Parameter identification of the Black-Scholes model driven by multiplicative fractional Brownian motion

Abstract

In this paper, we propose a parameter identification method based on deep learning network, which can jointly identify all parameters of the Black–Scholes (BS) model driven by multiplicative fractional Brownian motion (FBM) in a discrete sample trajectory. Firstly, the Convolutional Neural Network (CNN) is combined with the Bi-directional Gated Recurrent Unit (BiGRU) and the attention mechanism (AM) is integrated to construct the new identifier (CBANN). Then, the multiplicative FBM is constructed as the random effect of the BS model, and all the parameters of the model are identified by the new identifier. Finally, extensive numerical simulations are conducted for both known and unknown Hurst exponents, and two empirical studies are performed using real data. The results suggest that, compared to the PENN identifier and the maximum likelihood (ML) identifier, the proposed identifier can simultaneously identify all parameters in the model more quickly and accurately. Additionally, several advantages of the new identifier are discussed, including its strong generalization performance, flexibility in training set proportion settings, and the incorporation of an attention mechanism layer.