Numerical Analysis of Novel Finite Difference Methods

On an improved computational solution for the 3D HCIR PDE in finance

“…Recalling again that for these boundaries, we do not impose the reduced PDE (although many of the 2nd order derivative terms in (5) vanish once v, r ↓ 0). The system of SDEs (2) for v and r in (2) are of the same type and, thus, also the 1st-and 2nd-order derivatives ∂u ∂r , ∂ 2 u ∂r 2 and their coefficients in the PDE (5) are of the same form as for v. Therefore, at r ↓ 0, the HCIR model has degeneracy and no side condition is prescribed. Therefore, the obtained equations at such boundaries are taken into account for the boundary nodes.…”

“…wherein s max , v max , r max are three positive real constants and assumed to be large enough. Recently, the author in [11] generalized the numerical grid of nodes proposed previously in [15] to tackle (5). However, the investigated numerical method is only of linear convergence rate to estimate the 2nd derivative terms in (5).…”

“…Recently, the author in [11] generalized the numerical grid of nodes proposed previously in [15] to tackle (5). However, the investigated numerical method is only of linear convergence rate to estimate the 2nd derivative terms in (5). Here the aim is to apply the non-uniform computational grid generated in [11], but with higher order FD formulas on these non-equidistant grids so as to keep the second convergence rate along the fully discretized set of equations.…”

“…[10,18,22,24]. For some discussions concerning the problem of pricing option, an interested reader may refer to the work [5].…”

Asset pricing for an affine jump‐diffusion model using an FD method of lines on nonuniform meshes

“…Proof. See Briani et al 17 Since calculating the price of a European option is correspondent to calculating an expectation E t , several approaches such as Monte Carlo simulation, finite difference (FD) methods, or meshless methods are among the best choices for solving this problem (see Company et al 18 and Milovanović and von Sydow 19 for a detailed background). In Clift and Forsyth, 20 two asset contingent claims based on jump diffusion were computed applying a Markov chain technique that could be basically considered as an explicit FD scheme.…”