# Dupire pricing equation derivation vs Black Scholes PDE

I know the Dupire pricing equation is derived in similar way to Black Scholes PDE, but it is not exactly the same equation. Dupire equation reads:

$$\boxed{\frac{\partial C}{\partial T} = \frac{\sigma^2(K,T)}{2} \; K^2 \frac{\partial^2 C}{\partial K^2} - (r - q)K \frac{\partial C}{\partial K} - qC}$$

The main difference is that in BS equation the term multiplying the gamma is -1/2, wile in Dupire it is +1/2. Where does this difference comes from?

In John Hull book, the Black Scholes equation is derived in much intuitive way. Is there an intuitive way to derive Dupire equation as well?

The Local Volatility pricing PDE is a different beast. It allows you to find the price of a single instrument (e.g. a vanilla of fixed strike and maturity $$V(t,S) = C(t,S;K,T)$$ ) at different future times (t) for different spot levels (S). The LV PDE is a direct generalisation of the BS PDE: $$\frac{\partial V}{\partial t}(t,S) + \mu(t) S \frac{\partial V}{\partial S}(t,S) + \frac{1}{2} \sigma^2(t,S) S^2 \frac{\partial^2 V}{\partial S^2}(t,S) - r V(t,S) = 0$$