  The probabilistic equivalent formulation of Dupire's PDE is the Put-Call
duality equality. In local volatility models including exponential L\'{e}vy
jumps, we give a direct probabilistic proof for this result based on stochastic
flows arguments. This approach also enables us to check the probabilistic
equivalent formulation of various generalizations of Dupire's PDE recently
obtained by Pironneau by the adjoint equation technique in the case of complex
options.
