In this article, we propose a numerical algorithm for computing price of discrete single and double barrier option under the emph{Black-Scholes} model. In virtue of some general transformations, the partial differential equations of option pricing in different monitoring dates are converted into simple diffusion equations. The present method is fast compared to alterna...

Abstract. The purpose of this survey chapter is to present a transformation technique that can be used in analysis and numerical computation of the early exercise boundary for an American style of vanilla options that can be modelled by class of generalized Black-Scholes equations. We analyze qualitatively and quantitatively the early exercise boundary for a linear as well as a class of nonline...

This paper deals with the numerical solution of Black–Scholes option pricing partial differential equations by means of semidiscretization technique. For the linear case a fourth-order discretization with respect to the underlying asset variable allows a highly accurate approximation of the solution. For the nonlinear case of interest modeling option pricing with transaction costs, semidiscreti...

We study properties of solutions to fully nonlinear versions of the standard Black– Scholes partial differential equation. These equations have been introduced in financial mathematics in order to deal with illiquid markets or with stochastic volatility. We show that typical nonlinear Black–Scholes equations can be viewed as dynamic programming equation of an associated control problem. We esta...

Stochastic differential equations and the Black-Scholes PDE. We derived the BlackScholes formula by using arbitrage (risk-neutral) valuation in a discrete-time, binomial tree setting, then passing to a continuum limit. This section explores an alternative, continuoustime approach via the Ito calculus and the Black-Scholes differential equation. This material is very standard; I like Wilmott-How...

Using the Mellin transform a new method for solving the Black-Scholes equation is proposed. Our approach does not require either variable transformations or solving diffusion equations.

In this work, we apply He’s variotional iteration method for obtaining analytic solutions to nonlinear Black-Scholes equation with boundary conditions for European option pricing problem. The analytical solution of the equation is calculated in the form a convergent power series with easily computable components. The powerful VIM method is capable of handling both linear and non-linear equation...

Comparing with the linear Black–Scholes model, fractional option pricing models are constructed by taking account some more parameters like, for example, transaction cost, so that it becomes difficult to find exact analytical solution. In this paper, we analyze a nonlinear Black and Scholes solution using novel numerical method, based on mixture of efficient techniques. particular, combine (1) ...

We prove a sharp version of the Hopf boundary point lemma for Black-Scholes type equations. We also investigate the existence and the regularity of the spatial derivative of the solutions at the spatial boundary.