Solving multi-dimensional fractional Black–Scholes model using deep learning

Abstract

In this paper, we propose the multi-dimensional fractional Black–Scholes model (MDFBSM) with correlations between different assets for the first time and solve the MDFBSM employing deep learning method by using the reformulation of the backward stochastic differential equations (BSDEs). The fractional Black–Scholes model (FBSM) is an extension of the traditional Black–Scholes model, which adopts the fractional Brownian motion (FBM) to describe the dynamic changes of asset prices, so as to capture the long-term memory, thick tail, autocorrelation and hidden dynamic changes in the financial market. Its complexity and “curse of dimensionality” makes the MDFBSM very difficult to solve. Thus, this paper uses BSDEs to reformulate the partial differential equations (PDEs). Combining the TensorFlow framework with the gradient of the solution as the policy function and the error between the solution of the BSDE and the prescribed terminal condition as the loss function, we approximate the policy function of the model by minimizing the residuals of the PDEs through a neural network approach, thus overcoming the “curse of dimensionality” problem. To verify the validity of this paper, the historical data of 67 futures contracts in China are used for empirical analysis. And then, we find that our results can truly reflect the dynamics of asset prices in the real market.