Finite Element Error Estimates for a Nonlocal Problem in American Option Valuation

Superconvergence estimates of finite element methods for American options

“…For numerical approximations, approaches such as the binomial method, the Monte Carlo simulation method, the finite difference method, the finite element method, genetic algorithm approximation, the domain decomposition method are typical. See, for instance, [10], [21], [8], [9], [22], [20], [12], [2], [18], [3], [4], [30], [29], [36], [31], and the references therein. Although convergence analysis is given in some cases (see, for example, [3], [30], and [40] for finite element approximations, and [21], [22] for binomial methods and finite difference methods), to the best of the authors' knowledge there are no known error estimates for most of the numerical methods.…”

“…In practice, this is dealt with by numerically solving the new problem over a large but finite range (see, for example, [9], [20], [23], [26], [39]). Then, as mentioned in [3], two difficulties arise: (1) the computer simulations must be run over a "large" region and thus are relatively slow; (2) an artificial boundary value must be imposed, which affects the accuracy of the simulation. Especially, when the interest rate is greater than the dividend, the accuracy problem becomes serious because of the specific nature of the convection term in the Black-Scholes's partial differential equation.…”

“…Especially, when the interest rate is greater than the dividend, the accuracy problem becomes serious because of the specific nature of the convection term in the Black-Scholes's partial differential equation. In [3], a new nonlocal boundary condition is introduced to eliminate the above two difficulties, which is mathematically exact and allows us to reformulate the original problem as a variational inequality on a very narrow region without changing the solution. Numerical results show (see [3] and [4]) that compared with the existing computational methods, this approach provides very rapid option pricing.…”

“…In this paper we are concerned with sharp L 2 -, L ∞ -norm error estimates and an H 1 -norm global superconvergence estimate of finite element methods to enhance the finite element approximations given in [3] by means of superapproximation estimates between the finite element solution and an interpolating function of the exact solution, and an interpolation postprocessing technique. As a by-product of the superconvergence property, we illustrate that the approximation of higher accuracy can be used to form an a posteriori error estimator for this finite element method.…”

“…This paper is organized in the following way. Following [3], in Section 2 the American option is exactly reformulated as the linear complementarity form of the heat equation with a nonlocal boundary condition on a bounded domain. In addition, we give an equivalent variational inequality to the linear complementarity form and the semidiscrete finite element approximate formula.…”