Exotic option pricing models: continuous and discrete time considerations.

Summary Despite its greater complexity, the valuation of exotic options follows the same principle as that of traditional options, namely that a very broad class of approximations and process dissemination for the valuation of contingent assets is obtained from the same formalism. In continuous time, when the option is European, practitioners sometimes arrive at parametric and non-parametric formulas in analytical form with different starting assumptions, the common denominator of all these methods is that they assume the positivity of the density law of the distribution of the returns of the underlying assets, then thanks to the probability change theorems, the valuation is reduced to the calculations of density laws within a Gaussian random walk associated with a neutral risk diffusion. However, when the option is American, there is no analytical solution and practitioners often use interpolation techniques and approximation by options obtained by convolutions along the optimal exercise frontier. In discrete time, valuation methods with mesh (two or three support points) or without mesh (nearest neighbor method) are based on numerical simulations formulated in terms of stochastic partial differential equations or in variational integral form, requiring a nodal discretization of the domain and a generation of the point cloud covering the domain in the sense of a point density distribution, taking into account both the geometry of the domain and real information related to the nature of the diffusion equations. As a result, the discrete method seems more general and allows to value an important class of American options as well as many more complex exotic options on one or more underlying assets.