# For any twice differential continuous function C(T, K), does there exist a sigma(t, S) that can reproduce C(T, K)?

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.

• 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 Mar 11 '20 at 7:08