javad damirchi

The standard model, which determines option pricing, is the well-known Black-Scholes formula. Heston in addition to Cox-Ingersoll-Ross which is called CIR, respectively, implemented the models of stochastic volatility and interest rate to the standard option pricing model. The cost of transaction, which the Black-Scholes method overlooked, is another crucial consideration that must be made when trading a service or production. It is acknowledged that by employing the log-normal stock diffusion hypothesis with constant volatility, the Black-Scholes model for option pricing departs from reality. The standard log-normal stock price distribution used in the Black-Scholes model is insufficient to account for the leaps that regularly emerge in the discontinuous swings of stock prices. A jump-diffusion model, which combines a jump process and a diffusion process is a type of mixed model in the Black-Scholes model belief. Merton developed a jump model as a modification of jump models to better describe purchasing and selling behavior. In this study, the Heston-Cox-Ingersoll-Ross (HCIR) model with transaction costs is solved using the alternating direction implicit (ADI) approach and the Monte Carlo simulation assuming the underlying asset adheres to the jump-diffusion case, then the outcomes are compared to the analytical solution. In addition, the consistency of the numerical method is proven for the model.

In this research paper, a numerical method for one- and two- dimensional heat equation with nonlinear diffusion conductivity and source terms is proposed. In this work, the numerical technique is based on the polynomial differential quadrature method for discretization of the spatial domain. The resulting nonlinear system time depending ordinary differential equations is discretized by using the second order Runge–Kutta methods. The Chebyshev-Gauss-Lobatto points in this paper are used for collocation points in spatial discretization. We study accuracy in terms of L_∞ error norm and maximum absolute error along time levels. Finally, several test examples demonstrate the accuracy and efficiency of the proposed schemes. It is shown that the numerical schemes give better solutions. Moreover, the schemes can be easily applied to a wide class of higher dimension nonlinear diffusion equations.

‎The inverse problem considered in this paper is devoted to reconstruction of the unknown source term in parabolic equation from additional information which is given by measurements at final time‎. ‎The cost functional is introduced and existence of the minimizer for this functional is established‎. ‎The numerical algorithm to solve the inverse problem is based on the Ritz-Galerkin method with shifted Legendre polynomials as basis functions‎. ‎Finally‎, ‎some numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for test example‎.

‎In this paper‎, ‎an inverse problem of determining an unknown reaction coefficient in a one-dimensional time-fractional reaction-diffusion equation is considered‎. ‎This inverse problem is generally ill-posed‎. ‎For this reason‎, ‎the mollification regularization technique with the generalized cross-validation criteria will be employed to find an equivalent stable problem‎. ‎Afterward‎, ‎a finite difference marching scheme is introduced to solve this regularized problem‎. ‎The stability and convergence of the numerical solution are investigated‎. ‎In the end‎, ‎some numerical examples are presented to verify the ability and effectiveness of the proposed algorithm‎.

In this paper, a numerical method based on the method of fundamental solutions (MFS) is employed for solving some two dimensional direct and inverse heat conduction problems. Based on the fundamental solution to the heat equation and theoretical properties of these solutions, including linear independence and denseness, wih suitbale placement of source points, the MFS is introduced for solving two dimensional heat conduction problems. Since the resultant matrix of the MFS is ill-conditioned for solving direct and inverse problems, to regularize this matrix equation, we apply Tikhonov regularization technique, while the choice of the regularization parameter is based on L-curve critera to obtain a stable solution. Numerical results show the effectiveness and ability of the proposed method.