A note on a stationary problem for a Black-Scholes equation with transaction cost

In this paper, we consider the nonlinear Black-Scholes equation arising in certain option pricing models with transaction costs. Following the classical Leland approach and applying Itô’s Lemma, the stochastic model yields the nonlinear parabolic partial differential equation for the option price ….info:eu-repo/semantics/publishedVersio

High order compact finite difference schemes for a nonlinear Black-Scholes equation

A nonlinear Black-Scholes equation which models transaction costs arising in the hedging of portfolios is discretized semi-implicitly using high order compact finite difference schemes. In particular, the compact schemes of Rigal are generalized. The numerical results are compared to standard finite difference schemes. It turns out that the compact schemes have very satisfying stability and non-oscillatory properties and are generally more e±cient than the considered classical schemes.Option pricing, transaction costs, parabolic equations, compact finite difference discretizations

Numerical Solution of Nonlinear Black – Scholes Equation by Accelerated Genetic Algorithm

In this paper we using an accelerated genetic algorithm to find the numerical solution of the nonlinear versions of the standard Black–Scholes partial differential equation  with stochastic volatility (transaction coast) for European call option .  We study this equation with different models of volatility and comparison these solutions with the solution of linear model of Black-Scholes equation without transaction coast . Keywords: Nonlinear Black-Scholes Equation , Accelerated Genetic Algorithm , Option Pricin

Analytical and numerical results for American style of perpetual put options through transformation into nonlinear stationary Black-Scholes equations

We analyze and calculate the early exercise boundary for a class of stationary generalized Black-Scholes equations in which the volatility function depends on the second derivative of the option price itself. A motivation for studying the nonlinear Black Scholes equation with a nonlinear volatility arises from option pricing models including, e.g., non-zero transaction costs, investors preferences, feedback and illiquid markets effects and risk from unprotected portfolio. We present a method how to transform the problem of American style of perpetual put options into a solution of an ordinary differential equation and implicit equation for the free boundary position. We finally present results of numerical approximation of the early exercise boundary, option price and their dependence on model parameters

Convergence of a high-order compact finite difference scheme for a nonlinear Black-Scholes equation

A high-order compact finite difference scheme for a fully nonlinear parabolic differential equation is analyzed. The equation arises in the modeling of option prices in financial markets with transaction costs. It is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. The proof is based on a careful study of the discretization matrices and on an abstract convergence result due to Barles and Souganides.High-order compact finite differences, numerical convergence, viscosity solution, financial derivatives

Computing option pricing models under transaction costs

AbstractThis paper deals with the Barles–Soner model arising in the hedging of portfolios for option pricing with transaction costs. This model is based on a correction volatility function Ψ solution of a nonlinear ordinary differential equation. In this paper we obtain relevant properties of the function Ψ which are crucial in the numerical analysis and computing of the underlying nonlinear Black–Scholes equation. Consistency and stability of the proposed numerical method are detailed and illustrative examples are given