Adaptive neural network surrogate model for solving the implied volatility of time-dependent American option via Bayesian inference

Yiyuan Qian; Kai Zhang; Jingzhi Li; Xiaoshen Wang; Yiyuan Qian; Kai Zhang; Jingzhi Li; Xiaoshen Wang

doi:10.3934/era.2022119

Electronic Research Archive

2022, Volume 30, Issue 6: 2335-2355. doi: 10.3934/era.2022119

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Adaptive neural network surrogate model for solving the implied volatility of time-dependent American option via Bayesian inference

1.
Department of Mathematics, Jilin University, Changchun 130012, China
2.
Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, China
3.
Department of Mathematics and Statistics, University of Arkansas at Little Rock, Arkansas 72204, USA

Received: 12 September 2021 Revised: 28 November 2021 Accepted: 28 February 2022 Published: 25 April 2022

In this paper, we propose an adaptive neural network surrogate method to solve the implied volatility of American put options, respectively. For the forward problem, we give the linear complementarity problem of the American put option, which can be transformed into several standard American put option problems by variable substitution and discretization in the temporal direction. Thus, the price of the option can be solved by primal-dual active-set method using numerical transformation and finite element discretization in spatial direction. For the inverse problem, we give the framework of the general Bayesian inverse problem, and adopt the direct Metropolis-Hastings sampling method and adaptive neural network surrogate method, respectively. We perform some simulations of volatility in the forward model with one- and four-dimension to compare the point estimates and posterior density distributions of two sampling methods. The superiority of adaptive surrogate method in solving the implied volatility of time-dependent American options are verified.
- adaptive neural network surrogate method,
- primal–dual active-set method,
- far–field technique,
- Bayesian inverse problem,
- Metropolis–Hastings sampling
Citation: Yiyuan Qian, Kai Zhang, Jingzhi Li, Xiaoshen Wang. Adaptive neural network surrogate model for solving the implied volatility of time-dependent American option via Bayesian inference[J]. Electronic Research Archive, 2022, 30(6): 2335-2355. doi: 10.3934/era.2022119

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Abstract

In this paper, we propose an adaptive neural network surrogate method to solve the implied volatility of American put options, respectively. For the forward problem, we give the linear complementarity problem of the American put option, which can be transformed into several standard American put option problems by variable substitution and discretization in the temporal direction. Thus, the price of the option can be solved by primal-dual active-set method using numerical transformation and finite element discretization in spatial direction. For the inverse problem, we give the framework of the general Bayesian inverse problem, and adopt the direct Metropolis-Hastings sampling method and adaptive neural network surrogate method, respectively. We perform some simulations of volatility in the forward model with one- and four-dimension to compare the point estimates and posterior density distributions of two sampling methods. The superiority of adaptive surrogate method in solving the implied volatility of time-dependent American options are verified.

