Abstract:
This paper will discuss the importance of the Black-Scholes equation and its applications in finance. Also, the ways to solve the Black-Scholes equation will be discuss in length.

Abstract:
We determine the algebra of isovectors for the Black--Scholes equation. As a consequence, we obtain some previously unknown families of transformations on the solutions.

Abstract:
Motivated by the work of Segal and Segal on the Black-Scholes pricing formula in the quantum context, we study a quantum extension of the Black-Scholes equation within the context of Hudson-Parthasarathy quantum stochastic calculus. Our model includes stock markets described by quantum Brownian motion and Poisson process.

Abstract:
In common finance literature, Black-Scholes partial differential equation of option pricing is usually derived with no-arbitrage principle. Considering an asset market, Merton applied the Hamilton-Jacobi-Bellman techniques of his continuous-time consumption-portfolio problem, deriving general equilibrium relationships among the securities in the asset market. In special case where the interest rate is constant, he rederived the Black-Scholes partial differential equation from the general equilibrium asset market. In this work, I follow Cox-Ingersoll-Ross formulation to consider an economy which includes (1) uncertain production processes, and (2) the random technology change. Assuming a random production stochastic process of constant drift and variance, and assuming a random technology change to follow a log normal process, the equilibrium point of this economy will lead to the Black-Scholes partial differential equation for option pricing.

Abstract:
In this paper symmetry expansions for Black-Scholes equation are studied. Differently then the other studies in the literature for Black-Scholes equation, we expanded the equation into a parametric form by adding a coefficient a. By using this expansion the dimension of the solution spaces is increased by one and symmetry reduction will be done with the deterministic equations in the new increased solution spaces.

Abstract:
In our paper we build a reccurence from Black-Scholes equation and discretization of domain. From reccurencewe build 3 algorithms: one for a serial machine (A1), one for aPRAM parallel machine (A2) and one for message based parallelmachine (A3).

Abstract:
Option pricing with transaction costs leads to a nonlinear Black-Scholes type equation where the nonlinear term reflects the presence of transaction costs. Under suitable conditions, we prove the existence of weak solutions in a bounded domain and we extend the results to the whole domain using a diagonal process.

Abstract:
In this paper we apply the innovative Laplace transformation method introduced by Sheen, Sloan, and Thom\'ee (IMA J. Numer. Anal., 2003) to solve the Black-Scholes equation. The algorithm is of arbitrary high convergence rate and naturally parallelizable. It is shown that the method is very efficient for calculating various options. Existence and uniqueness properties of the Laplace transformed Black-Scholes equation are analyzed. Also a transparent boundary condition associated with the Laplace transformation method is proposed. Several numerical results for various options under various situations confirm the efficiency, convergence and parallelization property of the proposed scheme.

Abstract:
We analyze a generalized version of the Black-Scholes equation depending on a parameter $a\!\in \!(-\infty,0)$. It satisfies the martingale condition and coincides with the Black-Scholes equation in the limit case $a\nearrow 0$. We show that the generalized equation is exactly solvable in terms of Hermite polynomials and numerically compare its solution with the solution of the Black-Scholes equation.

Abstract:
The Operator Splitting method is applied to differential equations occurring as mathematical models in financial models. This paper provides various operator splitting methods to obtain an effective and accurate solution to the Black-Scholes equation with appropriate boundary conditions for a European option pricing problem. Finally brief comparisons of option prices are given by different models.