Deep PPDEs for Rough Local Stochastic Volatility

Jacquier, Antoine; Oumgari, Mugad

doi:10.2139/ssrn.3400035

Cited by 33 publications

(36 citation statements)

References 70 publications

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“…One case study involves the pricing of European options on 100 defaultable underlying assets. There are several recent papers, such as Henry-Labordère [2017], Sirignano andSpiliopoulos [2018], Chan-Wai-Nam et al [2019], Huré et al [2019] , Jacquier and Oumgari [2019], and Vidales et al [2019], who have developed this application of ANNs further.…”

Section: Solving Partial Differential Equationsmentioning

confidence: 99%

Neural networks for option pricing and hedging: a literature review

Ruf¹,

Wang²

2020

JCF

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Neural networks have been used as a nonparametric method for option pricing and hedging since the early 1990s. Far over a hundred papers have been published on this topic. This note intends to provide a comprehensive review. Papers are compared in terms of input features, output variables, benchmark models, performance measures, data partition methods, and underlying assets. Furthermore, related work and regularisation techniques are discussed.

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Section: Solving Partial Differential Equationsmentioning

confidence: 99%

Neural networks for option pricing and hedging: a literature review

Ruf¹,

Wang²

2020

JCF

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“…Remark 2.5. An alternative lift approach, in the spirit of [2,14,18,21,28], consists in introducing the double-indexed (controlled) processes…”

Section: Markovian Representationmentioning

confidence: 99%

Linear-quadratic control for a class of stochastic Volterra equations: Solvability and approximation

2021

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“…Recently, several new stochastic approximation methods for certain classes of high-dimensional nonlinear PDEs have been proposed and studied in the scientific literature. In particular, we refer, e.g., to [11,12,26,29,30,53] for BSDE-based approximation methods for PDEs in which nested conditional expectations are discretized through suitable regression methods, we refer, e.g., to [10,39,41,42] for branching diffusion approximation methods for PDEs, we refer, e.g., to [1][2][3][6][7][8]13,14,16,17,21,24,25,31,[34][35][36]40,43,48,50,52,[54][55][56][57][58]60,62,63] for deep learning based approximation methods for PDEs, and we refer to [4,5,20,28,46,47] for numerical simulations, approximation results, and extensions of the in…”

Section: Introductionmentioning

confidence: 99%

“…For MLP approximation methods it has been recently shown in [4,45,46] that such algorithms do indeed overcome the curse of dimensionality for certain classes of gradient-independent PDEs. Numerical simulations for deep learning based approximation methods for nonlinear PDEs in high dimensions are very encouraging (see, e.g., the above named references [1][2][3][6][7][8]13,14,16,17,21,24,25,31,[34][35][36]40,43,48,50,52,[54][55][56][57][58]60,62,63]) but so far there is only partial error analysis available for such algorithms (which, in turn, is strongly based on the above-mentioned error analysis for the MLP approximation method; cf. [44] and, e.g., [9,23,32,33,36,49,51,61,62]).…”

Section: Introductionmentioning

confidence: 99%

Overcoming the Curse of Dimensionality in the Numerical Approximation of Parabolic Partial Differential Equations with Gradient-Dependent Nonlinearities

2021

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Partial differential equations (PDEs) are a fundamental tool in the modeling of many real-world phenomena. In a number of such real-world phenomena the PDEs under consideration contain gradient-dependent nonlinearities and are high-dimensional. Such high-dimensional nonlinear PDEs can in nearly all cases not be solved explicitly, and it is one of the most challenging tasks in applied mathematics to solve high-dimensional nonlinear PDEs approximately. It is especially very challenging to design approximation algorithms for nonlinear PDEs for which one can rigorously prove that they do overcome the so-called curse of dimensionality in the sense that the number of computational operations of the approximation algorithm needed to achieve an approximation precision of size $${\varepsilon }> 0$$ ε > 0 grows at most polynomially in both the PDE dimension $$d \in \mathbb {N}$$ d ∈ N and the reciprocal of the prescribed approximation accuracy $${\varepsilon }$$ ε . In particular, to the best of our knowledge there exists no approximation algorithm in the scientific literature which has been proven to overcome the curse of dimensionality in the case of a class of nonlinear PDEs with general time horizons and gradient-dependent nonlinearities. It is the key contribution of this article to overcome this difficulty. More specifically, it is the key contribution of this article (i) to propose a new full-history recursive multilevel Picard approximation algorithm for high-dimensional nonlinear heat equations with general time horizons and gradient-dependent nonlinearities and (ii) to rigorously prove that this full-history recursive multilevel Picard approximation algorithm does indeed overcome the curse of dimensionality in the case of such nonlinear heat equations with gradient-dependent nonlinearities.

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Deep PPDEs for Rough Local Stochastic Volatility

Cited by 33 publications

References 70 publications

Neural networks for option pricing and hedging: a literature review

Neural networks for option pricing and hedging: a literature review

Linear-quadratic control for a class of stochastic Volterra equations: Solvability and approximation

Overcoming the Curse of Dimensionality in the Numerical Approximation of Parabolic Partial Differential Equations with Gradient-Dependent Nonlinearities

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