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

Abstract

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 determine fair prices of financial derivatives in the financial industry. The PDEs appearing in financial engineering applications are often nonlinear and high dimensional since the dimension typically corresponds to the number of considered financial assets. A major issue is that most approximation methods for nonlinear PDEs in the literature suffer under 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 have not been shown not to suffer under 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 those findings to a more general class of semilinear PDEs including as special cases semilinear Black-Scholes equations used for the pricing of financial derivatives with default risks. More specifically, we introduce an MLP algorithm for the approximation of solutions of semilinear Black-Scholes equations and prove that the computational effort of our method grows at most polynomially both in the dimension and the reciprocal of the prescribed approximation accuracy. This is, to the best of our knowledge, the first result showing that the approximation of solutions of semilinear Black-Scholes equations is a polynomially tractable approximation problem.

71 pages. arXiv admin note: text overlap with arXiv:1807.01212

References in corpus (8)

• A regression-based Monte Carlo method to solve backward stochastic differential equations
• A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations
• A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations
• Stratified regression Monte-Carlo scheme for semilinear PDEs and BSDEs with large scale parallelization on GPUs
• Runge-Kutta schemes for backward stochastic differential equations
• On nonlinear Feynman-Kac formulas for viscosity solutions of semilinear parabolic partial differential equations
• Numerical simulation of BSDEs using empirical regression methods: theory and practice
• Variations on branching methods for non linear PDEs

Cited by in corpus (19)

• Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning
• Deep splitting method for parabolic PDEs
• Neural Network Approximation: Three Hidden Layers Are Enough
• An overview on deep learning-based approximation methods for partial differential equations
• A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations
• SelectNet: Self-paced Learning for High-dimensional Partial Differential Equations
• Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations
• Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations
• On nonlinear Feynman-Kac formulas for viscosity solutions of semilinear parabolic partial differential equations
• Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations
• Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities
• Generalised multilevel Picard approximations
• Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing
• Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities
• Solving PDEs on Unknown Manifolds with Machine Learning
• On existence and uniqueness properties for solutions of stochastic fixed point equations
• Full history recursive multilevel Picard approximations for ordinary differential equations with expectations
• A Machine-Learning Method for Time-Dependent Wave Equations over Unbounded Domains
• Strong $L^p$-error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations