A weighted finite difference method for subdiffusive Black Scholes Model

Abstract

In this paper we focus on the subdiffusive Black Scholes model. The main part of our work consists of the finite difference method as a numerical approach to the option pricing in the considered model. We derive the governing fractional differential equation and the related weighted numerical scheme being a generalization of the classical Crank-Nicolson scheme. The proposed method has 2 − α order of accuracy with respect to time where α ∈ (0, 1) is the subdiffusion parameter, and 2 with respect to space. Further, we provide the stability and convergence analysis. Finally, we present some numerical results.

A computational weighted finite difference method for American and barrier options in subdiffusive Black-Scholes model

“…We also present the optimal choice of parameter θ in terms of conservation of unconditional stability/convergence and minimization of potential numerical error. The unconditional stability/convergence is the property that numerical scheme is stable/convergent independently of ∆t and ∆x [13].…”

“…To do so, we will approximate limits by finite numbers and derivatives by finite differences. We will proceed for a θ-convex combination of explicit (θ = 1) and implicit (θ = 0) discrete scheme, similarly as it was done for European options in [13]. We introduce parameter θ ∈ [0, 1] by optimization purposes -similarly as for the case α = 1, θ = 1 2 has the best properties in terms of the error and unconditional stability/convergence [12].…”

“…Let us treat x min and x max not as approximations of infinite values, but as logarithm of lower and supreme barriers H − and H + defined in double barrier option. We take a logarithm because of change of variables x = ln z made in Theorem 4.1 and in [13]. The initial û0 i = max exp ln…”

“…With increasing interest of fractional calculus and non-local differential operators the family of fractional Black-Scholes equations has emerged in the recent literature (see e.g. [13] and references therein).…”

“…The proposed FD method has 2 − α order of accuracy with respect to time, where α ∈ (0, 1) is the subdiffusion parameter, and 2 with respect to space. The paper is a continuation of [13], where the derivation of the governing fractional differential equation, similarly as the stability and convergence analysis can be found.…”