Fractional Black–Scholes Under Memory Effects: A Sixth-Order Local RBF–FD Scheme with Integrated Multiquadric Kernels

Abstract

In this work, a high-order meshless framework is developed for the numerical resolution of the temporal–fractional Black–Scholes equation arising in option pricing with long-memory effects. The spatial discretization is carried out with a local radial basis function produced finite difference (RBF–FD) method on seven-node stencils. Analytical differentiation weights are constructed by employing closed-form second integrations of a variant of the inverse multiquadric kernel, which yields sparse differentiation matrices. Explicit formulas are derived for both first- and second-order operators, and a detailed truncation error analysis confirms sixth-order convergence in space. Numerical experiments for European options discuss better accuracy per spatial node than standard finite difference schemes.

1. Introduction

The classical Black–Scholes structure considers that the price of a risky asset $S_t$ follows geometric Brownian motion [1]

$$d{S}_t=(r-q)S_{t}dt+\sigma S_{t}d{W}_t,$$

(1)

wherein $q\ge 0$ is the dividend yield, $W_t$ is a standard Brownian motion, $r\ge 0$ is the risk-free rate, and $\sigma >0$ is the volatility. By applying Itô’s formula to a sufficiently smooth contingent claim $u=u(S,t)$ and enforcing the self-financing hedging condition, one arrives at the classic Black–Scholes partial differential equation (PDE):

$${\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }t}}(S,t)+{\displaystyle \frac{1}{2}}{\sigma }^2{S}^2{\displaystyle \frac{{\mathsf{\partial }}^{2}u}{\mathsf{\partial }{S}^2}}(S,t)+(-q+r)S{\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }S}}(S,t)-ru(S,t)=0,$$

(2)

together with the terminal condition $u(S,T)=\mathsf{\Phi }\left(S\right)$ and suitable boundary conditions as $S\to 0$ and $S\to \infty$. However, empirical studies [2,3] indicate that the increments of $S_t$ are not independent, and the autocorrelation of volatility decays slowly over time. Such features contradict the Markovian assumption of (1) and suggest the presence of memory effects in asset prices [4].

Fractional generalizations of geometric Brownian motion have been proposed in the literature in order to incorporate memory, long-range dependence, and anomalous diffusion characteristics observed in financial time series. One classical approach is to replace the Brownian driver by a fractional Brownian motion with Hurst parameter $H\ne \frac{1}{2}$, leading to models with persistent or antipersistent increments; see, for example, the long-memory stochastic volatility frameworks studied in [5]. Another widely used idea is to modify the time evolution by replacing the classical first-order derivative with a Caputo fractional derivative of order $\alpha \in (0,1)$, which introduces a nonlocal memory kernel and yields a fractional diffusion-type model better aligned with empirical market data.

A natural generalization is to replace (1) by a fractional stochastic differential equation (SDE), see e.g., [6]. One formulation, consistent with subdiffusive market behavior, is [7]

$$d^{\alpha }{S}_t=(r-q)S_t{\left(dt\right)}^{\alpha }+\sigma S_{t}d{W}_t,0<\alpha \le 1,$$

(3)

where $d^{\alpha }$ is a differential operator of fractional order $\alpha$. The term ${\left(dt\right)}^{\alpha }$ reflects the fact that increments scale like $t^{\alpha }$ instead of t, producing a slower spread of randomness. From a probabilistic viewpoint, Equation (3) corresponds to a time-changed diffusion [8] whose mean-square displacement satisfies

$$\mathbb{E}\left[\right|S_t-S_0{|}^2]\propto t^{\alpha },\mathrm{instead}\mathrm{of}\mathbb{E}\left[\right|S_t-S_0{|}^2]\propto t.$$

For $\alpha <1$, the variance grows sublinearly in t, a phenomenon consistent with empirical volatility clustering. The long-memory effect emerges naturally by integrating (3) with respect to a kernel involving ${(t-\xi )}^{\alpha -1}$. Equivalent formulations of (3) can be written using fractional integrals. For instance, employing the Riemann–Liouville (RL) integral [9]

$$\left(I_0^{1-\alpha }f\right)\left(\tau \right)={\displaystyle \frac{1}{\mathsf{\Gamma }(1-\alpha )}}{\int }_0^{\tau }{(\tau -\xi )}^{-\alpha }f\left(\xi \right)d\xi ,$$

one may express the solution of (3) in the integral form [10]

$$S_{\tau }=S_0+\left(\mathsf{\Gamma }{\left(\alpha \right)}^{-1}\right){\int }_0^{\tau }{(\tau -\xi )}^{\alpha -1}\left[(r-q)S_{\xi }+\sigma S_{\xi }w\left(\xi \right)\right]d\xi ,$$

(4)

where $w\left(\xi \right)$ is white noise. The kernel ${(\tau -\xi )}^{\alpha -1}$ is weakly singular when $\xi \to \tau$, demonstrating that recent events have a stronger influence but earlier information still contributes with diminishing weight. The case $\alpha =1$ corresponds to ${(\tau -\xi )}^0$, leading to the classical integral of (1).

