In this paper we present a novel numerical method for a degenerate partial differential equation, called the Black–Scholes equation, governing option pricing. The method is based on a fitted finite volume spatial discretization and an implicit time stepping technique. To derive the error bounds for the spatial discretization of the method, we formulate it as a Petrov–Galerkin finite element method with each basis function of the trial space being determined by a set of two‐point boundary value problems defined on element edges. Stability of the discretization is proved and an error bound for the spatial discretization is established. It is also shown that the system matrix of the discretization is an M‐matrix so that the discrete maximum principle is satisfied by the discretization. Numerical experiments are performed to demonstrate the effectiveness of the method.

In this paper we propose a new arithmetic and a novel order relation for interval numbers. We shall show that this new interval arithmetic satisfies some operational properties and has the merit that it reduces uncertainties coming from classic ones. We shall also show that the order relation introduced in this paper is a linear or partial order (satisfying reflexivity, anti-symmetry and transitivity), and has the property of comparability. This is in contrast to the existing interval orders which do not have the property of comparability. Numerical examples on profit intervals and uncertain interval data will be presented to demonstrate the usefulness and applicability of these new arithmetic and order relations.