Optimal calculations for the space–time fractional derivative option pricing models with stochastic liquidity risk and volatility using a combination neural network

This work incorporates a stochastic liquidity risk, stochastic volatility, and Caputo-type fractional derivatives into European option pricing and establishes two novel space-time fractional hybrid models to capture the nonlinearity and non-stationarity of price evolution processes. The combination neural network algorithm with the defined nonlinear test solutions is designed to solve the fractional derivative models with the initial condition, Dirichlet and Robin boundary conditions. When the influence of a liquidity risk is removed, the studied models are reduced to the space-time fractional Heston models, and the pricing results are compared with the analytical formula of the classical Heston model. In the presence of a liquidity risk, the pricing models under the Caputo and Caputo-Fabrizio fractional derivatives are tested based on the market data. The applications and comparison results prove that the dynamical models demonstrated in the work have small prediction errors and can highly fit market data. The designed combination neural network can effectively handle the mixed problems of the high-dimensional fractional derivative equations and derive the optimal approximations under a stochastic liquidity risk and volatility.