30]	E. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, in Stochastic Partial Differential Equations and Their Applications, Springer, (1992), 200-217. doi: 10.1007/BFb0007334.
[31]	S. Peng, Filtration consistent nonlinear expectations and evaluations of contingent claims, Acta Mathematicae Applicatae Sinica, English Series, 20 (2004), 191-214. doi: 10.1007/s10255-004-0161-3.
[32]	S. Peng and F. Wang, BSDE, path-dependent pde and nonlinear feynman-kac formula, Science China Mathematics, 59 (2016), 19-36. doi: 10.1007/s11425-015-5086-1.
[33]	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.
[34]	J. Reizenstein and B. Graham, Algorithm 1004: The iisignature library: Efficient calculation of iterated-integral signatures and log signatures, ACM Transactions on Mathematical Software (TOMS), 46 (2020), Article No. 8, 21 pages. doi: 10.1145/3371237.
[35]	J. Ruan, Numerical Methods for High-Dimensional Path-Dependent PDEs Driven by Stochastic Volterra Integral Equations, Ph.D Thesis, University of Southern California, Volume12/etd-RuanJie-8638, 2020.
[36]	M. Sabate-Vidales, D. Šiška and L. Szpruch, Solving path dependent PDEs with LSTM networks and path signatures, preprint, 2020, arXiv: 2011.10630.
[37]	Y. F. Saporito and Z. Zhang, PDGM: A neural network approach to solve path-dependent partial differential equations, SIAM Journal on Financial Mathematics, 12 (2021), 912-940. doi: 10.1137/20M1329597.
[38]	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.
[39]	Y. Yu, B. Hientzsch and N. Ganesan, Backward deep bsde methods and applications to nonlinear problems, preprint, 2020, arXiv: 2006.07635.
[40]	J. Zhang, Backward Stochastic Differential Equations, Springer, 2017. doi: 10.1007/978-1-4939-7256-2.