It is worth recalling that the classical Black–Scholes framework has played a foundational role in modern quantitative finance. The analytic option pricing formula derived independently by Black and Scholes, and by Merton, represents one of the most influential developments in mathematical finance and contributed directly to the awarding of the 1997 Nobel Prize in Economics to Merton and Scholes. The underlying Black–Scholes equation relies on key structural assumptions, including memoryless price dynamics driven by Brownian motion and a constant volatility coefficient [11]. Although these assumptions lead to a tractable diffusion model and closed-form solutions, they do not always reflect the empirical features of real markets, where volatility clustering, long-range temporal dependence, and persistent deviations from Markovian behavior are frequently observed. Fractional extensions of the Black–Scholes equation incorporate memory effects through nonlocal time operators and therefore provide a more flexible modeling framework. In particular, the Caputo and modified RL formulations used in this work capture such effects in a mathematically consistent manner while preserving the fundamental structure of the original model.

Translating (3) into a risk-neutral valuation setting and applying a fractional Itô-type formula yields a temporal–fractional pricing PDE. Considering $\tau =T-t$, the time–fractional Black–Scholes equation takes the form [12]

$${}^{C}D_{\tau }^{\alpha }u(S,\tau )={\displaystyle \frac{1}{2}}{\sigma }^2{S}^2{\displaystyle \frac{{\mathsf{\partial }}^{2}u}{\mathsf{\partial }{S}^2}}(S,\tau )+(r-q)S{\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }S}}(S,\tau )-ru(S,\tau ),0<\alpha \le 1.$$

(5)

The unknown u is subject to the terminal payoff

$$u(S,0)=\mathsf{\Phi }\left(S\right),$$

and boundary conditions [13]:

$$u(0,\tau )=0,u(S,\tau )\sim S{e}^{-q\tau }\mathrm{as}S\to \infty .$$

(6)

For a European call, $\mathsf{\Phi }\left(S\right)=max(S-K,0)$, and for a put, $\mathsf{\Phi }\left(S\right)=max(K-S,0)$. Here $K>0$ denotes the strike price specified in the option contract, i.e., the fixed level at which the underlying asset may be bought (call) or sold (put) at maturity. Throughout this work, K is assumed to be known and constant, in accordance with the standard European option framework. It should be emphasized that the European setting adopted in (5) is idealized when compared with most real-world market features and contract specifications. In particular, the model excludes early-exercise rights, path-dependent payoffs, transaction costs, and liquidity effects. The present formulation is therefore intended as a mathematically controlled benchmark framework for assessing the accuracy of the proposed high-order numerical discretization under fractional memory effects, rather than as a fully description of all features observed in practical option markets.

To avoid notational ambiguity between fractional derivatives, we adopt the standard convention that ${}^{C}D_{\tau }^{\alpha }$ denotes the Caputo derivative and ${}^{\mathit{RL}}D_{\tau }^{\alpha }$ denotes the modified RL derivative of order $\alpha \in (0,1]$. With this convention, the Caputo operator is written as (for an auxiliary smooth suitable function f) [14]

$${}^{C}D_{\tau }^{\alpha }f\left(\tau \right)={\displaystyle \frac{1}{\mathsf{\Gamma }(1-\alpha )}}{\int }_0^{\tau }{(\tau -\xi )}^{-\alpha }{\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }\xi }}\left(\xi \right)d\xi .$$

(7)

Since ${\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }\xi }}$ appears inside the integral, if f is constant then ${}^{C}D_{\tau }^{\alpha }f\left(\tau \right)=0$, an essential property for financial interpretation: a static payoff does not “evolve” in time. Furthermore, applying the Laplace transform of (7) yields

$$\mathcal{L}\left\{{}^{C}D_{\tau }^{\alpha }f\right\}\left(s\right)=s^{\alpha }\tilde{f}\left(s\right)-s^{\alpha -1}f\left(0\right),$$

demonstrating that the fractional derivative replaces the multiplier s with $s^{\alpha }$ in the transformed PDE.

Another important definition is the modified RL operator, which we denote by ${}^{\mathit{RL}}D_{\tau }^{\alpha }$ [15]:

$${}^{\mathit{RL}}D_{\tau }^{\alpha }f\left(\tau \right)=\left(\mathsf{\Gamma }{(1-\alpha )}^{-1}\right){\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\tau }}{\int }_0^{\tau }{(\tau -\xi )}^{-\alpha }\left(f\left(\xi \right)-f\left(0\right)\right)d\xi .$$

(8)

Subtracting $f\left(0\right)$ inside the integral eliminates the singular term present in the standard RL derivative. When f has limited regularity, as in financial payoffs with kinks, Equation (8) provides a stable definition. If f is differentiable, substituting

$$f\left(\xi \right)-f\left(0\right)={\int }_0^{\xi }{f}^{\prime }\left(s\right)ds,$$

shows that (8) reduces to (7). Using (8) in (5) (for the PDE problem), we obtain an equivalent integral equation

$${\int }_0^{\tau }{(\tau -\xi )}^{-\alpha }\left(u(S,\xi )-u(S,0)\right)d\xi$$

$$=\mathsf{\Gamma }(1-\alpha ){\int }_0^{\tau }\left[{\displaystyle \frac{1}{2}}{\sigma }^2{S}^2{u}_{SS}(S,\xi )+(r-q)S{u}_S(S,\xi )-ru(S,\xi )\right]d\xi ,$$

(9)

which explicitly reveals the memory effect: the right-hand side integrates the instantaneous spatial derivatives over time, weighted by $\mathsf{\Gamma }(1-\alpha )$, and the left-hand side accumulates the differences $u(S,\xi )-u(S,0)$ with a kernel ${(\tau -\xi )}^{-\alpha }$. The price at time $\tau$ is thus determined by all previous states of u.

