Hi,

Suppose that a stock price follows standard GBM process with constant risk-free rate and constant vol. Suppose we also have some derivative with unknown payoff P(.) at expiration T which depends in some general way on the stock price process (it could be just ending stock price or path dependent or anything). Does vol and T always appear “coupled” as vol*sqrt(T) in the derivative price (which may or may not exist in an analytical closed-form)? Said differently, if I reduce vol by 0.5x and increase T by 4x will I always get the same derivative price and if not, under what conditions about the payoff function? Thanks for your input.