On Dupire Formula and Diffusion with Given Marginals

This paper presents relations between the drift and the diffusion coefficients of a diffusion in terms of the prices of a European call option, giving an extension to the Dupire formula. Due to the correspondence between the call option prices and the marginal distributions of the underlying process, a necessary condition for a diffusion with given marginal distributions is obtained. Some specific examples, including the the cases of Normal and Lognormal marginals, are considered. In particular, we construct fake Brownian motion diffusions, a family of diffusions, which all are Gaussian processes with Brownian marginals N(0, t) but not a Brownian motion unless σ2 = 1.