Option pricing in Hilbert space valued jump-diffusion models using partial integro-differential equations

Hilbert space-valued jump-diffusion models are employed for various markets and derivatives. Examples include swaptions, which depend on continuous forward curves, and basket options on stocks. Usually, no analytical pricing formulas are available for such products. Numerical methods, on the other hand, suffer from exponentially increasing computational effort with increasing dimension of the problem, the “curse of dimension.” In this paper, we present an efficient approach using partial integrodifferential equations. The key to this method is a dimension reduction technique based on a Karhunen–Loève expansion, which is also known as proper orthogonal decomposition. Using the eigenvectors of a covariance operator, the differential equation is projected to a low-dimensional problem. Convergence results for the projection are given, and the numerical aspects of the implementation are discussed. An approximate solution is computed using a sparse grid combination technique and discontinuous Galerkin discretization. The main goal of this article is to combine the different analytical and numerical techniques needed, presenting a computationally feasible method for pricing European options. Numerical experiments show the effectiveness of the algorithm.     «

Hilbert space-valued jump-diffusion models are employed for various markets and derivatives. Examples include swaptions, which depend on continuous forward curves, and basket options on stocks. Usually, no analytical pricing formulas are available for such products. Numerical methods, on the other hand, suffer from exponentially increasing computational effort with increasing dimension of the problem, the “curse of dimension.” In this paper, we present an efficient approach using partial i...     »