Numerical Fourier method and second-order Taylor scheme for backward SDEs in finance

Ruijter, Marjon; Oosterlee, Cornelis W.

doi:10.1016/j.apnum.2015.12.003

Cited by 28 publications

(9 citation statements)

References 24 publications

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“…Several methods leverage a connection between parabolic PDEs and backward stochastic differential equations (BSDEs). For example, [2,9,10,13,15,16,20,21,22,23,29,30,31,32,42,43,35,36,44,45,46,47,48,49,50,60,73,74,75,76,77,78,82,83,86,87,88,89,93,94,95,100,106,107,108] use discretizations of the associated first-order BSDEs and [14,24,41,53,72,109] use discretizations ...…”

Section: Introductionmentioning

confidence: 99%

Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks

Hutzenthaler¹,

Jentzen²,

Wurstemberger³

2020

Electron. J. Probab.

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Parabolic partial differential equations (PDEs) are widely used in the mathematical modeling of natural phenomena and man-made complex systems. In particular, parabolic PDEs are a fundamental tool to approximately determine fair prices of financial derivatives in the financial engineering industry. The PDEs appearing in financial engineering applications are often nonlinear (e.g., in PDE models which take into account the possibility of a defaulting counterparty) and high-dimensional since the dimension typically corresponds to the number of considered financial assets. A major issue in the scientific literature is that most approximation methods for nonlinear PDEs suffer from the so-called curse of dimensionality in the sense that the computational effort to compute an approximation with a prescribed accuracy grows exponentially in the dimension of the PDE or in the reciprocal of the prescribed approximation accuracy and nearly all approximation methods for nonlinear PDEs in the scientific literature have not been shown not to suffer from the curse of dimensionality. Recently, a new class of approximation schemes for semilinear parabolic PDEs, termed full history recursive multilevel Picard (MLP) algorithms, were introduced and it was proven that MLP algorithms do overcome the curse of dimensionality for semilinear heat equations. In this paper we extend and generalize those findings to a more general class of semilinear PDEs which includes as special cases the important examples of semilinear Black-Scholes equations used in pricing models for financial derivatives with default risks. In particular, we introduce an MLP algorithm for the approximation of solutions of semilinear Black-Scholes equations and prove, under the assumption that the nonlinearity in the PDE is globally Lipschitz continuous, that the computational effort of the proposed method grows at most polynomially in both the

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Section: Introductionmentioning

confidence: 99%

Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks

Hutzenthaler¹,

Jentzen²,

Wurstemberger³

2020

Electron. J. Probab.

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“…Different from Chau et al [22], we estimate the Gerber-Shiu function based on discrete observations over a finite interval. The Fourier-Cosine expansion method was used in different scenarios; we refer the interested readers to [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. The main goal of this paper is to estimate the Gerber-Shiu function by Fourier-Cosine series expansion based on a discretely observed sample of the aggregate claims process.…”

Section: Introductionmentioning

confidence: 99%

Estimating the Gerber-Shiu Function in Lévy Insurance Risk Model by Fourier-Cosine Series Expansion

Wang

2021

Mathematics

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In this paper, we propose an estimator for the Gerber–Shiu function in a pure-jump Lévy risk model when the surplus process is observed at a high frequency. The estimator is constructed based on the Fourier–Cosine series expansion and its consistency property is thoroughly studied. Simulation examples reveal that our estimator performs better than the Fourier transform method estimator when the sample size is finite.

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“…Indeed, a discretization of the backward equation in Equation 1.1b yields a sequence of recursively nested conditional expectations at each point in the discretized time window. Over the years, several methods have been proposed to tackle the solution of the FBSDE system using: PDE methods in [33]; forward Picard iterations in [5]; quantization techniques in [3]; chaos expansion formulas in [8]; Fourier cosine expansions in [40,41] and regression Monte Carlo approaches in [21,7,6]. These methods have shown great results in low-dimensional settings, however, the majority of them suffers from the curse of dimensionality, meaning that their computational complexity scales exponentially in the number of dimensions.…”

Section: Introductionmentioning

confidence: 99%

The One Step Malliavin scheme: new discretization of BSDEs implemented with deep learning regressions

Negyesi,

Andersson,

Oosterlee

2021

Preprint

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A novel discretization is presented for forward-backward stochastic differential equations (FBSDE) with differentiable coefficients, simultaneously solving the BSDE and its Malliavin sensitivity problem. The control process is estimated by the corresponding linear BSDE driving the trajectories of the Malliavin derivatives of the solution pair, which implies the need to provide accurate Γ estimates. The approximation is based on a merged formulation given by the Feynman-Kac formulae and the Malliavin chain rule. The continuous time dynamics is discretized with a theta-scheme. In order to allow for an efficient numerical solution of the arising semi-discrete conditional expectations in possibly high-dimensions, it is fundamental that the chosen approach admits to differentiable estimates. Two fully-implementable schemes are considered: the BCOS method as a reference in the one-dimensional framework and neural network Monte Carlo regressions in case of high-dimensional problems, similarly to the recently emerging class of Deep BSDE methods [23,27]. An error analysis is carried out to show L 2 convergence of order 1/2, under standard Lipschitz assumptions and additive noise in the forward diffusion. Numerical experiments are provided for a range of different semi-and quasi-linear equations up to 50 dimensions, demonstrating that the proposed scheme yields a significant improvement in the control estimations.

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Numerical Fourier method and second-order Taylor scheme for backward SDEs in finance

Cited by 28 publications

References 24 publications

Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks

Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks

Estimating the Gerber-Shiu Function in Lévy Insurance Risk Model by Fourier-Cosine Series Expansion

The One Step Malliavin scheme: new discretization of BSDEs implemented with deep learning regressions

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