Existence and uniqueness of (5) under Dirichlet boundary conditions have been studied using fractional maximum principles [16,17]. For example, one may show that if $u_1$ and $u_2$ are two classical solutions of (5) with identical terminal conditions $\mathsf{\Phi }$ and boundary conditions (6), then $u_1-u_2$ satisfies

$${}^{C}D_{\tau }^{\alpha }(u_1-u_2)-\mathcal{L}(u_1-u_2)=0,\mathcal{L}u:={\displaystyle \frac{1}{2}}{\sigma }^2{S}^2{u}_{SS}+(r-q)S{u}_S-ru,$$

and the fractional maximum principle implies ${max}_{S,\tau }(u_1-u_2)=0$, establishing uniqueness. Regularity results may be derived using semigroup theory and fractional Sobolev spaces; in particular, if $\mathsf{\Phi }$ is bounded and piecewise smooth, one can show

$$\parallel u(·,t_2)-u(·,t_1){\parallel }_{L^2(0,S_{max})}\le C{|t_2-t_1|}^{\alpha /2},$$

demonstrating Hölder continuity in $\tau$ with exponent $\alpha /2$ [18].

Because closed-form solutions to (5) require advanced special functions, numerical approximation is essential. The nonlocal term ${}^{C}D_{\tau }^{\alpha }$ generates a dense convolution in time, and the spatial term contains a degeneracy at $S=0$ due to the factor $S^2$. Classical finite difference (FD) schemes require careful stencil adaptation near $S=0$, while spectral methods typically impose global smoothness, see also the discussions [19].

Localized meshless methods [20] have emerged as powerful alternatives to conventional grid-based numerical schemes for solving PDEs, especially when the computational domain is irregular or when resolution must be concentrated in regions of steep gradients. Financial models, including the temporal–fractional Black–Scholes Equation (5), benefit from such flexibility because the underlying asset domain $[0,S_{max}]$ may require dense discretization near the degenerate boundary $S=0$ or near the strike $S=K$ where the solution exhibits low smoothness. Unlike classical FD or finite element methods that demand structured meshes or element connectivity, localized meshless methods rely solely on a set of nodes and an associated interpolation basis. Among the various approaches, the local radial basis function FD (RBF–FD) solver offers high-order accuracy, geometric flexibility, and sparse system matrices. In this paper, a modified multiquadric (MQ) basis function $\varphi \left(y\right)$ is considered, and its first and second integrals are studied explicitly:

$$I_1\left(y\right)={\int }_0^y\varphi \left(s\right)ds,I_2\left(y\right)={\int }_0^y{I}_1\left(s\right)ds.$$

Using $I_2\left(y\right)$, one may derive stable and smooth differentiation weights for approximating $\mathsf{\partial }u/\mathsf{\partial }S$ and ${\mathsf{\partial }}^{2}u/\mathsf{\partial }{S}^2$. The constructed analytical weights are then incorporated into the RBF–FD discretization of (5), leading to a fully discrete scheme which approximates ${}^{C}D_{\tau }^{\alpha }$ in time and the spatial derivatives by seven-node RBF–FD. The resulting scheme has high accuracy and excellent stability properties even for small $\alpha$, where numerical stiffness becomes important.

A further difficulty arises from the behavior of standard MQ kernels when they are used to approximate first and especially second spatial derivatives in fractional PDEs. High-order RBF–FD formulas require repeated differentiation of the generating kernel, and for the classical MQ basis, these derivatives contain rapidly growing powers of the shape parameter together with alternating signs. As a consequence, the local interpolation matrices may become nearly singular even on small seven-point stencils, and the resulting differentiation weights exhibit strong sensitivity to the shape parameter. This effect is amplified in the fractional Black–Scholes setting because the temporal operator produces a history convolution, so any spatial instability is propagated and accumulated over all time levels. The integrated MQ kernels employed in this work remove these high-order derivative instabilities by replacing direct differentiation with analytic antiderivatives.

Over the last decade, a wide spectrum of numerical techniques has been developed for the classical and temporal–fractional Black–Scholes Equation (5). FD discretizations combined with Caputo time-stepping remain the most common choice; for example, ref. [21] analyzed an implicit FD scheme based on the L1 formula in time and second-order stencils in space, while higher-order compact schemes in space have been proposed in [22,23]. Spectral and pseudo-spectral methods have also been constructed, in which the spatial derivatives in (5) are approximated by global basis expansions that yield spectrally accurate prices under suitable regularity assumptions, see e.g., [24]. Robust implicit time-stepping and stability analyses for time–fractional Black–Scholes PDEs have been carried out in [25], and a comprehensive review of fractional Black–Scholes models together with their numerical treatment is provided in [26]. In parallel, meshless and RBF–based approaches have been successfully applied both to the classical Black–Scholes equation and to its fractional extensions [27], demonstrating that localized RBF–FD discretizations can offer high-order accuracy on scattered nodes with sparse linear systems.

A key motivation for the present work is the known ill-conditioning of differentiation matrices generated by standard (inverse) MQ kernels when high-order RBF–FD formulas are required. The MQ-variant basis produces entries of the form ${(y^2+c^2)}^{-5/2}$, and repeated differentiation of this kernel introduces rapidly growing factors of $c^{-2}$ and alternating signs. On seven-point stencils, the resulting local interpolation matrices may approach singularity even for moderate $c/h$, and the computed weights for $u_S$ and $u_{SS}$ become extremely sensitive to perturbations in node placement. In the temporal–fractional Black–Scholes model, this defect is particularly problematic: the history term generated by the fractional derivative accumulates and amplifies any spatial instability across all previous time levels.

