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Generally, Dupire's formula is taking derivatives on the call option prices. Here it only uses information of the call options.

If now we have the data including both call and put options, is there a mathematical formula using all the information? Or is there a corresponding Dupire's formula for put options?

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    $\begingroup$ Just transform your puts into calls using the put-call parity? $\endgroup$
    – Alex
    Nov 20 '20 at 8:39
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It depends what you exactly call Dupire's formula. If you take the original formula, valid under zero interest rates and dividends (or equivalently, considering undiscounted option prices on the forwards), which reads $$\sigma_L^2 = 2 \frac{ \frac{\partial C}{\partial T} }{K^2 \frac{\partial^2 C}{\partial K^2}}\,.$$

Then the formula for a put is the same, as the implied density is the same: $\frac{\partial^2 C}{\partial K^2} = \frac{\partial^2 P}{\partial K^2}$. Using the Put-Call parity formula $C-P = F-K$, which also allows to derive the equations in the more general case, we obtain

$$\sigma_L^2 = 2 \frac{ \frac{\partial P}{\partial T} }{K^2 \frac{\partial^2 P}{\partial K^2}}\,.$$

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