Ensemble deep neural network method for solving free boundary American style stochastic volatility models

We present an ensemble deep learning method for solving free boundary American-style stochastic volatility models. Our solution framework for such free boundary problems—where the early exercise boundary surface is a function of time and volatility—is developed so as to obtain the value function and Greeks simultaneously. To this end, we first use the Landau transformation to fix the free boundary, and we normalize the value function and the time domain. We then develop a novel ensemble auxiliary operator (EANO) involving a suite of configurations based on an ensemble neural network output (EDNN). The early exercise boundary surface, value function, delta sensitivity, vega, gamma, vomma and vanna are predicted from the EDNN, EANO, and the derivatives of the EANO after training. The performance of our neural network configuration is validated by comparison with existing methods and examples. Results show that our ensemble learning method achieves good predictive potential with a small dataset and training time. In particular, it performs well when the correlation factor \(\rho \) $\rho$ is high, a situation which has been shown in the literature to give rise to computational error and numerical instability arising from the mixed derivative term.