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Random neural networks for rough volatility

Antoine Jacquier, Zan Zuric

TL;DR

The paper tackles solving path-dependent PDEs arising from rough volatility by recasting option pricing into BSPDE/BSDE form and introducing a reservoir-based Random-Weight Neural Network (RWNN) solver. By fixing hidden RWNN weights and training only the final readouts, the approach yields a convex, scalable least-squares framework with explicit error bounds in terms of the number of hidden units and time discretisation. The authors develop both Markovian and non-Markovian schemes, deriving gradient and Hessian formulas for ReLU RWNNs and establishing convergence results for the resulting RWNN approximations, including a rigorous bound on the overall error. Numerical experiments on Black-Scholes and rough Bergomi models demonstrate favorable convergence rates, competitive pricing accuracy, and notable training speedups, highlighting a practical pathway to high-dimensional, non-Markovian PDEs in finance.

Abstract

We construct a deep learning-based numerical algorithm to solve path-dependent partial differential equations arising in the context of rough volatility. Our approach is based on interpreting the PDE as a solution to an BSDE, building upon recent insights by Bayer, Qiu and Yao, and on constructing a neural network of reservoir type as originally developed by Gonon, Grigoryeva, Ortega. The reservoir approach allows us to formulate the optimisation problem as a simple least-square regression for which we prove theoretical convergence properties.

Random neural networks for rough volatility

TL;DR

The paper tackles solving path-dependent PDEs arising from rough volatility by recasting option pricing into BSPDE/BSDE form and introducing a reservoir-based Random-Weight Neural Network (RWNN) solver. By fixing hidden RWNN weights and training only the final readouts, the approach yields a convex, scalable least-squares framework with explicit error bounds in terms of the number of hidden units and time discretisation. The authors develop both Markovian and non-Markovian schemes, deriving gradient and Hessian formulas for ReLU RWNNs and establishing convergence results for the resulting RWNN approximations, including a rigorous bound on the overall error. Numerical experiments on Black-Scholes and rough Bergomi models demonstrate favorable convergence rates, competitive pricing accuracy, and notable training speedups, highlighting a practical pathway to high-dimensional, non-Markovian PDEs in finance.

Abstract

We construct a deep learning-based numerical algorithm to solve path-dependent partial differential equations arising in the context of rough volatility. Our approach is based on interpreting the PDE as a solution to an BSDE, building upon recent insights by Bayer, Qiu and Yao, and on constructing a neural network of reservoir type as originally developed by Gonon, Grigoryeva, Ortega. The reservoir approach allows us to formulate the optimisation problem as a simple least-square regression for which we prove theoretical convergence properties.
Paper Structure (26 sections, 12 theorems, 141 equations, 3 figures, 4 tables, 2 algorithms)

This paper contains 26 sections, 12 theorems, 141 equations, 3 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Random-weight neural network (RWNN)
  3. Derivatives of ReLu-RWNN
  4. Randomised least squares (RLS)
  5. The Markovian case
  6. Random weighted neural network scheme
  7. Algorithm
  8. The non-Markovian case
  9. Random neural network scheme
  10. Algorithm
  11. Convergence analysis
  12. Numerical results
  13. Example: Black-Scholes
  14. Convergence rate
  15. ATM Call option
  16. ...and 11 more sections

Key Result

Lemma 2.4

For any linear function $\ell(\mathbf{x})=\mathrm{A} \mathbf{x} + \bm$, with $\mathrm{A}\in\mathbb{R}^{K\times d}$ and $\bm\in\mathbb{R}^K$, then

Figures (3)

  • Figure 1: Empirical convergence of the MSE from Corollary \ref{['cor:convergence']} under Black-Scholes in terms of the number of hidden nodes (for a fixed grid and number of MC samples). Error bars mark 0.1 and 0.9 quantiles of 20 separate runs of the algorithm. The slope coefficient of the dashed line is obtained through regression of the means of individual runs, while the solid line represents $1/K$ convergence and is shown as a reference.
  • Figure 2: Computational time vs number of dimensions, as in Table \ref{['tab:BS_MSE_time']}.
  • Figure 3: Empirical convergence of MSE under rBergomi in terms of the number of hidden nodes. Error bars mark 0.1 and 0.9 quantiles of 20 separate runs of the algorithm. The slope coefficient of the dashed line is obtained through regression of the means of individual runs, while the solid line represents $1/K$ convergence and is shown as a reference.

Theorems & Definitions (34)

  • Definition 2.1: Neural network
  • Definition 2.2: Single layer RWNN
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5: Approximate limit, Evans1992
  • Definition 2.6: Approximate differentiability, Evans1992
  • Remark 2.7
  • Remark 2.8: Evans1992
  • Lemma 2.9
  • ...and 24 more