ABSTRACT
Abstract In this chapter, we present the key tool of this book: the heat kernel expansion on a Riemannian manifold. In the first section, in order to introduce this technique naturally, we remind the reader of the link between the multi-dimensional Kolmogorov equation and the value of a European option. In particular, an asymptotic implied volatility in the short-time limit will be obtained if we can find an asymptotic expansion for the multidimensional Kolmogorov equation. This is the purpose of the heat kernel expansion. Rewriting the Kolmogorov equation as a heat kernel equation on a Riemannian manifold endowed with an Abelian connection, we can apply Hadamard-DeWitt’s theorem giving the short-time asymptotic solution to the Kolmogorov equation. An extension to the time-dependent heat kernel will also be presented as this case is particularly important in finance in order to include term structures. In the next chapters, we will present several applications of this technique, for example the calibration of local and stochastic volatility models.
