Solving path dependent PDEs with LSTM networks and path signatures

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.

References in corpus (12)

• DGM: A deep learning algorithm for solving partial differential equations
• Solving high-dimensional partial differential equations using deep learning
• Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations
• Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations
• Functional Itô calculus and stochastic integral representation of martingales
• Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning
• Deep splitting method for parabolic PDEs
• Learning from the past, predicting the statistics for the future, learning an evolving system
• Discretely sampled signals and the rough Hoff process
• Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations
• Weak approximation of martingale representations
• Deep learning calibration of option pricing models: some pitfalls and solutions

Cited by in corpus (3)

• Neural networks-based algorithms for stochastic control and PDEs in finance
• Unbiased deep solvers for linear parametric PDEs
• Deep Signature FBSDE Algorithm