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

Sebastian Becker, Arnulf Jentzen, Marvin S. Müller, Philippe von Wurstemberger

TL;DR

A new numerical approximation strategy for parametric approximation problems including the parametric financial pricing problems described above is introduced and it is illustrated 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 L∞$L^\infty$ ‐norm.

Abstract

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. In this paper we introduce a new approximation strategy for parametric approximation problems including the parametric financial pricing problems described above. 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 artificial neural networks (ANNs) but to learn random variables appearing in MC approximations. We numerically test the LRV strategy on various parametric problems with convincing results when compared with standard MC simulations, Quasi-Monte Carlo simulations, SGD-trained shallow ANNs, and SGD-trained deep ANNs. Our numerical simulations strongly indicate that the LRV strategy might be capable to overcome the curse of dimensionality in the $L^\infty$-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 lower bounds established in the scientific literature 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.

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

TL;DR

A new numerical approximation strategy for parametric approximation problems including the parametric financial pricing problems described above is introduced and it is illustrated 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 L∞ ‐norm.

Abstract

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. In this paper we introduce a new approximation strategy for parametric approximation problems including the parametric financial pricing problems described above. 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 artificial neural networks (ANNs) but to learn random variables appearing in MC approximations. We numerically test the LRV strategy on various parametric problems with convincing results when compared with standard MC simulations, Quasi-Monte Carlo simulations, SGD-trained shallow ANNs, and SGD-trained deep ANNs. Our numerical simulations strongly indicate that the LRV strategy might 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 lower bounds established in the scientific literature 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.
Paper Structure (56 sections, 10 theorems, 152 equations, 4 figures, 14 tables)

This paper contains 56 sections, 10 theorems, 152 equations, 4 figures, 14 tables.

Table of Contents

  1. Introduction
  2. Learning the random variables (LRV) strategy in the case of Monte Carlo approximations
  3. Parametric expectations involving vector valued random variables
  4. Approximative pricing of European options in the Black-Scholes model
  5. Monte Carlo approximations
  6. Replacing the random variables in Monte Carlo approximations
  7. Random loss functions for fixed random variables in Monte Carlo approximations
  8. Learning the random variables with stochastic gradient descent
  9. Description of the proposed approximation algorithm
  10. LRV strategy in the case of Monte Carlo-Euler-Maruyama approximations
  11. Parametric expectations involving vector valued stochastic processes
  12. Temporal discretizations of the Brownian motion
  13. Euler-Maruyama approximations for parametric stochastic differential equations
  14. Monte Carlo approximations
  15. Replacing the random variables in Monte Carlo approximations
  16. ...and 41 more sections

Key Result

Lemma 2.1

Let $\xi, T, \sigma \in ( 0, \infty )$, $r \in \mathbb{R}$, let $\mathfrak{N} \colon \mathbb{R} \to \mathbb{R}$ satisfy for all $x \in \mathbb{R}$ that $\mathfrak{N}( x ) = \int_{ - \infty }^x \frac{ 1 }{ \sqrt{ 2 \pi } } \, \exp({ - \frac{ y^2 }{ 2 } }) \operatorname{d\!} y$, let $( \Omega, \mathc Then

Figures (4)

  • Figure 1: Numerical simulations for the LRV strategy in case of the Black-Scholes model for European call options on one underlying described in \ref{['simul:BS1']} (5-dimensional approximation problem). See \ref{['fig:BS_plot_2']} below for the legend.
  • Figure 2: Smallest estimated $L^2(\lambda_{\mathfrak{P}};\mathbb{R})$-errors over $\mathfrak{M}$ for different choices of training parameters. Each dot corresponds to the minimum of a line of the same color in \ref{['fig:BS_plot_1']}. The convergence rate 1/2 is inspired by the convergence rate 1/2 of the MC method which is strongly related to SGD.
  • Figure 3: Histograms and sample moments of realizations of learned random variables$\Theta^{(1)}_{140000}, \Theta^{(2)}_{140000}, \ldots, \Theta^{(\mathfrak{M})}_{140000}$ in the LRV strategy in case of the Black-Scholes model for European call options on one underlying described in \ref{['simul:BS1']}. See \ref{['fig:Histograms_MC']} for the moments of the standard normal distribution.
  • Figure 4: Histograms and sample moments of realizations of random variables appearing in the MC method and QMC method in case of the Black-Scholes model for European call options on one underlying described in \ref{['simul:BS1']}

Theorems & Definitions (13)

  • Lemma 2.1
  • proof : Proof of \ref{['cor:black_scholes']}
  • Lemma 6.2
  • Lemma 6.3
  • Lemma 6.4
  • Lemma 6.5
  • Lemma 6.6
  • Lemma 6.7
  • Lemma 6.8
  • Lemma 7.1
  • ...and 3 more