Neural network approximation for superhedging prices

Biagini, Francesca; Gonon, Lukas; Reitsam, Thomas

doi:10.1111/mafi.12363

Cited by 3 publications

(1 citation statement)

References 25 publications

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“…(2021), Biagini et al. (2021) for parametric pricing results for European options, we refer to, for example, Andersson and Oosterlee (2021), Salvador et al. (2020), Becker et al.…”

Section: Numerical Examplesmentioning

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.

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“…(2021), Biagini et al. (2021) for parametric pricing results for European options, we refer to, for example, Andersson and Oosterlee (2021), Salvador et al. (2020), Becker et al.…”

Section: Numerical Examplesmentioning

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

View full text Add to dashboard Cite

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Option pricing in the Heston model with physics inspired neural networks

Hainaut,

Casas

2024

Ann Finance

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Valuation of guaranteed minimum accumulation benefits (GMABs) with physics-inspired neural networks

Hainaut

2024

Ann. actuar. sci.

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Guaranteed minimum accumulation benefits (GMABs) are retirement savings vehicles that protect the policyholder against downside market risk. This article proposes a valuation method for these contracts based on physics-inspired neural networks (PINNs), in the presence of multiple financial and biometric risk factors. A PINN integrates principles from physics into its learning process to enhance its efficiency in solving complex problems. In this article, the driving principle is the Feynman–Kac (FK) equation, which is a partial differential equation (PDE) governing the GMAB price in an arbitrage-free market. In our context, the FK PDE depends on multiple variables and is difficult to solve using classical finite difference approximations. In comparison, PINNs constitute an efficient alternative that can evaluate GMABs with various specifications without the need for retraining. To illustrate this, we consider a market with four risk factors. We first derive a closed-form expression for the GMAB that serves as a benchmark for the PINN. Next, we propose a scaled version of the FK equation that we solve using a PINN. Pricing errors are analyzed in a numerical illustration.

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Neural network approximation for superhedging prices

Cited by 3 publications

References 25 publications

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

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

Option pricing in the Heston model with physics inspired neural networks

Valuation of guaranteed minimum accumulation benefits (GMABs) with physics-inspired neural networks

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