European option pricing under a generalized fractional Brownian motion Heston exponential Hull–White model with transaction costs by the Deep Galerkin Method

We propose a new financial model called the generalized fractional Brownian motion Heston exponential Hull–White model, which has stochastic volatility and interest rate, long memory, and heavy tail distribution. Based on the market price of the volatility and delta hedging strategies, we propose a partial differential equation (PDE) to obtain the European option price. To do this, portfolio changes contain long one position of the European call option and shares of the underlying assets (stock, zero coupon bond, volatility), where we use the mentioned model to obtain the price. Due to transaction costs, the resulting equation is a fully nonlinear PDE, which we use the Deep Galerkin Method (DGM) to solve it. Also, we present the proof of the convergence of the method to this class of equations, which includes two parts: the convergence of the loss function to zero and the convergence of the neural network to the exact solution of the equation. We finally present numerical results to show the model and method’s effectiveness.