The most important application of the Itô calculus, derived from the Itô lemma, in financial mathematics is the pricing of options. The most famous result in this area is the Black-Scholes formulae for pricing European vanilla call and put options. As a consequence of the formulae, both in theoretical and practical applications, Robert Merton and Myron Scholes were awarded the Nobel Prize for E...

In this paper, Heir-equations method is applied to investigate nonclassical symmetries and new solutions of the Black-Scholes equation. Nonlinear self-adjointness is proved and infinite number of conservation laws are computed by a new conservation laws theorem.

This paper considers the pricing of a European option using a B, S -market in which the stock price and the asset in the riskless bank account both have hereditary price structures described by the authors of this paper 1999 . Under the smoothness assumption of the payoff function, it is shown that the infinite dimensional Black-Scholes equation possesses a unique classical solution. A spectral...

The objective of this paper is twofold. Firstly, to derive time-fractional evolution equation modeling the No-Arbitrage premium Asian option (with arithmetic and geometric averages ) contingent upon an underlying asset that satisfies fractional stochastic differential equation, in a setting when strike price fixed floating. Secondly, we have computed four versions put-call parities for options,...

In this paper, the pricing of a European call option on the underlying asset is performed by using a Monte Carlo method, one of the powerful simulation methods, where the price development of the asset is simulated and value of the claim is computed in terms of an expected value. The proposed approach, applied in Monte Carlo simulation, is based on the Black-Scholes equation which generally def...

in this paper, heir-equations method is applied to investigate nonclassical symmetries and new solutions of the black-scholes equation. nonlinear self-adjointness is proved and infinite number of conservation laws are computed by a new conservation laws theorem.

A Computational Analysis of the Black-Scholes Equations by Yifan Wang This paper explores the most decorated option pricing model in recent history of the financial industry: the Black-Scholes Equation. We will first study the framework of the Black-Scholes Equation in detail by introducing its object of evaluation, distinguished assumptions, and deduction of the Black-Scholes partial different...

In this paper we develop a numerical approach to a fractional-order differential linear complementarity problem (LCP) arising in pricing European and American options under a geometric Lévy process. The (LCP) is first approximated by a penalized nonlinear fractional Black-Scholes (fBS) equation. To numerically solve this nonlinear (fBS), we use the horizontal method of lines to discretize the t...

We construct a three-point compact finite difference scheme on a non-uniform mesh for the time-fractional Black-Scholes equation. We show that for special graded meshes used in finance, the Tavella-Randall and the quadratic meshes the numerical solution has a fourth-order accuracy in space. Numerical experiments are discussed. Introduction The Black-Scholes-Merton model for option prices is an ...