The integrated MQ-variant kernels used in this study act as an intrinsic regularization mechanism. Instead of differentiating the kernel directly, one constructs the local interpolant using its first and second antiderivatives. These antiderivatives are smooth, slowly varying functions whose derivatives recover the required operators without introducing large oscillatory factors. The associated local interpolation matrices remain well conditioned, and the resulting differentiation weights preserve the polynomial moment cancellations needed for sixth-order accuracy. In this sense, the integrated-kernel approach replaces the unstable action of repeatedly differentiating sharp MQ profiles by the stable action of differentiating smoother primitives, thereby enhancing both numerical stability and high-order accuracy across the spatial domain.

At the modeling level, a fundamental choice concerns the definition of the fractional time derivative used to represent memory effects. In this work, the Caputo fractional derivative is adopted throughout, as it allows the initial condition to be imposed in the classical sense and admits a clear financial interpretation in terms of past price evolution. For functions with sufficient temporal regularity, the Caputo formulation coincides with the modified RL derivative introduced in (8), and both lead to equivalent integral representations of the fractional Black–Scholes equation.

It is emphasized that this modeling choice is independent of the proposed spatial discretization. The integrated RBF–FD scheme developed in this paper is designed to approximate the spatial differential operator with high accuracy and stability and can, in principle, be coupled with any consistent discretization of the fractional time derivative. In the present study, the Caputo operator is discretized by a standard L1 scheme in time, while the main methodological contribution focuses on the construction of sixth-order accurate, well-conditioned spatial operators based on analytically integrated kernels.

Section 2 describes the local RBF–FD formulation and its theoretical properties. Section 3 presents the computation of the analytical weights for the second integrated MQ-variant kernel. Section 4 applies the proposed spatial analytical weights method to the full discretization of the fractional Black–Scholes PDE (5) and Section 5 provides numerical evidence of convergence and financial relevance. The conclusion is presented in the Section 6.

2. The Methodology of Localized Meshless Methods

Consider a computational domain $\mathsf{\Omega }\subset \mathbb{R}$, which in our application corresponds to the interval $[0,S_{max}]$. Let ${\left\{y_i\right\}}_{i=1}^N$ denote a set of distinct nodes placed inside $\mathsf{\Omega }$ without the requirement of forming a structured mesh. For a sufficiently smooth function $g:\mathsf{\Omega }\to \mathbb{R}$, the key idea of the RBF–FD approach is to estimate g by a localized radial basis representation over each stencil. For a fixed node $y_c$, denote the m nearest nodes to $y_c$ inside $\{y_1,y_2,\dots ,y_m\}$, with $m\ll N$ [28]. Over this stencil, the function g is interpolated as a linear superposition of translates of a chosen RBF $\varphi (·)$, i.e.,

$$g\left(y\right)\approx \sum _{j=1}^m{\lambda }_j\varphi (\parallel y-y_j\parallel ),$$

(10)

where ${\left\{{\lambda }_j\right\}}_{j=1}^m$ are coefficients to be determined and $\parallel ·\parallel$ is the Euclidean norm. Here the basis function $\varphi (·)$ is a general RBF. Later, we will focus on a specific RBF in this work. The representation (10) is purely local: no information from nodes outside the stencil is needed. This locality is a defining advantage because it yields a sparse differentiation matrix and avoids the dense systems typical of global RBF approximations.

To determine the coefficients ${\lambda }_j$, one enforces interpolation at the stencil points. Let $A={\left[a_{ij}\right]}_{i,j=1}^m$ with $a_{ij}=\varphi (\parallel y_i-y_j\parallel )$ and $g={(g\left(y_1\right),\dots ,g\left(y_m\right))}^T$. Then $\lambda =A^{-1}g$. Once the coefficients are known, differentiation of (10) yields RBF–FD formulas for spatial derivatives. For example, differentiating with respect to y and evaluating at the center node $y=y_c$ gives

$${\displaystyle \frac{\mathsf{\partial }g}{\mathsf{\partial }y}}\left(y_c\right)\approx \sum _{j=1}^m{\lambda }_j{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}\varphi (\parallel y_c-y_j\parallel ).$$

(11)

Since $\lambda =A^{-1}g$, the derivative (11) depends linearly on the values $g\left(y_i\right)$ and therefore can be written as

$${\displaystyle \frac{\mathsf{\partial }g}{\mathsf{\partial }y}}\left(y_c\right)\approx \sum _{j=1}^m{w}_{c,j}^{\left(1\right)}g\left(y_j\right),$$

(12)

where the weights $w_{c,j}^{\left(1\right)}$ are given by

$$w_{c,j}^{\left(1\right)}=\sum _{\mathsf{\ell }=1}^m{\left(A^{-1}\right)}_{\mathsf{\ell },j}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}\varphi (\parallel y_c-y_{\mathsf{\ell }}\parallel ).$$

Expression (12) is the RBF–FD analogue of an FD stencil and represents the derivative at $y_c$ as a weighted combination of function values at neighboring knots [29]. For second-order derivatives, one differentiates (10) twice and obtains

$${\displaystyle \frac{{\mathsf{\partial }}^{2}g}{\mathsf{\partial }{y}^2}}\left(y_c\right)\approx \sum _{j=1}^m{w}_{c,j}^{\left(2\right)}g\left(y_j\right),$$

(13)

with weights

$$w_{c,j}^{\left(2\right)}=\sum _{\mathsf{\ell }=1}^m{\left(A^{-1}\right)}_{\mathsf{\ell },j}{\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{y}^2}}\varphi (\parallel y_c-y_{\mathsf{\ell }}\parallel ).$$

Since $m\ll N$ and each node generates exactly one row of weights, the global differentiation matrices formed by assembling all stencils are extremely sparse. This sparsity is crucial for large-scale simulations in finance, where N may be in the tens or hundreds of thousands.

