Hints for an extension of the early exercise premium formula for American options

There exists a non-closed formula for the American put option price and non-trivial computations are required to solve it. Strong efforts have been made to propose efficient numerical techniques but few have strong mathematical reasoning to ascertain why they work well. We present an extension of the American put price aiming to catch weaknesses of the numerical methods based on their non-fulfillment of the smooth pasting condition.