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.