[Paper Review] Solving path dependent PDEs with LSTM networks and path signatures

Marc Sabate‐Vidales, David Šiška+1 more|arXiv (Cornell University)|Nov 20, 2020

Model Reduction and Neural Networks40 references4 citations

TL;DR

This paper proposes a novel deep learning framework combining LSTM networks and path signatures to solve parametric families of path-dependent PDEs (PPDEs) arising in financial derivatives pricing and hedging. By leveraging the martingale representation theorem, the method enables unbiased estimation with confidence intervals, achieving high accuracy in pricing and hedging strategies for lookback and autocallable options across stochastic volatility models.

ABSTRACT

Using a combination of recurrent neural networks and signature methods from the rough paths theory we design efficient algorithms for solving parametric families of path dependent partial differential equations (PPDEs) that arise in pricing and hedging of path-dependent derivatives or from use of non-Markovian model, such as rough volatility models in Jacquier and Oumgari, 2019. The solutions of PPDEs are functions of time, a continuous path (the asset price history) and model parameters. As the domain of the solution is infinite dimensional many recently developed deep learning techniques for solving PDEs do not apply. Similarly as in Vidales et al. 2018, we identify the objective function used to learn the PPDE by using martingale representation theorem. As a result we can de-bias and provide confidence intervals for then neural network-based algorithm. We validate our algorithm using classical models for pricing lookback and auto-callable options and report errors for approximating both prices and hedging strategies.

Motivation & Objective

• To develop efficient deep learning algorithms for solving parametric families of path-dependent PDEs (PPDEs) that arise in non-Markovian financial models, such as rough volatility models.
• To address the challenge of infinite-dimensional path dependence by encoding paths into finite-dimensional representations using path signatures and LSTMs.
• To enable fast model calibration by learning the solution across the full parameter space, including model parameters and path histories.
• To provide accurate approximations of the gradient ∇ωF, which corresponds to the hedging strategy in derivative pricing.
• To de-bias the neural network approximation and provide a posteriori confidence intervals using the martingale representation theorem.

Proposed method

• The solution to the PPDE is represented probabilistically via the Feynman–Kac formula, enabling Monte Carlo sampling for training.
• Path signatures are used to encode the continuous path ω ∈ C([0,T]; ℝ^d) into a finite-dimensional vector, capturing path-dependent information efficiently.
• LSTM networks are employed to model non-anticipative functionals of the path, enabling sequence modeling of asset price histories.
• The martingale representation theorem is applied to construct a loss function that allows for de-biasing and confidence interval estimation in the training objective.
• Two training algorithms are proposed: one using conditional expectation (Algorithm 2) and another using the martingale representation (Algorithm 3) for improved variance control.
• The method jointly learns the PPDE solution F(t, ω; β) and its path derivative ∇ωF, which corresponds to the hedging strategy in financial applications.

Experimental results

Research questions

• RQ1 Can LSTM networks combined with path signatures effectively approximate solutions to parametric families of path-dependent PDEs in high-dimensional path spaces?
• RQ2 Can the martingale representation theorem be leveraged to construct a training objective that enables de-biasing and confidence interval estimation in deep learning-based PPDE solvers?
• RQ3 How does the proposed method compare in accuracy and efficiency to feedforward networks for path-dependent option pricing and hedging?
• RQ4 To what extent can the method support fast model calibration by learning the solution across a continuous parameter space β ∈ B?
• RQ5 Can the method accurately recover the hedging strategy ∇ωF for complex derivatives such as lookback and autocallable options?

Key findings

• For the lookback option under geometric Brownian motion, the method achieved an integral error of 6.47×10⁻³ and a hedging error of 3.6×10⁻² using the martingale representation with path signature input.
• In the Heston model for autocallable options, the method achieved an integral error of 1.4×10⁻² and a hedging error of 1.8×10⁻², with a correlation coefficient of 0.948 between predicted and Monte Carlo prices.
• The LSTM-based model with path signature input outperformed feedforward networks in accuracy, particularly in capturing path-dependent dynamics and hedging sensitivities.
• The use of the martingale representation theorem enabled the construction of unbiased estimators and valid confidence intervals for the neural network predictions.
• The framework successfully learned the full parametric family of solutions, enabling rapid model calibration and real-time pricing across different model parameters.
• The method demonstrated robustness across different volatility levels and path-dependent payoff structures, including barrier and auto-call features.