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.

本文的主要目的是将 Dupire 公式推广和推广到 Margrabe 期权的情况。上述金融衍生品不是普通期权的特例，因此，从理论和实践的角度来看，我们的概括都很重要。我们使用先进的随机方法、概率论和施瓦茨分布理论来为分布空间中的广义 Dupire 公式提供数学上严格的证明。我们将获得的结果应用于经典的 Dupire 局部波动率设置。此外，我们讨论了它在利率互换中的应用。