We introduce two algorithms in order to find the exact solution of the nonlinear Time-fractional Partial differential equation, in this research work. Those algorithms are proposed in the following structure: The Modi...We introduce two algorithms in order to find the exact solution of the nonlinear Time-fractional Partial differential equation, in this research work. Those algorithms are proposed in the following structure: The Modified Homotopy Perturbation Method (MHPM), The Homotopy Perturbation and Sumudu Transform Method. The results achieved using the both methods are the same. However, we calculate the approached theoretical solution of the Black-Scholes model in the form of a convergent power series with a regularly calculated element. Finally, we propose a descriptive example to demonstrate the efficiency and the simplicity of the methods.展开更多
In this study,we prove that modified diffusion equations,including the generalized Burgers'equation with variable coefficients,can be derived from the Black-Scholes equation with a time-dependent parameter based o...In this study,we prove that modified diffusion equations,including the generalized Burgers'equation with variable coefficients,can be derived from the Black-Scholes equation with a time-dependent parameter based on the propagator method known in quantum and statistical physics.The extension for the case of a local fractal derivative is also addressed and analyzed.展开更多
This paper addresses a finite difference approximation for an infinite dimensional Black-Scholesequation obtained by Chang and Youree (2007).The equation arises from a consideration ofan European option pricing proble...This paper addresses a finite difference approximation for an infinite dimensional Black-Scholesequation obtained by Chang and Youree (2007).The equation arises from a consideration ofan European option pricing problem in a market in which stock prices and the riskless asset prices havehereditary structures.Under a general condition on the payoff function of the option,it is shown thatthe pricing function is the unique viscosity solution of the infinite dimensional Black-Scholes equation.In addition,a finite difference approximation of the viscosity solution is provided and the convergenceresults are proved.展开更多
In this paper, we first present an option pricing model of stochastic differential equations driven by the G-Lévy process under the G-expectation framework, and prove the generalized Black-Scholes equations. Then...In this paper, we first present an option pricing model of stochastic differential equations driven by the G-Lévy process under the G-expectation framework, and prove the generalized Black-Scholes equations. Then, we present the algorithm for the time-homogeneous Poisson process versus the non-time-homogeneous Poisson process. Finally, we provide an explicit solution of generalized Black-Scholes equations and simulate it numerically with Matlab software.展开更多
In this paper the Black Scholes differential equation is transformed into a parabolic heat equation by appropriate change in variables. The transformed equation is semi-discretized by the Method of Lines (MOL). The ev...In this paper the Black Scholes differential equation is transformed into a parabolic heat equation by appropriate change in variables. The transformed equation is semi-discretized by the Method of Lines (MOL). The evolving system of ordinary differential equations (ODEs) is integrated numerically by an L-stable trapezoidal-like integrator. Results show accuracy of relative maximum error of order 10–10.展开更多
文摘We introduce two algorithms in order to find the exact solution of the nonlinear Time-fractional Partial differential equation, in this research work. Those algorithms are proposed in the following structure: The Modified Homotopy Perturbation Method (MHPM), The Homotopy Perturbation and Sumudu Transform Method. The results achieved using the both methods are the same. However, we calculate the approached theoretical solution of the Black-Scholes model in the form of a convergent power series with a regularly calculated element. Finally, we propose a descriptive example to demonstrate the efficiency and the simplicity of the methods.
基金The authors would like to thank the anonymous referees for their useful comments and valuable suggestions.
文摘In this study,we prove that modified diffusion equations,including the generalized Burgers'equation with variable coefficients,can be derived from the Black-Scholes equation with a time-dependent parameter based on the propagator method known in quantum and statistical physics.The extension for the case of a local fractal derivative is also addressed and analyzed.
基金supported by a grant W911NF-04-D-0003 from the US Army Research Office
文摘This paper addresses a finite difference approximation for an infinite dimensional Black-Scholesequation obtained by Chang and Youree (2007).The equation arises from a consideration ofan European option pricing problem in a market in which stock prices and the riskless asset prices havehereditary structures.Under a general condition on the payoff function of the option,it is shown thatthe pricing function is the unique viscosity solution of the infinite dimensional Black-Scholes equation.In addition,a finite difference approximation of the viscosity solution is provided and the convergenceresults are proved.
文摘In this paper, we first present an option pricing model of stochastic differential equations driven by the G-Lévy process under the G-expectation framework, and prove the generalized Black-Scholes equations. Then, we present the algorithm for the time-homogeneous Poisson process versus the non-time-homogeneous Poisson process. Finally, we provide an explicit solution of generalized Black-Scholes equations and simulate it numerically with Matlab software.
文摘In this paper the Black Scholes differential equation is transformed into a parabolic heat equation by appropriate change in variables. The transformed equation is semi-discretized by the Method of Lines (MOL). The evolving system of ordinary differential equations (ODEs) is integrated numerically by an L-stable trapezoidal-like integrator. Results show accuracy of relative maximum error of order 10–10.
