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Unsupervised Learning-based Calibration Scheme for Rough Volatility Models

Changqing Teng, Guanglian Li

TL;DR

The paper tackles calibrating rough volatility models where non-Markovian dynamics hinder PDE-based pricing. It introduces an unsupervised Deep BSDE scheme that treats model parameters as trainable weights and solves the associated BSPDE/BSDE pricing problem forward in time, using terminal mispricing as the loss. A key theoretical result bounds the pricing discrepancy $F(\theta)$ by the learning loss $\mathcal{L}(\theta;\nu)$, and neural networks' universal approximation ensures this loss can be driven arbitrarily close to zero. Numerical experiments on simulated data and historical S&P 500 data with the rBergomi model demonstrate accurate calibration and substantial computational efficiency gains over traditional Monte Carlo methods, supporting online market-fitting applications and suggesting extensions to time-dependent parameters.

Abstract

Existing deep learning-based calibration scheme for rough volatility models predominantly rely on supervised learning frameworks, which incur significant computational costs due to the necessity of generating massive synthetic training datasets. In this work, we propose a novel unsupervised learning-based calibration scheme for rough volatility models that eliminates the data generation bottleneck. Our approach leverages the backward stochastic differential equation (BSDE) representation of the pricing function derived by Bayer et al. \cite{bayer2022pricing}. By treating model parameters as trainable variables, we simultaneously approximate the BSDE solution and optimize the parameters within a unified neural network training process, with the terminal misfit as the loss. We theoretically establish that the mean squared error between the model-implied prices and market data is bounded by the loss function. Furthermore, we prove that the loss can be minimized to an arbitary degree, depending on the model's market fitting capacity and the universal approximation capability of neural networks. Numerical experiments for both simulated and historical S\&P 500 data based on rough Bergomi (rBergomi) model demonstrate the efficiency and accuracy of the proposed scheme.

Unsupervised Learning-based Calibration Scheme for Rough Volatility Models

TL;DR

The paper tackles calibrating rough volatility models where non-Markovian dynamics hinder PDE-based pricing. It introduces an unsupervised Deep BSDE scheme that treats model parameters as trainable weights and solves the associated BSPDE/BSDE pricing problem forward in time, using terminal mispricing as the loss. A key theoretical result bounds the pricing discrepancy by the learning loss , and neural networks' universal approximation ensures this loss can be driven arbitrarily close to zero. Numerical experiments on simulated data and historical S&P 500 data with the rBergomi model demonstrate accurate calibration and substantial computational efficiency gains over traditional Monte Carlo methods, supporting online market-fitting applications and suggesting extensions to time-dependent parameters.

Abstract

Existing deep learning-based calibration scheme for rough volatility models predominantly rely on supervised learning frameworks, which incur significant computational costs due to the necessity of generating massive synthetic training datasets. In this work, we propose a novel unsupervised learning-based calibration scheme for rough volatility models that eliminates the data generation bottleneck. Our approach leverages the backward stochastic differential equation (BSDE) representation of the pricing function derived by Bayer et al. \cite{bayer2022pricing}. By treating model parameters as trainable variables, we simultaneously approximate the BSDE solution and optimize the parameters within a unified neural network training process, with the terminal misfit as the loss. We theoretically establish that the mean squared error between the model-implied prices and market data is bounded by the loss function. Furthermore, we prove that the loss can be minimized to an arbitary degree, depending on the model's market fitting capacity and the universal approximation capability of neural networks. Numerical experiments for both simulated and historical S\&P 500 data based on rough Bergomi (rBergomi) model demonstrate the efficiency and accuracy of the proposed scheme.

Paper Structure

This paper contains 15 sections, 3 theorems, 82 equations, 3 figures, 9 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. BSPDE model
  3. Unsupervised Learning-based Calibration method
  4. Deep learning based scheme
  5. Calibration task
  6. Convergence analysis
  7. Numerical experiments
  8. Numerical accuracy
  9. Calibration with historical data
  10. Model parameters as scalars
  11. Model parameters as time-varying functions
  12. Conclusion
  13. Acknowledgment.
  14. mSOE scheme for time-dependent rBergomi model
  15. Computational Complexity:

Key Result

Theorem 2.1

Suppose Assumptions assump: variance process and assump: payoff hold. Let $\left(u, \psi\right)$ be a weak solution of the BSPDE eq: BSPDE such that there is a constant $C_0 \in (0, \infty)$ satisfying for each $t \in [0, T]$ Then the following holds a.s., for $0\leq t \leq s \leq T$ and $x \in \mathbb{R}$, where $(Y_s^{t, x}, Z_s^{t,x}, \tilde{Z}_s^{t, x})$ is the unique solution of the followin

Figures (3)

  • Figure 1: Plot of error variation against the number of iterations using Algorithm \ref{['alg: calibration_1']}, with $h = 1/20$.
  • Figure 2: The heatmaps of the maean relative pricing error and the standard deviation of relative pricing error, with $h = 1/20$.
  • Figure 3: Calibrated model parameters under different parameterizations.

Theorems & Definitions (6)

  • Theorem 2.1: Stochastic Feynman-Kac formula bayer2022pricing
  • Remark 2.1: Functional dependence of BSDE solutions
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof