I would always presume that the portfolio consists of 1 stock and 1 risk-free asset. And that the $r, \alpha,\sigma$ are all non-zero, but might be time-dependent. When I say "any" option, I am actually referring to an option of the above type.
I understand that "any" European option can be perfectly dynamically hedged. In the discrete-time case, you can solve for the coefficients of each part of the portfolio backwards. Since this can be done in the discrete-time case, intuitively I believe that this can also be done in the continuous-time case.
But when I look at some type of option whose payoff depends on the stock price in the past in an arbitrary way (e.g. some option that is even more complicated than exotic option/American option), I can't find any intuition to convince myself that this option can be perfectly dynamically hedged, even in the discrete-time case.
In Chapter 5 of Steven E. Shreve's book "Stochastic Calculus for Finance 2 Continuous-Time Models", he defines a complete market as a market where every derivate security can be hedged. It looks like that the existence of such a definition has an implication that "a properly designed option can be dynamically hedged". I wonder where is the intuition from?