Dupire Equation: Interpretation

The backward BS equation applies to a given call option and relates its time derivative to its convexity. 

It is a heat equation that defines the price at a given time as the discounted expectation of the call price an instant later. 


According to the forward Dupire equation, the cost of extending the maturity of a call depends on the probability of being at the strike at maturity and on the level of volatility there. It can be seen as relating the price of a calendar spread to the price of a butterfly spread.