No arbitrage assumption implies the differentiability of the derivative pricing function

The no-arbitrage assumption implies that the price of an asset must be a semimartingale. In this article, we characterize the class of functions that map Itô processes to continuous semimartingales, as well as those that map continuous Markov semimartingales to continuous Markov semimartingales. This class of functions generalizes the conventional Sobolev space by adopting a weaker notion of derivatives. In particular, the functions must be weakly differentiable with respect to the input process. From a financial perspective, our results show that the no-arbitrage assumption implies that any derivative price is differentiable with respect to the underlying asset, provided that the underlying noise is a continuous Markov semimartingale. Moreover, we demonstrate that Malliavin differentiability of the input process (the underlying) implies Malliavin differentiability of the output process (the derivative price).