Neural network smoothing of American options payoff with grid refinement strategies

For an American call or put option, at the expiry time, the value function is a ramp function, the Delta sensitivity is a Heaviside function and Gamma is a Dirac delta measure. In addition, the first derivative with respect to the early exercise boundary is singular and relates to Theta sensitivity at the free boundary. This research explores the possibility of whether deep learning can efficiently account for the payoff condition and learn these irregularities. To this end, we introduce a smoothed modular learning approximant (SMLA) that combines key expressions, and a regulator representing a smoothed terminal condition for the value function. The SMLA's derivatives at the terminal time correspond to smoothed Heaviside and Dirac delta functions. With appropriate grid refinement strategies, our neural network solver can predict the early exercise boundary and its derivative, value function, and Greeks. The predictive performance of SMLA is illustrated by examples, and accurate results are achieved even for extreme maturity time and volatility values