Fitted finite volume method for a generalized Black–Scholes equation transformed on finite interval

Abstract

A generalized Black-Scholes equation is considered on the semiaxis. It is transformed on the interval (0, 1) in order to make the computational domain finite. The new parabolic operator degenerates at the both ends of the interval and we are forced to use the Gärding inequality rather than the classical coercivity. A fitted finite volume element space approximation is constructed. It is proved that the time θ-weighted full discretization is uniquely solvable and positivity-preserving. Numerical experiments, performed to illustrate the usefulness of the method, are presented.

American option pricing problem transformed on finite interval

“…These results are consistent with the estimate (10), showing profit in the order of convergence when computing with large values of ξ . We stress that theoretical considerations suggest first order of spatial convergence [1,19] while some of the numerical results in [17] and also Table 2 motivate investigation of the superconvergence property.…”

“…In the neighbourhood of S = ∞ the convergence rate deteriorates. This region is, in general, not of practical interest but one may threat this defect, for example, by the power-graded mesh in [17].…”

“…In this section we focus on the spatial discretization of the following penalized problem (7) by the FVM, developed by Wang [19] and further in [1,17].…”

“…The finite element space V h is introduced by specifying it's basis {φ i } N i=0 [17] and for any N, and arrive at the following semi-discretization of Equation (13) in the space V h :…”

“…Section 3 presents the monotonic penalty method and analysis of the penalized problem. A fitted finite volume method (FVM) [17] is considered in Section 4 in order to overcome the degeneration and, further, the time stepping is performed by the θ-method as we prove solvability of the corresponding semi-discrete and fully discrete problems. Numerical experiments in Section 6 illustrate the stability, convergence and positivity of the generated numerical solution.…”