Dupire Equation: Introduction

The Dupire equation is a PDE that links the contemporaneous prices of European call options of all strikes and maturities to instantaneous volatility of the price process, assumed to be a function of price and time only. 


The main application of the equation is to compute local volatilities from market option prices to build a local volatility model, which many major banks currently use for option pricing. 


If we assume that the price process S follows the SDE: Dupire 1

Then if C(S, t, K, T) denotes the price at time t of the European call of strike K and maturity T for an underlying price of S that pays (ST - K)+ at time T, 

C satisfies, for a fixed (S,t), the Dupire equation:  Dupire 2     and the boundary conditions: C(S, t, K, T) = (ST - K)+

It is commonly named the forward equation, as it indicates how current call prices are affected by an increase in maturity. 

This can be contrasted with the classical backward BS PDE that applies to a European call of fixed strike and maturity: 

Bs pde


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