ABSTRACT
On the other hand, according to the Dupire theory (see e.g. [3] for the case of European options), the fair price of an American put option with fixed underlying stock price S, at fixed time t, as a function of the expiration time T and the option strike price k, is the unique solution of the (degenerate) forward parabolic obstacle problem (written in the free boundary problem form):
together with the initial condition
The trading significance of the free boundary now is that for the given current price of the underlying stock S, only options with strikes below the free boundary should be considered for trading. It is remarkable that the same volatility function or, with different arguments, appears in both problems (as well as in the underlying stock price evolution SDE above). Also notice that in either one of the problems the underlying stock price appreciation rate a(t, S) does not appear (but instead the known interest rate r does), while volatility cr is of the deciding importance (options with different underlying volatilities have significantly different prices). It is then of paramount importance to have a precise estimate of the volatility cr.
