The put-call symmetry states that C(S,t;X,r,q) = P(X,t;S,q,r)
, and that this works for American options. According to my notes, this is 'model dependent' because it depends on the assumption that the underlying price follows geometric brownian motion.
However, I don't understand why this would matter: Considering that the payoffs are the same, shouldn't the two options have the same value even if the stock price doesn't follow geometric brownian motion?