Pricing and Hedging American-Style Options with Deep Learning

Becker, S.; Cheridito, Patrick; Jentzen, Arnulf

doi:10.3390/jrfm13070158

Cited by 54 publications

(43 citation statements)

References 24 publications

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“…For example, such approaches include approximating the Snell envelope or continuation values (cf., e.g., [89,5,28,75]), computing optimal exercise boundaries (cf., e.g., [2]) and dual methods (cf., e.g., [80,51]). Whereas in [51,66] artificial neural networks with one hidden layer were employed to approximate continuation values, more recently numerical approximation methods for American and Bermudan option pricing that are based on deep learning were introduced, cf., for example, [86,85,9,42,10,72,30]. More precisely, in [86,85] deep neural networks are used to approximately solve the corresponding obstacle partial differential equation problem, in [9] the corresponding optimal stopping problem is tackled directly with a deep learning-based algorithm, [42] applies an extension of the deep backward stochastic differential equation (BSDE) solver from [50,37] to the corresponding reflected BSDE problem, [30] suggests a different deep learning-based algorithm that relies on discretising BSDEs and in [10,72] deep neural network-based variants of the classical algorithm introduced by Longstaff & Schwartz [75] are examined.…”

Section: Introductionmentioning

confidence: 99%

Solving high-dimensional optimal stopping problems using deep learning

Becker¹,

Cheridito²,

Jentzen

et al. 2021

Eur. J. Appl. Math

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Nowadays many financial derivatives, such as American or Bermudan options, are of early exercise type. Often the pricing of early exercise options gives rise to high-dimensional optimal stopping problems, since the dimension corresponds to the number of underlying assets. High-dimensional optimal stopping problems are, however, notoriously difficult to solve due to the well-known curse of dimensionality. In this work, we propose an algorithm for solving such problems, which is based on deep learning and computes, in the context of early exercise option pricing, both approximations of an optimal exercise strategy and the price of the considered option. The proposed algorithm can also be applied to optimal stopping problems that arise in other areas where the underlying stochastic process can be efficiently simulated. We present numerical results for a large number of example problems, which include the pricing of many high-dimensional American and Bermudan options, such as Bermudan max-call options in up to 5000 dimensions. Most of the obtained results are compared to reference values computed by exploiting the specific problem design or, where available, to reference values from the literature. These numerical results suggest that the proposed algorithm is highly effective in the case of many underlyings, in terms of both accuracy and speed.

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Section: Introductionmentioning

confidence: 99%

Solving high-dimensional optimal stopping problems using deep learning

Becker¹,

Cheridito²,

Jentzen

et al. 2021

Eur. J. Appl. Math

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“…Additionally, the recent advances in machine learning has incentivized research in this particular area and allowed for extensions of these techniques to high-dimensional problems (cf. [KKT10], [BCJ19], [BCJ20], [GMZ20], [RW20]). The present article follows this stream of the literature and provides a variance-reduction technique that can be applied on top of numerous Monte Carlo based algorithms, therefore providing a powerful way to speed up these methods.…”

Section: Introductionmentioning

confidence: 99%

“…[GMZ20]) and (deep) neural network based stopping-time algorithms (cf. [BCJ19], [BCJ20]), respectively, with our JDOI variance reduction technique. Investigating these types of algorithms could be part of future research.…”

Section: Introductionmentioning

confidence: 99%

JDOI Variance Reduction Method and the Pricing of American-Style Options

2021

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The present article revisits the Diffusion Operator Integral (DOI) variance reduction technique originally proposed in [HP02] and extends its theoretical concept to the pricing of American-style options under (timehomogeneous) Lévy stochastic differential equations. The resulting Jump Diffusion Operator Integral (JDOI) method can be combined with numerous Monte Carlo based stopping-time algorithms, including the ubiquitous least-squares Monte Carlo (LSMC) algorithm of Longstaff and Schwartz (cf. [Car96], [LS01]). We exemplify the usefulness of our theoretical derivations under a concrete, though very general jump-diffusion stochastic volatility dynamics and test the resulting LSMC based version of the JDOI method. The results provide evidence of a strong variance reduction when compared with a simple application of the LSMC algorithm and proves that applying our technique on top of Monte Carlo based pricing schemes provides a powerful way to speed-up these methods.

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“…Motivated by the universal approximation theorems [8,9], nowadays ANNs are also being used to approximate solutions to ordinary differential equations (ODEs) or partial differential equations (PDEs) [5,[10][11][12]. Our contribution to this field consists of solving some PDEs that appear in computational finance applications with ANNs, following the unsupervised learning methodology introduced by [13] and refined in [14].…”

Section: Introductionmentioning

confidence: 99%

“…The authors in [14] extended the class of PDE solutions that may be approximated by these unsupervised learning methods, by translating the PDEs to a suitably weighted minimization problem for the ANNs to solve. Moreover, in [8,9] American options were formulated as optimal stopping problems, where optimal stopping decisions were learned and so-called ANN regression was used to estimate the continuation values. This is an example of the unsupervised learning approach to solve a specific formulation of options with early-exercise features.…”

Section: Introductionmentioning

confidence: 99%

Financial Option Valuation by Unsupervised Learning with Artificial Neural Networks

2020

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Artificial neural networks (ANNs) have recently also been applied to solve partial differential equations (PDEs). The classical problem of pricing European and American financial options, based on the corresponding PDE formulations, is studied here. Instead of using numerical techniques based on finite element or difference methods, we address the problem using ANNs in the context of unsupervised learning. As a result, the ANN learns the option values for all possible underlying stock values at future time points, based on the minimization of a suitable loss function. For the European option, we solve the linear Black–Scholes equation, whereas for the American option we solve the linear complementarity problem formulation. Two-asset exotic option values are also computed, since ANNs enable the accurate valuation of high-dimensional options. The resulting errors of the ANN approach are assessed by comparing to the analytic option values or to numerical reference solutions (for American options, computed by finite elements). In the short note, previously published, a brief introduction to this work was given, where some ideas to price vanilla options by ANNs were presented, and only European options were addressed. In the current work, the methodology is introduced in much more detail.

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Pricing and Hedging American-Style Options with Deep Learning

Cited by 54 publications

References 24 publications

Solving high-dimensional optimal stopping problems using deep learning

Solving high-dimensional optimal stopping problems using deep learning

JDOI Variance Reduction Method and the Pricing of American-Style Options

Financial Option Valuation by Unsupervised Learning with Artificial Neural Networks

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