Stochastic flow approach to Dupire’s formula

The equivalent probabilistic formulation of Dupire’s PDE is the put-call duality equality. In local volatility models including exponential Levy jumps, we give a direct probabilistic proof for this result based on stochastic flow arguments. This approach also enables us to check the equivalent probabilistic formulation of various generalizations of Dupire’s PDE recently obtained by Pironneau [C. R. Acad. Sci. Paris Ser. I 344(2) 127–133 (2007)] by the adjoint equation technique in the case of complex options.