Equations (12) and (13) show that, once the local interpolation coefficients are eliminated, spatial derivatives at the stencil center can be written entirely in terms of nodal values of the target function. In particular, for a sufficiently smooth function g and for $k=1,2$, the local RBF–FD approximation takes the unified form

$${\displaystyle \frac{{\mathsf{\partial }}^{k}g}{\mathsf{\partial }{S}^k}}\left(y_i\right)\approx \sum _{j=1}^m{w}_{i,j}^{\left(k\right)}g\left(y_j\right),$$

where the weights $w_{i,j}^{\left(k\right)}$ depend only on the relative stencil geometry and on the chosen kernel, but not on the function g itself.

One choice as a basis kernel (which we focus on in this work) is the following variant of the inverse MQ kernel, expressed as [30]:

$$\varphi \left(y\right)={(y^2+c^2)}^{-5/2},c>0.$$

(14)

In particular, when the shape parameter c assumes small values or when higher-order derivatives are required, issues of severe ill-conditioning and diminished stability may emerge [31].

It is important to note that the stencil size m also affects approximation quality. Larger m generally yields more accurate differentiation weights but results in denser rows in the global matrix. In financial PDEs, one must approximate first and second spatial derivatives accurately near $S=0$, where the coefficient ${\sigma }^2{S}^2$ degenerates, and near the strike $S=K$, where the payoff has limited smoothness. Small stencils such as $m=5$ may fail to capture the curvature of u near these regions. In contrast, $m=7$ provides a good compromise between accuracy and sparsity. With seven nodes, one can construct stable and high-order formulas for $u_S$ and $u_{SS}$, see also [32,33].

To make the construction of the RBF–FD differentiation weights more transparent, we begin by introducing the interpolation function, the kernel, and all parameters at the moment they first appear. Let $u=u(S,\tau )$ denote the option value, and fix a spatial location $y_i$ (corresponding to $S_i$ in the original coordinates). On the seven-point stencil

$${\mathcal{S}}_i=\{y_{i-3},y_{i-2},y_{i-1},y_i,y_{i+1},y_{i+2},y_{i+3}\},$$

(15)

we introduce an auxiliary interpolation function $g\left(y\right)={\sum }_{j=-3}^3{\lambda }_{i+j}{\varphi }_2(y-y_{i+j})$, where ${\varphi }_2$ is our integrated inverse MQ kernel, which will be defined later in the next section.

Figure 1 illustrates how a seven-node RBF–FD stencils overlap with a sample 9-point grid. For an interior point, such as the node $y_5$, a symmetric stencil $\{y_2,y_3,\dots ,y_8\}$ is used. Near boundaries, the method automatically employs one-sided stencils, for example $\{y_1,y_2,\dots ,y_7\}$ for $y_2$ and $\{y_3,y_4,\dots ,y_9\}$ for $y_9$, while preserving the seven-node structure required for the derivation of the analytical weights.

We end this section by noting that since the spatial domain is one-dimensional, the Euclidean norm used in the general RBF Formulation (10) reduces to the absolute value, i.e., $\parallel y-y_j\parallel =|y-y_j|$. For this reason, absolute values are used throughout the present section to simplify notation, without altering the underlying interpolation structure.

3. Constructing New Weighting Coefficients

At the beginning of this section, we clarify the role of the auxiliary function g and the kernel $\varphi$, since both participate in the construction of the seven-point differentiation weights. Throughout the weight-derivation process, g denotes a generic smooth target function whose derivatives (such as $g^{\prime }\left(y_i\right)$ or $g^{″}\left(y_i\right)$) we wish to approximate at the stencil center $y_i$. In the actual fractional Black–Scholes application, one sets $g(·)=u(·,\tau )$, but the derivation of the weights must be carried out for an arbitrary g so that the resulting formulas are universal and independent of the particular PDE solution. Therefore, even though the MQ kernel used for interpolation is given explicitly, the function g whose derivative is sought is not known analytically; only its nodal values $\left\{g\left(y_{i+j}\right)\right\}$ are available on the stencil. This is the reason the derivative $g^{\prime }\left(y_i\right)$ or $g^{″}\left(y_i\right)$ must be approximated. In addition, since the target PDE problem consists of partial derivatives up to the second order, approximations of the first and second derivatives would be enough. Hence, higher-order derivatives can be approximated in a similar pattern and be used for solving PDEs, including higher-order derivative terms.

To avoid notational ambiguity, we first consider the kernel (14) and the local RBF interpolant on the stencil $\{y_{i-3},y_{i-2},\dots ,y_{i+3}\}$ as

$$P\left(y\right)=\sum _{j=-3}^3{\lambda }_{i+j}\varphi \left(\right|y-y_{i+j}\left|\right),P\left(y_{i+j}\right)=g\left(y_{i+j}\right).$$

Here $P\left(y\right)$ is introduced only as a local auxiliary interpolant on the seven-point stencil, serving the same conceptual role as the local representation (10) but specialized to the fixed stencil and the chosen kernel (${\varphi }_2$); it is used to explain how RBF–FD weights arise by enforcing exactness of the differential operator on the basis functions. In practice, one does not need to construct $P\left(y\right)$ explicitly: the differentiation weights are obtained directly by requiring that the discrete operator reproduces the action of $\mathsf{\partial }/\mathsf{\partial }y$ or ${\mathsf{\partial }}^2/\mathsf{\partial }{y}^2$ on the stencil basis, which leads to the usual small linear system for the weights and is algebraically equivalent to differentiating the local interpolant and eliminating the coefficients.

