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Neural variance reduction for stochastic differential equations

P. D. Hinds, M. V. Tretyakov

TL;DR

The work tackles high-variance Monte Carlo estimation in SDE-based pricing by introducing neural SDEs with neural-network–parameterized control variates to learn variance-reducing terms on the fly. By leveraging the Feynman–Kac link to PDEs/PIDEs, it derives optimality conditions for zero-variance control variates in Brownian and Lévy settings and develops a two-pass, online training algorithm that minimally biases the estimator. Empirical results across GBM, Heston, Merton, and higher-dimensional Lévy models show substantial speedups, up to around 40× in some cases, and competitive performance relative to MLMC and crude CV, with notable gains from transfer learning. The approach provides a flexible, black-box variance-reduction tool for pricing under complex noise structures, including infinite-activity jumps, while maintaining unbiased MC estimates and offering practical online training without extensive pretraining.

Abstract

Variance reduction techniques are of crucial importance for the efficiency of Monte Carlo simulations in finance applications. We propose the use of neural SDEs, with control variates parameterized by neural networks, in order to learn approximately optimal control variates and hence reduce variance as trajectories of the SDEs are being simulated. We consider SDEs driven by Brownian motion and, more generally, by Lévy processes including those with infinite activity. For the latter case, we prove optimality conditions for the variance reduction. Several numerical examples from option pricing are presented.

Neural variance reduction for stochastic differential equations

TL;DR

The work tackles high-variance Monte Carlo estimation in SDE-based pricing by introducing neural SDEs with neural-network–parameterized control variates to learn variance-reducing terms on the fly. By leveraging the Feynman–Kac link to PDEs/PIDEs, it derives optimality conditions for zero-variance control variates in Brownian and Lévy settings and develops a two-pass, online training algorithm that minimally biases the estimator. Empirical results across GBM, Heston, Merton, and higher-dimensional Lévy models show substantial speedups, up to around 40× in some cases, and competitive performance relative to MLMC and crude CV, with notable gains from transfer learning. The approach provides a flexible, black-box variance-reduction tool for pricing under complex noise structures, including infinite-activity jumps, while maintaining unbiased MC estimates and offering practical online training without extensive pretraining.

Abstract

Variance reduction techniques are of crucial importance for the efficiency of Monte Carlo simulations in finance applications. We propose the use of neural SDEs, with control variates parameterized by neural networks, in order to learn approximately optimal control variates and hence reduce variance as trajectories of the SDEs are being simulated. We consider SDEs driven by Brownian motion and, more generally, by Lévy processes including those with infinite activity. For the latter case, we prove optimality conditions for the variance reduction. Several numerical examples from option pricing are presented.
Paper Structure (24 sections, 2 theorems, 92 equations, 1 figure, 11 tables, 2 algorithms)

This paper contains 24 sections, 2 theorems, 92 equations, 1 figure, 11 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. SDEs driven by Brownian motion
  3. Numerical algorithm
  4. SDEs driven by Lévy processes
  5. Numerical algorithm
  6. Numerical Experiments
  7. Diffusion models
  8. Geometric Brownian motion
  9. Multi-dimensional geometric Brownian motion
  10. Heston Model
  11. Non-singular Lévy measure
  12. Merton model
  13. Singular Lévy measure
  14. One-dimensional European call
  15. Four-dimensional call-on-max option
  16. ...and 9 more sections

Key Result

Theorem 1

If $\mu$ and $G$ are such that for all $(t, x) \in [s, T] \times \mathbb{R}^d$, then $\emph{Var}\Gamma_{s, x}(T) = 0$.

Figures (1)

  • Figure 1: The true optimal control variate and the learned control variate for a European call option with strike $K=1$ under the Black-Scholes model with parameters in Table \ref{['tab:gbm_cv']}.

Theorems & Definitions (9)

  • Theorem 1: Milstein & Schoenmakers milstein_monte_2002
  • Remark 1
  • Theorem 2
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6