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Application of Itô processes and Schwartz distributions to local volatility for Margrabe options

The main aim of this paper is a generalization and extension of the Dupire formula to the case of the Margrabe option. The aforementioned financial derivative is not a special case of a plain vanilla option, and therefore, our generalization is significant from both a theoretical and practical point of view. We use advanced stochastic methods, the theory of probability, and the theory of Schwartz distributions to provide a mathematically rigorous proof of the generalized Dupire formula in the space of distributions. We apply the obtained result in the classical Dupire local volatility setting. Furthermore, we discuss its application to interest rate swaptions.