References

[1]	M. Bergounioux, K. Ito, K. Kunisch, Primal–dual strategy for constrained optimal control problem, SIAM J. Control Optim., 37 (1999), 1176–1194. https://doi.org/10.1137/S0363012997328609c doi: 10.1137/S0363012997328609c
[2]	Y. Gao, J. Li, Y. Song, C. Wang, K. Zhang, Alternating direction based method for optimal control problem constrained by Stokes equation, J. Inverse Ill–posed Probl., 29 (2021), 249–266. https://doi.org/10.1515/jiip-2020-0101 doi: 10.1515/jiip-2020-0101
[3]	M. Hintermuller, K. Ito, K. Kunisch, The primal–dual active set strategy as a semi–smooth newton method, SIAM J. Control Optim., 13 (2003), 865–888. https://doi.org/10.1137/S1052623401383558 doi: 10.1137/S1052623401383558
[4]	H. Song, K. Zhang, Y. Li, Finite element and discontinuous Galerkin methods with perfect matched layers for American option, Numer. Math-Theory Methods Appl., 10 (2017), 829–851. https://doi.org/10.4208/nmtma.2017.0020 doi: 10.4208/nmtma.2017.0020
[5]	K. Zhang, H. Song, J. Li, Front–fixing FEMs for the pricing of American options based on a PML technique, Appl. Anal., 94 (2015), 903–931. https://doi.org/10.1080/00036811.2014.907563 doi: 10.1080/00036811.2014.907563
[6]	K. Ishihara, Projected successive overrelaxation method for finite–element solutions to the Dirichlet problem for a system of nonlinear elliptic equations, J. Comput. Appl. Math., 38 (1991), 185–200. https://doi.org/10.1016/0377-0427(91)90170-O doi: 10.1016/0377-0427(91)90170-O
[7]	D. Calvetti, E. Somersalo, Inverse problems: from regularization to Bayesian inference, Wiley Interdiscip Rev. Comput. Stat., 10 (2018), e127. https://doi.org/10.1002/wics.1427 doi: 10.1002/wics.1427
[8]	G. Ju, C. Chen, R. Chen, J. Li, K. Li, S. Zhang, Numerical simulation for 3D flow in flow channel of aeroengine turbine fan based on dimension splitting method, Electron. Res. Archive, 28 (2020), 837–851. https://doi.org/10.3934/era.2020043 doi: 10.3934/era.2020043
[9]	M. Li, L. Zhu, J. Li, K. Zhang, Design optimization of interconnected porous structures using extended triply periodic minimal surfaces, J. Comput. Phys., 425 (2021), 109909. https://doi.org/10.1016/j.jcp.2020.109909 doi: 10.1016/j.jcp.2020.109909
[10]	A. M. Stuart, Inverse problems: a Bayesian perspective, Acta Numerica, 19 (2010), 451–559. https://doi.org/10.1017/S0962492910000061 doi: 10.1017/S0962492910000061
[11]	C. Robert, G. Casella, Monte Carlo Statistical Methods, Springer–Verlag, New York, 2013. https://doi.org/10.1007/978-1-4757-4145-2
[12]	M. Xiong, L. Chen, J. Ming, J. Shin, Accelerating the Bayesian inference of inverse problems by using data–driven compressive sensing method based on proper orthogonal decomposition, Electron. Res. Archive, 29 (2021), 3383–3403. https://doi.org/10.3934/era.2021044 doi: 10.3934/era.2021044
[13]	B. D. Flury, Acceptance–rejection sampling made easy, SIAM Rev., 32 (1990), 474–476. https://doi.org/10.1137/1032082 doi: 10.1137/1032082
[14]	R. E. Liesenfeld, Importance sampling in structural systems, Struct. Saf., 6 (1989), 3–10. https://doi.org/10.1016/0167-4730(89)90003-9 doi: 10.1016/0167-4730(89)90003-9
[15]	D. Van Ravenzwaaij, P. Cassey, S. D. Brown, A simple introduction to Markov Chain Monte CCarlo sampling, Psychon. Bull. Rev., 25 (2018), 143–154. https://doi.org/10.3758/s13423-016-1015-8 doi: 10.3758/s13423-016-1015-8
[16]	D. Galbally, K. Fidkowski, K. Willcox, O. Ghattas, Non–linear model reduction for uncertainty quantilcation in large-scale inverse problems, Int. J. Numer. Methods Eng., 81 (2010), 1581–1608. https://doi.org/10.1002/nme.2746 doi: 10.1002/nme.2746
[17]	Y. M. Marzouk, H. N. Najm, Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems, J. Comput. Phys., 228 (2009), 1862–1902. https://doi.org/10.1016/j.jcp.2008.11.024 doi: 10.1016/j.jcp.2008.11.024
[18]	L. Yan, Y. Zhang, Convergence analysis of surrogate-based methods for Bayesian inverse problems, Inverse Probl., 33 (2017), 125001. https://doi.org/10.1088/1361-6420/aa9417 doi: 10.1088/1361-6420/aa9417
[19]	J. Berner, P. Grohs, A. Jentzen, Analysis of the generalization error: empirical risk minimization over deep artifcial neural networks overcomes the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations, SIAM J. Math. Data Sci., 2 (2020), 631–657. https://doi.org/10.1137/19M125649X doi: 10.1137/19M125649X
[20]	P. Grohs, F. Hornung, A. Jentzen, P. V. Wurstemberger, A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations, arXiv preprint, (2019), arXiv: 1809.02362.
[21]	J. Li, Y. M. Marzouk, Adaptive construction of surrogates for the Bayesian solution of inverse problems, SIAM J. Sci. Comput., 36 (2014), A1163–A1186. https://doi.org/10.1137/130938189 doi: 10.1137/130938189
[22]	A. D. Homes, H. Yang, A front–fixing finite element method for the valuation of American options, SIAM J. Sci. Comput., 30 (2008), 2158–2180. https://doi.org/10.1137/070694442 doi: 10.1137/070694442
[23]	H. Song, Q. Zhang, R. Zhang, A fast numerical method for the valuation of American lookback put options, Commun. Nonlinear Sci. Numer. Simul., 27 (2015), 302–313. https://doi.org/10.1016/j.cnsns.2015.03.010 doi: 10.1016/j.cnsns.2015.03.010
[24]	T. Deveney, E. Mueller, T. Shardlow, A deep surrogate approach to efficient Bayesian inversion in PDE and integral equation models, arXiv preprint, (2019), arXiv: 1910.01547.
[25]	Y. Li, J. M. G. Taylor, M. R. Elliott, A Bayesian approach to surrogacy assessment using principal stratification in clinical trials, Biometrics, 66 (2010), 523–531. https://doi.org/10.1111/j.1541-0420.2009.01303.x doi: 10.1111/j.1541-0420.2009.01303.x
[26]	L. Yan, T. Zhou, Adaptive multi–fidelity polynomial chaos approach to Bayesian inference in inverse problems, J. Comput. Phys., 381 (2019), 110–128. https://doi.org/10.1016/j.jcp.2018.12.025 doi: 10.1016/j.jcp.2018.12.025
[27]	P. S. Stanimirović, B. Ivanov, H. Ma, D. Mosić, A survey of gradient methods for solving nonlinear optimization, Electron. Res. Archive, 28 (2020), 1573–1624. https://doi.org/10.3934/era.2020115 doi: 10.3934/era.2020115
[28]	Y. Lecun, L. Bottou, G. B. Orr, Neural Networks: Tricks of the Trade, Springer–Verlag, Berlin, Heidelberg, 1998. https://doi.org/10.1007/3-540-49430-8
[29]	D. P. Kingma, J. Ba, Adam: A method for stochastic optimization, arXiv preprint, (2014), arXiv: 1412.6980.

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