Solution of nonlinear equation representing a generalization of the Black-Scholes model using ADM

Abstract

In this work the Adomian decomposition method (ADM) is used to solve the non-linear equation that represents the generalized model of Black-Scholes, that is to say that considers the volatility as a nonconstant function. The efficiency of this method is illustrated by investigating the convergence results for this type of models. The numerical results show the reliability and accuracy of the ADM.

From Black-Scholes to Hamilton-Jacobi

“…Where H(DV ) is the Hamiltonian associated with the Black-Scholes differential equation. From 7and (8) we have V t + H(DV ) = 0,…”

“…Where V = f (S, t) is the value of a call option, S represents the price of the underlying asset, r the risk-free rate, σ the volatility of the asset price, E the strike and T the maturation time of the option. The solution of this problem is classical when the volatility is a constant and can be obtained from the heat equation and appears related in (13), and when a volatility is no-constant in [8].…”

Modelling Fractional Advection–Diffusion Processes via the Adomian Decomposition

“…These difficulties have led, in the past decade, to the formulation of innovative methodologies to obtain the required solutions, avoiding discretization and linearization, such as the Adomian Decomposition Method [37][38][39][40][41], homotopy perturbation method [42], He's variational iteration method [43], the homotopy analysis method [44], the Galerkin method [45], and the collocation method [46]. Among these, the Adomian Decomposition Method (ADM), a semi-analytical method, acquired a prestigious position due to its effective and simple procedures for obtaining numerical solutions, still maintaining high accuracy solutions of a wide class of partial differential equations, linear or nonlinear, homogeneous or inhomogeneous, with constant coefficients or with variable coefficients, both integer and fractional [47,48].…”

Entropy method for generalized Poisson–Nernst–Planck equations

“…Its rigorous asymptotic analysis was carried our in [7,8]. For a broad class of other relevant transport equations we refer to [6,12,15,22], to [13] for stochastic systems, and to [17,25] for variational principles.…”