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?

  • $\begingroup$ My intuition is that if you can compute a delta function that is $C^0$ then you can hedge it. Problems would arise when there is a discontinuity or when Delta is undefined. $\endgroup$ – noob2 Nov 2 '16 at 16:30
  • $\begingroup$ @noob2 Do you mean you compute the Delta function from the option price in market? $\endgroup$ – Ethan Nov 2 '16 at 16:31
  • $\begingroup$ No, I mean to derive it analytically from the model assumptions. $\endgroup$ – noob2 Nov 2 '16 at 16:33
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    $\begingroup$ As long as your market model is complete you can always hedge an option. In that case all latent risks can indeed be neutralised by trading in either a specific instrument or a specific combination of them. Are you familiar with the concept of state prices? $\endgroup$ – Quantuple Nov 2 '16 at 18:27
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    $\begingroup$ @Ethan I don't know any math, but I would try to find out how the price changes in response to small increments in various "things" (what Quantuple calls state variables) and hope that there are securities traded that correspond to all those "things". $\endgroup$ – noob2 Nov 3 '16 at 12:12

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