From Stochastic to Rough Volatility : A New Deep Learning Perspective on the Hedging

The classical Black-Scholes model describes the market behavior assuming that the volatility process is a constant, and the Heston model extends it to the case where the volatility is a stochastic process. Then, the rough Bergomi (rBergomi) model was proposed to improve the description of the implied volatility, where the volatility is a fractional Brownian motion. However, the classical calibration and hedging techniques are difficult to apply under the rBergomi model due to the high cost caused by its non-Markovianity. In this paper, we are the first to propose the gated recurrent unit neural network (GRU-NN) architecture for hedging under volatility with different regularities. One advantage is that the gating network signals embedding in our architecture can control how the present input and previous memory are used to update the current activation. These gates are updated adaptively in the learning process, and thus it outperforms the conventional deep learning techniques in the non-Markovian environment, such as the convolutional neural network (CNN). The hedging performance of those models under our neural networks is compared with the S&P 500 option data with maturities from Dec 31, 2021 to Dec 15, 2023 and the FTSE 100 option data with maturities from Jan 21, 2022 to Dec 15, 2023. We find that the rBergomi model outperforms the other two models in multiple characteristics. Therein, the associated parameters in the rough Bergomi model are calibrated by the market data using the fully connected neural networks (fCNN) combined with the Levenberg-Marquardt algorithm