A semilinear Black and Scholes partial differential equation for valuing American options: approximate solutions and convergence

Abstract

In [7], we proved that the American (call/put) option valuation problem can be stated in terms of one single semilinear Black and Scholes partial differential equation set in a fixed domain. The semilinear Black and Scholes equation constitutes a starting point for designing and analyzing a variety of "easy to implement" numerical schemes for computing the value of an American option. To demonstrate this feature, we propose and analyze an upwind finite difference scheme of "predictor-corrector type" for the semilinear Black and Scholes equation. We prove that the approximate solutions generated by the predictor-corrector scheme respect the early exercise constraint and that they converge uniformly to the American option value. A numerical example is also presented. Besides the predictor-corrector schemes, other methods for constructing approximate solution sequences are discussed and analyzed.

A new predictor–corrector scheme for valuing American puts

“…Zhao et al [34] used a compact finite difference method to transform the original PDE to a set of ordinary differential equations (ODEs) to cope with the movement of this highly nonlinear optimal exercise boundary. More recently, Benth et al [5] adopted an upwind finite difference scheme to solve a semilinear Black-Scholes equation. Cho et al [9] used a set of linearized equations with a function space parameter estimation FDM to solve the option values and optimal exercise prices.…”

A semilinear equation for the American option in a general jump market

“…Thus, the equation is also referred to as the nonhomogeneous Black and Scholes equation in the literature. Simple examples of such schemes for the Black and Scholes market were studied and analyzed in Benth et al [21]. The SLBS equation is also related to so called penalty schemes, which have been studied in connection to American option pricing in [38,75,58], as some of these schemes can be seen as approximations to the semilinear equation (1.1).…”

“…The SLBS equation is also related to so called penalty schemes, which have been studied in connection to American option pricing in [38,75,58], as some of these schemes can be seen as approximations to the semilinear equation (1.1). We refer to [21] for a rigorous derivation of this connection in the pure diffusion case. The design and analysis of numerical schemes for the nonlocal semilinear equation (1.1), which is outside the scope of this paper, will be the topic of future work.…”

Convergence of a fitted finite volume method for the penalized Black–Scholes equation governing European and American Option pricing

“…We comment that the papers either lack investigation of convergence rates or assume unrealistic step-size restrictions. Another notable work is [8] in which the authors propose and analyze an upwind finite difference method of predictor-corrector type. Under a certain step-size restriction, they prove the convergence in the L ∞ loc -sense, but are not able to establish a rate of convergence.…”

“…(9) reduces to the standard linear Black-Scholes Eq. (8). It has been shown in [30] that the solution of (9) converges to that of (1)-(3) at the rate of order O(λ −k/2 ).…”