Deep learning and American options via free boundary framework
Chinonso Nwankwo, Nneka Umeorah, Tony Ware, Weizhong Dai
We propose a deep learning method for solving the American options model with
a free boundary feature. To extract the free boundary known as the early
exercise boundary from our proposed method, we introduce the Landau
transformation. For efficient implementation of our proposed method, we further
construct a dual solution framework consisting of a novel auxiliary function
and free boundary equations. The auxiliary function is formulated to include
the feed forward deep neural network (DNN) output and further mimic the far
boundary behaviour, smooth pasting condition, and remaining boundary conditions
due to the second-order space derivative and first-order time derivative.
Because the early exercise boundary and its derivative are not a priori known,
the boundary values mimicked by the auxiliary function are in approximate form.
Concurrently, we then establish equations that approximate the early exercise
boundary and its derivative directly from the DNN output based on some linear
relationships at the left boundary. Furthermore, the option Greeks are obtained
from the derivatives of this auxiliary function. We test our implementation
with several examples and compare them to the highly accurate sixth-order
compact scheme with left boundary improvement. All indicators show that our
proposed deep learning method presents an efficient and alternative way of
pricing options with early exercise features.