Conclusions and outlook, Acknowledgments, and References

Table of Links

Abstract and 1. Introduction

2. Bermudan option pricing and hedging
3. Sparse Hermite polynomial expansion and gradient
4. Algorithm and complexity
5. Convergence analysis
6. Numerical examples
7. Conclusions and outlook, Acknowledgments, and References

7. Conclusions and outlook. In this work, we have proposed a novel gradient-enhanced least squares Monte Carlo (G-LSM) method that employs sparse Hermite polynomials as the ansatz space to price and hedge American options. The method enjoys low complexity for the gradient evaluation, ease of implementation and high accuracy for high-dimensional problems. We analyzed rigorously the convergence of G-LSM based on the BSDE technique, stochastic and Malliavin calculus. Extensive benchmark tests clearly show that it outperforms least squares Monte Carlo (LSM) in high dimensions with almost the same cost and it can also achieve competitive accuracy relative to the deep neural networks-based methods.

Acknowledgments. The authors acknowledge the support of research computing facilities offered by Information Technology Services, the University of Hong Kong.

REFERENCES

[1] B. Adcock, S. Brugiapaglia, and C. G. Webster, Sparse Polynomial Approximation of HighDimensional Functions, SIAM, Philadelphia, PA, 2022.

[2] C. Bayer, M. Eigel, L. Sallandt, and P. Trunschke, Pricing high-dimensional Bermudan options with hierarchical tensor formats, SIAM Journal on Financial Mathematics, 14 (2023), pp. 383–406.

[3] S. Becker, P. Cheridito, and A. Jentzen, Deep optimal stopping, The Journal of Machine Learning Research, 20 (2019), pp. 2712–2736.

[4] S. Becker, P. Cheridito, and A. Jentzen, Pricing and hedging American-style options with deep learning, Journal of Risk and Financial Management, 13 (2020), p. 158.

[5] B. Bouchard and X. Warin, Monte-Carlo valuation of American options: facts and new algorithms to improve existing methods, in Numerical Methods in Finance: Bordeaux, June 2010, Springer, Berlin, 2012, pp. 215–255.

[6] Y. Chen and J. W. Wan, Deep neural network framework based on backward stochastic differential equations for pricing and hedging American options in high dimensions, Quantitative Finance, 21 (2021), pp. 45–67.

[7] W. E, 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), pp. 349–380.

[8] N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng, and M.-C. Quenez, Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s, The Annals of Probability, 25 (1997), pp. 702–737.

[9] F. Fang and C. W. Oosterlee, A Fourier-based valuation method for Bermudan and barrier options under Heston’s model, SIAM Journal on Financial Mathematics, 2 (2011), pp. 439–463.

[10] C. Gao, S. Gao, R. Hu, and Z. Zhu, Convergence of the backward deep bsde method with applications to optimal stopping problems, SIAM Journal on Financial Mathematics, 14 (2023), pp. 1290–1303.

[11] E. Gobet and C. Labart, Error expansion for the discretization of backward stochastic differential equations, Stochastic Processes and their Applications, 117 (2007), pp. 803–829.

[12] C. Hur´e, H. Pham, and X. Warin, Deep backward schemes for high-dimensional nonlinear PDEs, Mathematics of Computation, 89 (2020), pp. 1547–1579.

[13] P. Kovalov, V. Linetsky, and M. Marcozzi, Pricing multi-asset American options: A finite element method-of-lines with smooth penalty, Journal of Scientific Computing, 33 (2007), pp. 209–237.

[14] B. Lapeyre and J. Lelong, Neural network regression for Bermudan option pricing, Monte Carlo Methods and Applications, 27 (2021), pp. 227–247.

[15] F. Longstaff and E. Schwartz, Valuing American options by simulation: a simple least-squares approach, The Review of Financial Studies, 14 (2001), pp. 113–147.

[16] M. Ludkovski, Kriging metamodels and experimental design for Bermudan option pricing, Journal of Computational Finance, 22 (2018), pp. 37–77.

[17] X. Luo, Error analysis of the Wiener–Askey polynomial chaos with hyperbolic cross approximation and its application to differential equations with random input, Journal of Computational and Applied Mathematics, 335 (2018), pp. 242–269.

[18] A. S. Na and J. W. L. Wan, Efficient pricing and hedging of high-dimensional American options using deep recurrent networks, Quantitative Finance, 23 (2023), pp. 631–651.

[19] A. M. Reppen, H. M. Soner, and V. Tissot-Daguette, Deep stochastic optimization in finance, Digital Finance, 5 (2023), pp. 91–111.

[20] R. Seydel and R. Seydel, Tools for computational finance, vol. 3, Springer, 2006.

[21] J. Sirignano and K. Spiliopoulos, DGM: A deep learning algorithm for solving partial differential equations, Journal of Computational Physics, 375 (2018), pp. 1339–1364.

[22] J. Tsitsiklis and B. Van Roy, Regression methods for pricing complex American-style options, IEEE Transactions on Neural Networks, 12 (2001), pp. 694–703.

[23] 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 preprint arXiv:1807.06622, (2018).

[24] Y. Wang and R. Caflisch, Pricing and hedging American-style options: a simple simulation-based approach, The Journal of Computational Finance, 13 (2009), pp. 95–125.

[25] J. Yang and G. Li, On sparse grid interpolation for American option pricing with multiple underlying assets, arXiv preprint arXiv:2309.08287, (2023).

[26] J. Zhang, Backward stochastic differential equations, Springer, 2017.