Analytical solutions for the Black-Scholes equation

In this paper, the Black-Sholes equation (BS) has been applied successfully with the Cauchy-Euler method and the method of separation of variables and new analytical solutions have been found. The linear partial differential equation (PDE) transformed to linear ordinary differential equation (ODE) as well. We acquired three types of solutions including hyperbolic, trigonometric and rational solutions. Descriptions of these methods are given and the obtained results reveal that three methods are tools for exploring partial differential models

Solución Numérica del Modelo de Heston con Reversión a la Media

En este trabajo se propone una variación del modelo de Heston (1993) en la que se considera el precio del activo subyacente como un proceso de reversión a la media. A partir de esto, se realiza una transformación al problema que permite construir un esquema de diferencias finitas explícitas para la valoración de opciones según este modelo. El esquema es de segundo orden en el espacio y de primer orden en el tiempo. Se presenta un análisis sobre las condiciones para la positividad del esquema. Se prueba la estabilidad condicional en el sentido de von Neumann realizando un análisis de Fourier del problema y se verifica la convergencia del mismo. Se presentan algunos resultados de experimentos numéricos para la valoración de opciones call europea

An efficient numerical method based on exponential B-splines for time-fractional Black-Scholes equation governing European options

In this paper a time-fractional Black-Scholes model (TFBSM) is considered to study the price change of the underlying fractal transmission system. We develop and analyze a numerical method to solve the TFBSM governing European options. The numerical method combines the exponential B-spline collocation to discretize in space and a finite difference method to discretize in time. The method is shown to be unconditionally stable using von-Neumann analysis. Also, the method is proved to be convergent of order two in space and $2-\mu$ is time, where $\mu$ is order of the fractional derivative. We implement the method on various numerical examples in order to illustrate the accuracy of the method, and validation of the theoretical findings. In addition, as an application, the method is used to price several different European options such as the European call option, European put option, and European double barrier knock-out call option.Comment: 34 pages, 12 figure

On Analytical Solutions of the Black-Scholes Equation

This work presents a theoretical analysis for the Black-Scholes equation. Given a terminal condition, the analytical solution of the Black-Scholes equation is obtained by using the Adomian approximate decomposition technique. The mathematical technique employed in this work also has significance in studying some other problems in finance theory