Dupire pricing equation derivation vs Black Scholes PDE

Question

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?

Answer 1

The equation you mention is called Dupire Forward PDE. It allows you to compute the price of all vanillas of various strikes and maturities in one go, given the current spot price.

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: $$ \frac{\partial V}{\partial t}(t,S) + \mu(t) S \frac{\partial V}{\partial S}(t,S) + \frac{1}{2} \sigma^2(t,S) S^2 \frac{\partial^2 V}{\partial S^2}(t,S) - r V(t,S) = 0 $$

Answer 2

This is due to the connection between the BS PDE and the dual of that equation, at some point you substitute d/dt with -d/dT.

So K becomes S, dividend yield q becomes risk-free rate r, maturity time T becomes t, d/dT becomes -d/dt. Then the Dupire PDE becomes the Black-Scholes-Merton equation.

See: https://arxiv.org/abs/1912.10380 The Black-Scholes-Merton dual equation, by Shuxin Guo and Qiang Liu, 2019