Exact and numerical solution of Black–Scholes matrix equation

In this paper a new method for solving Black–Scholes equation is proposed. The approach is based on the Mellin transform. A numerical procedure for the approximation of the solution is given.     

Conservation laws for the Black–Scholes equation

In the search for solutions to the important partial differential equation due to Black, Scholes and Merton potential symmetries are very useful as new solutions of the equation can be obtained as a result. These potential symmetries require that the equation be written in conserved form, ie. we need to determine conservation laws for the equation. We calculate the conservation laws utilizing the point symmetries of the equation following the method of Kara and Mahomed [A.H. Kara, F.M. Mahomed, The relationship between symmetries and conservation laws, Int. J. Theor. Phys. 39 (2000) 23–40].     

A superconvergent fitted finite volume method for Black–Scholes equations governing European and American option valuation

We develop a superconvergent fitted finite volume method for a degenerate nonlinear penalized Black–Scholes equation arising in the valuation of European and American options, based on the fitting idea in Wang [IMA J Numer Anal 24 (2004), 699–720]. Unlike conventional finite volume methods in which the dual mesh points are naively chosen to be the midpoints of the subintervals of the primal mesh, we construct the dual mesh judiciously using an error representation for the flux interpolation so that both the approximate flux and solution have the second‐order accuracy at the mesh points without any increase in computational costs. As the equation is degenerate, we also show that it is essential to refine the meshes locally near the degenerate point in order to maintain the second‐order accuracy. Numerical results for both European and American options with constant and nonconstant coefficients will be presented to demonstrate the superconvergence of the method. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1190–1208, 2015     

Applications to give an analytical solution to the Black Scholes equation

A simple ?-function and its dynamic equation is presented. An application to give an analytical solution to the Black Scholes equation is presented.     

Fractional model and solution for the Black‐Scholes equation

This work presents a new model of the fractional Black‐Scholes equation by using the right fractional derivatives to model the terminal value problem. Through nondimensionalization and variable replacements, we convert the terminal value problem into an initial value problem for a fractional convection diffusion equation. Then the problem is solved by using the Fourier‐Laplace transform. The fundamental solutions of the derived initial value problem are given and simulated and display a slow anomalous diffusion in the fractional case.     

A numerical method for European Option Pricing with transaction costs nonlinear equation

This paper deals with the construction of a finite difference scheme and the numerical analysis of its solution for a nonlinear Black–Scholes partial differential equation modelling stock option pricing in the realistic case when transaction costs arising in the hedging of portfolios are taken into account. The analysed model is the Barles–Soner one for which an appropriate fully nonlinear numerical method has not still applied. After construction of the numerical solution, consistency and stability are studied and some illustrative examples are included.     

Convergence of a discrete Oort–Hulst–Safronov equation

A discrete version of the Oort–Hulst–Safronov (OHS) coagulation equation is studied. Besides the existence of a solution to the Cauchy problem, it is shown that solutions to a suitable sequence of those discrete equations converge towards a solution to the OHS equation. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical algorithms for solutions of Korteweg–de Vries equation

The nonlinear Korteweg–de Vries (KdVE) equation is solved numerically using both Lagrange polynomials based differential quadrature and cosine expansion‐based differential quadrature methods. The first test example is travelling single solitary wave solution of KdVE and the second test example is interaction of two solitary waves, whereas the other three examples are wave production from solitary waves. Maximum error norm and root mean square error norm are computed, and numerical comparison with some earlier works is done for the first two examples, the lowest four conserved quantities are computed for all test examples. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

On theoretical pricing of options with fuzzy estimators

In this paper we present an application of a new method of constructing fuzzy estimators for the parameters of a given probability distribution function, using statistical data. This application belongs to the financial field and especially to the section of financial engineering. In financial markets there are great fluctuations, thus the element of vagueness and uncertainty is frequent. This application concerns Theoretical Pricing of Options and in particular the Black and Scholes Options Pricing formula. We make use of fuzzy estimators for the volatility of stock returns and we consider the stock price as a symmetric triangular fuzzy number. Furthermore we apply the Black and Scholes formula by using adaptive fuzzy numbers introduced by Thiagarajah et al. [K. Thiagarajah, S.S. Appadoo, A. Thavaneswaran, Option valuation model with adaptive fuzzy numbers, Computers and Mathematics with Applications 53 (2007) 831–841] for the stock price and the volatility and we replace the fuzzy volatility and the fuzzy stock price by possibilistic mean value. We refer to both cases of call and put option prices according to the Black & Scholes model and also analyze the results to Greek parameters. Finally, a numerical example is presented for both methods and a comparison is realized based on the results.     

