Nonconforming Least-Squares Spectral Element Method for European Options

Abstract

a b s t r a c tSeveral methods have been proposed in the literature for solving the Black-Scholes equation for European Options. The method proposed in the current study achieves spectral accuracy in both space and time. The method is based on minimization of a functional given in terms of the sum of squares of the residuals in the partial differential equation and initial condition in different Sobolev norms, and a term which measures the jump in the function and its derivatives across inter-element boundaries in appropriate fractional Sobolev norms. To obtain values of the solution and its derivatives the initial condition is mollified and the computed solution is post processed. Error estimates are obtained for this method. Specific numerical examples are given to show the efficiency of this method.

Spectral element method for parabolic interface problems

“…Here, (v) τ and ∂v ∂ν denote the tangential and normal derivatives onγ m same as defined in [8,11,12]. As defined in (3.8), we choose our approximate solution to be the unique F…”

“…Dutt et al [8] proposed h − version and p − version least-squares spectral element methods for parabolic partial differential equations (PDE) with smooth coefficients on bounded domains. Recently, we proposed the least-squares spectral element method for parabolic initial value problems with non-smooth data in [11,12]. The method proposed in this paper is a nonconforming least-squares spectral element method (see [8,11,12,13,14,15]).…”

“…Recently, we proposed the least-squares spectral element method for parabolic initial value problems with non-smooth data in [11,12]. The method proposed in this paper is a nonconforming least-squares spectral element method (see [8,11,12,13,14,15]). Sobolev spaces of different orders in space and time to formulate the results are given in [17].…”

Spectral Method and Spectral Element Method for Three Dimensional Linear Elliptic System: Analysis and Application

“…Specifically, the proposed formulation is based on minimizing the quadratic form which consists the sum of the squares of a weighted squared norm of the residuals in the partial differential equation, the sum of the residuals in the boundary conditions in fractional Sobolev norms and enforce the continuity along the interelement boundary by adding a term which measures the sum of the squares of the jump in the function and its derivatives in fractional Sobolev norms. We refer to [6][7][8][9][10][11][12]15,[17][18][19][20][21][22] for the analysis and application of LSSEM to solve various types of partial differential equation in 2-D and 3-D.…”

Fractional model and solution for the Black‐Scholes equation

“…Theoretical analysis for the Black-Scholes equation attracts much interest, and some analytical and numerical methods have been proposed. [3][4][5][6][7][8][9] In Dehghan and Pourghanbar 10 approximate analytical methods were presented for solution of the Black-Scholes equation for valuation of barrier option. In Kazemi et al, 11 an asymptotic analytic approach was used to approximate the price of an American call option close to expiry.…”