The idea of the present work is that, instead of differentiating (14) directly, we construct weights by differentiating the integrated kernel. Specifically, we introduce the analytic antiderivatives

$${\varphi }_1\left(y\right)=\int \varphi \left(s\right)ds={\displaystyle \frac{3c^{2}y+2y^3}{3c^4{(c^2+y^2)}^{3/2}}},{\varphi }_2\left(y\right)=\int {\varphi }_1\left(s\right)ds={\displaystyle \frac{c^2+2y^2}{3c^4\sqrt{c^2+y^2}}},$$

(16)

whose explicit closed forms are derived later in this section. The role of ${\varphi }_1$ and ${\varphi }_2$ is to provide smoother generating functions for the local interpolation matrix and its derivatives, thereby mitigating the classical ill-conditioning associated with standard MQ differentiation. Once $P\left(y\right)$ is expressed in terms of ${\varphi }_2$, differentiating the interpolant at $y=y_i$ produces a linear relation

$$P^{\prime }\left(y_i\right)=\sum _{j=-3}^3{b}_{i+j}g\left(y_{i+j}\right),$$

from which the desired RBF–FD weights $\left\{b_{i+j}\right\}$ follow.

To clarify the smoothness enhancement produced by replacing the modified inverse MQ kernel $\varphi \left(y\right)={(y^2+c^2)}^{-5/2}$ with its second integral ${\varphi }_2\left(y\right)$ given in (16), Figure 2 displays both profiles on the same vertical scale. While $\varphi \left(y\right)$ exhibits a sharp peak near $y=0$ and decays rapidly, the integrated kernel ${\varphi }_2\left(y\right)$ is flatter and possesses smoother curvature transitions. This reduction in steepness lowers the magnitude of higher-order derivatives of the basis functions evaluated on each stencil, which directly decreases the amplification of local truncation errors in the RBF–FD weight computation. In turn, the eigenvalues of the corresponding local matrix $A\left(c\right)$ vary over a tighter spectral range, improving the conditioning of the system and stabilizing the resulting differentiation operators.

3.1. Approximation of the First Derivative

The seven-point RBF–FD approximation to the first derivative is defined in the standard form

$${\widehat{g}}^{\prime }\left(y_i\right)=\sum _{j=-3}^3{b}_{i+j}g\left(y_{i+j}\right),$$

(17)

where the weights $b_{i+j}$ depend only on the relative locations

$${\left\{y_{i+j}\right\}}_{j=-3}^3={\{y_i+jh\}}_{j=-3}^3,h>0.$$

(18)

We used the notation of hat, i.e., $\widehat{g}$ to show that it is the RBF–FD approximation on our stencil. On a symmetric stencil (18), antisymmetry of the first-derivative operator implies

$$b_i=0,b_{i-j}=-b_{i+j}\mathrm{for}j=1,2,3,$$

(19)

so only the three positive-side weights $b_{i+1},b_{i+2},b_{i+3}$ must be determined. In the integrated-kernel formulation, these weights are found by enforcing exactness of (17) on the seven local traces of the functions generated by ${\varphi }_2(·-y_{i+\mathsf{\ell }})$, $\mathsf{\ell }=-3,\dots ,3$, after analytically differentiating the corresponding local interpolant at $y_i$. This produces a $7\times 7$ linear system whose structure is Toeplitz-plus-border and whose right-hand side contains only closed forms built from (16). Direct solution by exact algebra is possible, but lengthy. A robust algebraic route is to expand each entry of the local system in Taylor series with respect to h up to a high order and to solve symbolically for the weights using a pseudo-inverse, followed by algebraic simplification [34,35].

Carrying out this program to tenth order in h and then simplifying the result about the small parameter h while keeping c explicit yields the compact formulas

$$\begin{array}{cc}\hfill b_{i-3}& =-b_{i+3}=-{\displaystyle \frac{11192299}{71680}}{\displaystyle \frac{h^3}{c^4}}+{\displaystyle \frac{3953}{3360}}{\displaystyle \frac{h}{c^2}}-{\displaystyle \frac{1}{60h}},\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill b_{i-2}& =-b_{i+2}=\phantom{-}{\displaystyle \frac{12571299}{17920}}{\displaystyle \frac{h^3}{c^4}}-{\displaystyle \frac{3953}{840}}{\displaystyle \frac{h}{c^2}}+{\displaystyle \frac{3}{20h}},\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill b_{i-1}& =-b_{i+1}=-{\displaystyle \frac{13398699}{14336}}{\displaystyle \frac{h^3}{c^4}}+{\displaystyle \frac{3953}{672}}{\displaystyle \frac{h}{c^2}}-{\displaystyle \frac{3}{4h}},\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill b_i& =0.\hfill \end{array}$$

(23)

The structure (20)–(23) makes three features transparent. First, antisymmetry (19) is exact. Second, the leading $1/h$ contributions coincide with the classical seven-point centered FD stencil for the first derivative on a uniform grid; polynomial reproduction through degree five is achieved already at the $1/h$ level. Third, the additional $h/c^2$ and $h^3/c^4$ corrections control high-order dispersion and stabilize the weights for moderate c without destroying the polynomial moment cancellations that determine the order of accuracy. Here, the term ‘stabilize the weights’ refers to the fact that using the twice-integrated MQ basis ${\varphi }_2$ produces a local interpolation matrix with improved conditioning, leading to smoothly varying and non-oscillatory RBF–FD coefficients.

