Dupire equation ($C_T=\frac{1}{2}{K}^2\sigma (T,K{)}^2{C}_{KK}-rK{C}_K$)

For a non-dividend-paying stock and constant interest rate $r$, discounted call prices satisfy $C_T=\frac{1}{2}{K}^2\sigma (T,K{)}^2{C}_{KK}-rK{C}_K$. Strike differentiation recovers the discounted terminal density as $C_{KK}$; differentiating the discounted payoff identity supplies the maturity derivative. Put-call parity implies the same equation for put prices.

Where $C_{KK}>0$, rearranging the Dupire equation gives $\sigma (T,K{)}^2=2(C_T+rK{C}_K)/(K^2{C}_{KK})$. The strike curvature represents discounted density, while the adjusted maturity derivative gives the local diffusion contribution.