The pricing of vulnerable options with double Mellin transforms

Many options traded in the over-the-counter markets are subject to default risks resulting from the probability that the option writer could not honor its contractual obligations. There have been growing concerns about financial derivatives subject to default risks, in particular, since the Global Financial Crisis and Eurozone crisis. This paper uses double Mellin transforms to study European vulnerable options under constant as well as stochastic (the Hull–White) interest rates. We obtain explicitly an analytic closed form pricing formula in each interest rate case so that the pricing of the options can be computed both accurately and efficiently.     

Isomorphism between the solution spaces of a discrete convolution equation and a convolution equation on the space of entire functions

In this paper, we solve the problem of reconstructing an arbitrary solution of a homogeneous convolution equation from its values at integer points of the real axis.     

Numerical solution of evolutionary integral equations with completely monotonic kernel by Runge–Kutta convolution quadrature

We study the numerical solutions of the initial boundary value problems for the Volterra‐type evolutionary integal equations, in which the integral operator is a convolution product of a completely monotonic kernel and a positive definite operator, such as an elliptic partial‐differential operator. The equation is discretized in time by the Runge–Kutta convolution quadrature. Error estimates are derived and numerical experiments reported. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq31: 105–142, 2015     

Test for dispersion constancy in stochastic differential equation models

In this paper, we propose a constancy test for volatility in It processes based on discretely sampled data. The test statistic constitutes an integration of the Ljung–Box test statistic and the kurtosis statistic in the Jarque–Bera test. It is shown that under regularity conditions, the proposed test asymptotically follows a chi‐square distribution under the null hypothesis of constant volatility. To evaluate the test, empirical sizes and powers were examined through a simulation study. Analysis of real data including ultra‐high frequency transaction data and interest rates was also conducted for illustration. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical solution of the nonlinear Klein–Gordon equation using radial basis functions

The nonlinear Klein–Gordon equation is used to model many nonlinear phenomena. In this paper, we propose a numerical scheme to solve the one-dimensional nonlinear Klein–Gordon equation with quadratic and cubic nonlinearity. Our scheme uses the collocation points and approximates the solution using Thin Plate Splines (TPS) radial basis functions (RBF). The implementation of the method is simple as finite difference methods. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the good accuracy of the presented scheme.     

Numerical solution of linear integro-differential equation by using sine–cosine wavelets

We use the continuous sine–cosine wavelets on the interval [0, 1] to solve the linear integro-differential equation. To do so, we construct the quadrature formulae for the calculation of inner products of any functions, which are required in the approximation for the integro-differential equation. Then, we reduced the integro-differential equation to the solution of linear algebraic equations.     

Numerical solution of the Bagley–Torvik equation by the Bessel collocation method

In this article, a numerical technique is presented for the approximate solution of the Bagley–Torvik equation, which is a class of fractional differential equations. The basic idea of this method is to obtain the approximate solution in a generalized form of the Bessel functions of the first kind. For this purpose, by using the collocation points, the matrix operations and a generalization of the Bessel functions of the first kind, this technique transforms the Bagley–Torvik equation into a system of the linear algebraic equations. Hence, by solving this system, the unknown Bessel coefficients are computed. The reliability and efficiency of the proposed scheme are demonstrated by some numerical examples. Copyright © 2012 John Wiley & Sons, Ltd.     

A numerical scheme for the solution of viscous Cahn–Hilliard equation

In this paper, we present a numerical scheme for the solution of viscous Cahn–Hilliard equation. The scheme is based on Adomian's decomposition approach and the solutions are calculated in the form of a convergent series with easily computable components. Some numerical examples are presented. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Numerical solution of the Rosenau–KdV–RLW equation by operator splitting techniques based on B‐spline collocation method

In the present study, the operator splitting techniques based on the quintic B‐spline collocation finite element method are presented for calculating the numerical solutions of the Rosenau–KdV–RLW equation. Two test problems having exact solutions have been considered. To demonstrate the efficiency and accuracy of the present methods, the error norms L₂ and L_∞ with the discrete mass Q and energy E conservative properties have been calculated. The results obtained by the method have been compared with the exact solution of each problem and other numerical results in the literature, and also found to be in good agreement with each other. A Fourier stability analysis of each presented method is also investigated.     

Optimal control problem for the modified Swift–Hohenberg equation in 3D case

In this paper, for 3D modified Swift–Hohenberg equation, the optimal control problem is considered, the existence of optimal solution is proved, and the optimality system is established. Copyright © 2015 John Wiley & Sons, Ltd.