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In the Dupire's paper, he assumes that there exits a function $\sigma(t,S)$ that can reproduce $C(T, K)$. My question is that: is the assumption true for any twice differential continuous function $C(T, K)$? Since Dupire's paper came out in 1994, is there any literature formally prove the assumption? I did some research and did not find any correlated materials.

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  • $\begingroup$ Note that $C(K,T)$ is not any function, it's a European vanilla price. As such the second derivative of $C(K,T)$ with respect to strike is related to the risk-neutral PDF at K (Breeden-Litzenger identity). Without a well defined pdf for the future distribution of an underlying price, you cannot hope to go much further $\endgroup$ – Quantuple Mar 11 at 7:08

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