The analytic order of accuracy follows from a moment-matching argument. Expanding $g\left(y_{i+j}\right)$ in Taylor series about $y_i$ and inserting into (17) gives

$${\widehat{g}}^{\prime }\left(y_i\right)=\sum _{m=0}^{\infty }{\displaystyle \frac{h^m}{m!}}\left(\sum _{j=-3}^3{j}^m{b}_{i+j}\right)g^{\left(m\right)}\left(y_i\right).$$

(24)

Let $M_m:={\sum }_{j=-3}^3{j}^m{b}_{i+j}$ denote the discrete moments. Antisymmetry implies $M_{2\mathsf{\ell }}=0$ for $\mathsf{\ell }=0,1,2,3$ and $M_0=0$ in particular, while N must equal 1 to recover $g^{\prime }\left(y_i\right)$. For the weights (20)–(23) one verifies

$$M_0=0,N=-{\displaystyle \frac{1}{h}},M_2=0,M_3={\displaystyle \frac{14775}{16}}{\displaystyle \frac{h^3}{c^4}},M_4=0,$$

$$M_5={\displaystyle \frac{h}{896}}{\displaystyle \frac{252992c^2-29439897h^2}{c^4}},$$

(25)

and $M_6=0$; the sign of N depends only on the indexing convention and can be absorbed by a consistent orientation of the stencil. Substituting (25) into (24) yields

$${\widehat{g}}^{\prime }\left(y_i\right)-g^{\prime }\left(y_i\right)=\underset{{\displaystyle \mathcal{O}\left(h^6/c^4\right)}}{\underbrace{{\displaystyle \frac{h^3}{6}}{M}_3{g}^{\left(3\right)}\left(y_i\right)}}+\underset{{\displaystyle \mathcal{O}\left(h^6/c^2\right)}}{\underbrace{{\displaystyle \frac{h^5}{120}}{M}_5{g}^{\left(5\right)}\left(y_i\right)}}$$

$$+\underset{{\displaystyle 0}}{\underbrace{{\displaystyle \frac{h^6}{720}}{M}_6{g}^{\left(6\right)}\left(y_i\right)}}+\underset{{\displaystyle \mathcal{O}\left(h^6\right)}}{\underbrace{{\displaystyle \frac{h^7}{5040}}{M}_7{g}^{\left(7\right)}\left(y_i\right)}}+\cdots ,$$

(26)

so the leading order is $\mathcal{O}\left(h^6\right)$ even when c is fixed, and the defect constants are explicitly controlled by c. In particular, the $\mathcal{O}(h^6/c^2)$ and $\mathcal{O}(h^6/c^4)$ terms vanish if c scales like $c=\kappa h^{-\beta }$ with any $\beta >0$, but such scaling is not required to secure sixth order. The precise constant in front of $h^6$ can be obtained by carrying the algebra one order further; see (27) below.

We now summarize the discussions into a theorem.

Theorem 1. 

Let $g\in C^7\left([y_i-3h,y_i+3h]\right)$ and let the seven-point weights $b_{i+j}$ be given by (20)–(23). Then the RBF–FD approximation (17) satisfies

$${\widehat{g}}^{\prime }\left(y_i\right)-g^{\prime }\left(y_i\right)=\mathcal{C}\left(c\right)h^6{g}^{\left(7\right)}\left(y_i\right)+\mathcal{R}(h,c),$$

where $\mathcal{C}\left(c\right)$ is a bounded function of $c>0$ and the remainder satisfies

$$\left|\mathcal{R}(h,c)\right|\le C{h}^6\sum _{m=3}^5{\parallel g^{\left(m\right)}\parallel }_{\infty }{\Xi }_m\left(c\right),$$

with explicit ${\Xi }_m\left(c\right)=\mathcal{O}\left(c^{-2}\right)$ or $\mathcal{O}\left(c^{-4}\right)$ as $c\to \infty$. In particular, for any fixed $c>0$ the approximation is sixth order:

$$|{\widehat{g}}^{\prime }\left(y_i\right)-g^{\prime }\left(y_i\right)|\le C\left(c\right)h^6{\parallel g\parallel }_{C^7}.$$

Proof. 

Taylor expansion (24) with moments (25) implies

$${\widehat{g}}^{\prime }\left(y_i\right)-g^{\prime }\left(y_i\right)={\displaystyle \frac{h^3}{6}}{M}_3{g}^{\left(3\right)}\left(y_i\right)+{\displaystyle \frac{h^5}{120}}{M}_5{g}^{\left(5\right)}\left(y_i\right)+{\displaystyle \frac{h^7}{5040}}{M}_7{g}^{\left(7\right)}\left(y_i\right)+\cdots .$$

Since $M_3=\mathcal{O}(h^3/c^4)$ and $M_5=\mathcal{O}\left(c^{-2}\right)+\mathcal{O}(h^2/c^4)$, the first two defect terms are $\mathcal{O}(h^6/c^4)$ and $\mathcal{O}(h^6/c^2)$, respectively, while the $m=7$ term is $\mathcal{O}\left(h^6\right)$ with a c-independent coefficient determined by the exact stencil geometry. All higher terms are $\mathcal{O}\left(h^8\right)$ or smaller. Grouping the $m=3$ and $m=5$ contributions into $\mathcal{R}(h,c)$ and the $m=7$ term into $\mathcal{C}\left(c\right)h^6{g}^{\left(7\right)}\left(y_i\right)$ yields the claim. Boundedness of $\mathcal{C}\left(c\right)$ on $(0,\infty )$ follows from the analyticity of the local system entries in c and h and the nondegeneracy of the seven-point stencil. The constant $C\left(c\right)$ relies on bounds for $g^{\left(m\right)}$ for $3\le m\le 7$ uniformly on the stencil.

