ABSTRACT
Abstract In the previous chapters, we have been focusing on (geometric) approximation methods particularly useful for the calibration of local and stochastic volatility models. However, in the case of particular models, we can have an exact solution for the Kolmogorov and the Black-Scholes equations. In this chapter we provide an extensive classification of one and two dimensional diffusion processes which admit an exact solution to the Kolmogorov (and hence Black-Scholes) equation (in terms of hypergeometric functions). By identifying the one-dimensional solvable processes with the class of integrable superpotentials introduced recently in supersymmetric Quantum Mechanics, we obtain new analytical solutions. In particular, by applying supersymmetric transformations on a known solvable diffusion process (such as the Natanzon process for which the solution is given by a hypergeometric function), we obtain a hierarchy of new solutions. These solutions are given by a sum of hypergeometric functions. For two-dimensional processes, more precisely stochastic volatility models, the classification is achieved for a specific class called gauge-free models including the Heston model, the 3/2-model and the geometric Brownian model. We then present a new exact stochastic volatility model belonging to this class. In addition to our geometrical framework, we will use tools from functional analysis in particular the spectral decomposition of (unbounded) linear operators.
