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I understand that the BS equation can be explained by a replicating portfolio, e.g., short an option and long $\Delta$ shares of the underlier [Bergomi's Stochastic Volatility Modeling]. I also understand how to derive the Dupire's formula using Fokker-Planck equation or via a probabilistic approach [Derman and Kani (1998)].

My question is: Is there a replicating portfolio method to arrive at the Dupire's formula?

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    $\begingroup$ Black Scholes equation is a pricing equation (Worth=Cost of replication). Whereas Dupire's formula is a calibration equation (put this as your local vol function given a vanilla market). There is no comparison between the two things, there is no way you can make a replicating portfolio and get Dupire's formula. The 'Black Scholes analog' of Dupire calibration is calibrating BS lognormal volatility to a particular vanilla option - you wouldn't use replication for that. $\endgroup$
    – user121416
    Nov 6, 2021 at 23:54
  • $\begingroup$ Actually I think you may be able to derive the Dupire equation as a replicating portfolio in an economy where the time runs backwards. Recall the duality between delta and the price of a digital and gamma and the density. $\endgroup$
    – user34971
    Nov 7, 2021 at 11:10

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As @user121416 mentioned, Dupire's formula is meant to be a calibration equation rather than a pricing formula. However, note that when using it, you're still able to use a similar replicating portfolio method as in BS.

The difference is the following, and lets think of a tree for simplicity, whereas for determining your delta in BS the same constant volatility is assumed at every node in the tree (i.e. the possible outcomes or future possible scenarios), Dupire (and then Derman and Kani) are telling you that every node of the tree shouldn't use that same volatility, but the volatility assumed by the market.

However, in both approaches the underlying hedging strategy is the same.

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