  Volatility models of price fluctuations are well studied in the econometrics
literature, with more than 50 years of theoretical and empirical findings. The
recent advancements in neural networks (NN) in the deep learning field have
naturally offered novel econometric modeling tools. However, there is still a
lack of explainability and stylized knowledge about volatility modeling with
neural networks; the use of stylized facts could help improve the performance
of the NN for the volatility prediction task. In this paper, we investigate how
the knowledge about the "physics" of the volatility process can be used as an
inductive bias to design or constrain a cell state of long short-term memory
(LSTM) for volatility forecasting. We introduce a new type of $\sigma$-LSTM
cell with a stochastic processing layer, design its learning mechanism and show
good out-of-sample forecasting performance.
