Unbiased Deep Solvers for Linear Parametric PDEs

Vidales, Marc Sabate; Šiška, David; Szpruch, Łukasz

doi:10.1080/1350486x.2022.2030773

Cited by 6 publications

(8 citation statements)

References 34 publications

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“…By providing an efficient means of approximating this calculation for a range of parameter choices, neural networks speed up the process of calibration, allowing a more efficient use of data. (Andreou et al, 2010;Bayer et al, 2019;Sabate-Vidales et al, 2018). These methods often depend on simulating option values from the handcrafted model, under a range of parameter values.…”

Section: Numericalmentioning

confidence: 99%

“…As mentioned above, using machine learning as a numerical tool introduces only modest model risks, while potentially providing significant speed and accuracy benefits. In Sabate-Vidales et al (2018, the authors developed deep learning algorithms for solving parametric families of (path-dependent) partial differential equations (P)PDEs that arise in pricing and hedging. The key idea in these works is to use a probabilistic representation of the (P)PDE, and learn both the solution and its gradient simultaneously.…”

Section: Machine Learning As a Numerical Toolmentioning

confidence: 99%

“…Hutchinson et al (1994) trained a neural network on simulated data to learn the Black-Scholes option pricing formula. A number of efficient algorithms have recently been developed to approximate parametric pricing operators with flexible modelling assumptions (for example, see Horvath et al (2020); Jacquier and Oumgari (2019); Ferguson and Green (2018); McGhee (2018); Sabate-Vidales et al (2018). This in turn can eliminate the calibration bottlenecks commonly found in using realistic pricing models.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Black-box model risk in finance

Cohen¹,

Snow²,

Szpruch³

2021

Preprint

Self Cite

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Machine learning models are increasingly used in a wide variety of financial settings. The difficulty of understanding the inner workings of these systems, combined with their wide applicability, has the potential to lead to significant new risks for users; these risks need to be understood and quantified. In this sub-chapter, we will focus on a well studied application of machine learning techniques, to pricing and hedging of financial options. Our aim will be to highlight the various sources of risk that the introduction of machine learning emphasises or de-emphasises, and the possible risk mitigation and management strategies that are available.

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Section: Numericalmentioning

confidence: 99%

Section: Machine Learning As a Numerical Toolmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Black-box model risk in finance

Cohen¹,

Snow²,

Szpruch³

2021

Preprint

Self Cite

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“…Many computational problems from engineering and science can be cast as certain parametric approximation problems (cf., e.g., Berner et al., 2020; Bhattacharya et al., 2021; Chkifa et al., 2015; Cohen & DeVore, 2015; Flandoli et al., 2021; Heinrich & Sindambiwe, 1999; Khoo et al., 2021; Kutyniok et al., 2021; O'Leary‐Roseberry et al., 2022; Vidales et al., 2018 and references mentioned therein). In particular, parametric PDEs are of fundamental importance in various applications, where one is not only interested in an approximation of the solution of the approximation problem at one fixed (space‐time) point but where one is interested to evaluate the approximative solution again and again as, for instance, in financial engineering where prices of financial products are computed approximately many times each trading day with (slightly) different parameters in each calculation.…”

Section: Introductionmentioning

confidence: 99%

“…For methods which are specifically designed for parametric PDEs we refer to, for example, Vidales et al. (2018), Khoo et al. (2021), Bhattacharya et al.…”

Section: Introductionmentioning

confidence: 99%

Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing

Becker,

Jentzen,

Müller

et al. 2023

Mathematical Finance

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In financial engineering, prices of financial products are computed approximately many times each trading day with (slightly) different parameters in each calculation. In many financial models such prices can be approximated by means of Monte Carlo (MC) simulations. To obtain a good approximation the MC sample size usually needs to be considerably large resulting in a long computing time to obtain a single approximation. A natural deep learning approach to reduce the computation time when new prices need to be calculated as quickly as possible would be to train an artificial neural network (ANN) to learn the function which maps parameters of the model and of the financial product to the price of the financial product. However, empirically it turns out that this approach leads to approximations with unacceptably high errors, in particular when the error is measured in the ‐norm, and it seems that ANNs are not capable to closely approximate prices of financial products in dependence on the model and product parameters in real life applications. This is not entirely surprising given the high‐dimensional nature of the problem and the fact that it has recently been proved for a large class of algorithms, including the deep learning approach outlined above, that such methods are in general not capable to overcome the curse of dimensionality for such approximation problems in the ‐norm. In this article we introduce a new numerical approximation strategy for parametric approximation problems including the parametric financial pricing problems described above and we illustrate by means of several numerical experiments that the introduced approximation strategy achieves a very high accuracy for a variety of high‐dimensional parametric approximation problems, even in the ‐norm. A central aspect of the approximation strategy proposed in this article is to combine MC algorithms with machine learning techniques to, roughly speaking, learn the random variables (LRV) in MC simulations. In other words, we employ stochastic gradient descent (SGD) optimization methods not to train parameters of standard ANNs but instead to learn random variables appearing in MC approximations. In that sense, the proposed LRV strategy has strong links to Quasi‐Monte Carlo (QMC) methods as well as to the field of algorithm learning. Our numerical simulations strongly indicate that the LRV strategy might indeed be capable to overcome the curse of dimensionality in the ‐norm in several cases where the standard deep learning approach has been proven not to be able to do so. This is not a contradiction to the established lower bounds mentioned above because this new LRV strategy is outside of the class of algorithms for which lower bounds have been established in the scientific literature. The proposed LRV strategy is of general nature and not only restricted to the parametric financial pricing problems described above, but applicable to a large class of approximation problems. In this article we numerically test the LRV strategy in the case of the pricing of European call options in the Black‐Scholes model with one underlying asset, in the case of the pricing of European worst‐of basket put options in the Black‐Scholes model with three underlying assets, in the case of the pricing of European average put options in the Black‐Scholes model with three underlying assets and knock‐in barriers, as well as in the case of stochastic Lorentz equations. For these examples the LRV strategy produces highly convincing numerical results when compared with standard MC simulations, QMC simulations using Sobol sequences, SGD‐trained shallow ANNs, and SGD‐trained deep ANNs.

show abstract

Black-Box Model Risk in Finance

Cohen¹,

Snow²,

Szpruch³

2023

Machine Learning and Data Sciences for Financial Markets

View full text Add to dashboard Cite

Machine learning models are increasingly used in a wide variety of financial settings. The difficulty of understanding the inner workings of these systems, combined with their wide applicability, has the potential to lead to significant new risks for users; these risks need to be understood and quantified. In this chapter, we will focus on a well studied application of machine learning techniques, to pricing and hedging of financial options. Our aim will be to highlight the various sources of risk that the introduction of machine learning emphasises or de-emphasises, and the possible risk mitigation and management strategies that are available.

show abstract

Unbiased Deep Solvers for Linear Parametric PDEs

Cited by 6 publications

References 34 publications

Black-box model risk in finance

Black-box model risk in finance

Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing

Black-Box Model Risk in Finance

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