This paper aims to find numerical solutions of the non-linear Black-Scholes partial differential equation (PDE), which often appears in financial markets, for European option pricing in the appearance of the transaction costs. Here we exploit the transformations for the computational purpose of a non-linear Black-Scholes PDE to modify as a non-linear parabolic type PDE with reliable initial and boundary conditions for call and put options. Several schemes are derived rigorously using the finite volume method (FVM) and finite difference method (FDM), which is the novelty of this paper. Stability and consistency analysis assure the convergence of these schemes. We apply these schemes to various volatility models, such as the Leland, Boyle and Vorst, Barles and Soner, and Risk-adjusted pricing methodology (RAPM). All the schemes are tested numerically. The convergence of the obtained results is observed, and we find that they are also reliable. Finally, we display all the approximate results together with the exact values through graphical and tabular representations.
Article History KeywordsBlack-Scholes equation European call option European put option Du Fort-Frankel finite difference method (DF3DM) Galerkin weighted residual method (GWRM) Modified legendre polynomials.The main objective of this paper is to find the approximate solutions of the Black-Scholes (BS) model by two numerical techniques, namely, Du Fort-Frankel finite difference method (DF3DM), and Galerkin weighted residual method (GWRM) for both (call and put) type of European options. Since both DF3DM and GWRM are the most familiar numerical techniques for solving partial differential equations (PDE) of parabolic type, we estimate options prices by using these techniques. For this, we first convert the Black-Scholes model into a modified parabolic PDE, more specifically, in DF3DM, the first temporal vector is calculated by the Crank-Nicolson method using the initial boundary conditions and then the option price is evaluated. On the other hand, in GWRM, we use piecewise modified Legendre polynomials as the basis functions of GWRM which satisfy the homogeneous form of the boundary conditions. We may observe that the results obtained by the present methods converge fast to the exact solutions. In some cases, the present methods give more accurate results than the earlier results obtained by the adomian decomposition method [14]. Finally, all approximate solutions are shown by the graphical and tabular representations.Contribution/Originality: The paper's primary contribution is finding that the approximate results of Black-Scholes model by DF3DM, and GWRM with modified Legendre polynomials as basis functions.
Numerical solution of ordinary differential equations is the most important technique which is widely used for mathematical modelling in science and engineering. The differential equation that describes the problem is typically too complex to precisely solve in real-world circumstances. Since most ordinary differential equations are not solvable analytically, numerical computations are the only way to obtain information about the solution. Many different methods have been proposed and used is an attempt to solve accurately various types of ordinary differential equations. Among them, Runge-Kutta is a well-known and popular method because of their good efficiency. This paper contains an analysis for the computations of the modified Runge-Kutta method for nonlinear second order initial value problems. This method is wide quite efficient and practically well suited for solving linear and non-linear problems. In order to verify the accuracy, we compare numerical solution with the exact solution. We also compare the performance and the computational effort of this method. In order to achieve higher accuracy in the solution, the step size needs to be small. Finally, we take some examples of non-linear initial value problems (IVPs) to verify proposed method. The results of that example indicate that the convergence, stability analysis, and error analysis which are discussed to determine the efficiency of the method.
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