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Options, Stochastic volatility, SDEs, PDEs, Duhamel’s principle, 60H10, 91B24, C02, G13,

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This work is devoted to the study of the existence and smoothness of the marginal densities of the solution of one-dimensional backward stochastic differential equations. Under monotonicity conditions of a function of the coefficients, we obtain that the smoothness properties of the forward process influencing the backward equation, transfer to the densities of the solution. Once established these conditions, we apply the result to study the tail behavior of the solution process. (2000): 60H10. Mathematics Subject Classification

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