A Family of Generating Functions with an Application in Finance

We introduce the definition of a family of generating functions (FGF) of Toeplitz matrices, which is a generalization of the generating function of Toeplitz matrix. The FGF has an important application in pricing derivatives. The pricing of a European call option leads under certain assumptions to a partial integro-differential equation (PIDE) without a convection term. We then use the FGF to analyze the convergence rate of the preconditioned conjugate gradient (PCG) method with Strang's circulant preconditioner for solving Toeplitz systems arising from a discretization of the PIDE. We show that if the FGF has certain properties, then the spectrum of the preconditioned matrix is clustered around 1, and moreover the smallest eigenvalue of the preconditioned matrix is uniformly bounded away from 0. It follows that the convergence rate of the PCG method is superlinear