In fact, by continuing the symbolic expansion one more order, the local truncation error can be written explicitly as

$$e_1\left(y_i\right):={\widehat{g}}^{\prime }\left(y_i\right)-g^{\prime }\left(y_i\right)=\left({\alpha }_3{\displaystyle \frac{g^{\left(3\right)}\left(y_i\right)}{c^4}}+{\alpha }_5{\displaystyle \frac{g^{\left(5\right)}\left(y_i\right)}{c^2}}+{\alpha }_7{g}^{\left(7\right)}\left(y_i\right)\right)h^6+\mathcal{O}\left(h^7\right),$$

(27)

with computable constants

$${\alpha }_3={\displaystyle \frac{14775}{96}},{\alpha }_5={\displaystyle \frac{252992}{107520}}-{\displaystyle \frac{29439897}{107520}}{\displaystyle \frac{h^2}{c^2}},{\alpha }_7=\mathrm{a}\mathrm{bounded}\mathrm{stencil}\text{-}\mathrm{dependent}\mathrm{constant}.$$

Formula (27) exhibits two useful design freedoms. First, if c is chosen proportional to $h^{-\beta }$ with $\beta \ge 1$, then the c-dependent coefficients are $\mathcal{O}\left(h^4\right)$ or smaller and the $g^{\left(7\right)}$ term dominates. Second, one can minimize the leading error by selecting c to cancel the $g^{\left(5\right)}$ contribution at a prescribed h, thereby tuning the dispersion of the first-derivative operator without compromising stability.

The weights (20)–(23) arise from an integrated-kernel construction. For completeness, we sketch the algebraic system that determines them. Let $\mathsf{\Psi }\left(y\right)=[{\varphi }_2(y-y_{i-3}),\dots ,{\varphi }_2(y-y_{i+3})]$ and define the local interpolant $P\left(y\right)=\mathsf{\Psi }\left(y\right)\lambda$ with coefficients $\lambda \in {\mathbb{R}}^7$ obtained by enforcing $P\left(y_{i+\mathsf{\ell }}\right)=g\left(y_{i+\mathsf{\ell }}\right)$ for $\mathsf{\ell }=-3,-2,\dots ,3$. Differentiating yields $P^{\prime }\left(y_i\right)={\mathsf{\Psi }}^{\prime }\left(y_i\right)\lambda$. Eliminating $\lambda ={\left(A\right)}^{-1}{g}_{\mathrm{loc}}$ with $A_{\mathsf{\ell }m}={\varphi }_2(y_{i+\mathsf{\ell }}-y_{i+m})$ and $g_{\mathrm{loc}}={[g\left(y_{i-3}\right),\dots ,g\left(y_{i+3}\right)]}^T$ gives

$$P^{\prime }\left(y_i\right)={\mathsf{\Psi }}^{\prime }\left(y_i\right)A^{-1}{g}_{\mathrm{loc}}=\sum _{m=-3}^3\left(\sum _{\mathsf{\ell }=-3}^3{\mathsf{\Psi }}_{\mathsf{\ell }}^{\prime }\left(y_i\right){\left(A^{-1}\right)}_{\mathsf{\ell }m}\right)g\left(y_{i+m}\right),$$

so $b_{i+m}={\sum }_{\mathsf{\ell }=-3}^3{\mathsf{\Psi }}_{\mathsf{\ell }}^{\prime }\left(y_i\right){\left(A^{-1}\right)}_{\mathsf{\ell }m}$. Expanding A and ${\mathsf{\Psi }}^{\prime }$ in powers of h, solving symbolically for $A^{-1}$, and simplifying the result produces (20)–(23). This route bypasses direct differentiation of $\varphi$ and, thanks to (16), preserves smoothness and reduces cancellation errors in the small h regime. The proof is complete.    □

One practical remark concerns implementation in the fractional Black–Scholes setting. Because the time discretization for the fractional operator ${}^{C}D_{\tau }^{\alpha }$ already induces a history convolution, it is essential that the spatial operator be both accurate and sparse. The seven-point formulas yield rows with exactly seven nonzeros in the global matrix and therefore admit fast sparse Krylov solvers per time step.

3.2. Approximation of the Second Derivative

This part develops the seven-point RBF–FD differentiation operator for the second derivative on the uniform sub-stencil (18) using the integrated MQ kernel ${\varphi }_2$. As in Section 2, the approximation of a sufficiently smooth target g is written in the algebraic form

$${\widehat{g}}^{″}\left(y_i\right)=\sum _{j=-3}^3{\overline{b}}_{i+j}g\left(y_{i+j}\right),$$

(28)

with symmetric weights for a centered second derivative,

$${\overline{b}}_i={\overline{b}}_i,{\overline{b}}_{i-j}={\overline{b}}_{i+j}\mathrm{for}j=1,2,3.$$

(29)

Hence, this results in the $7\times 7$ system

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{{\left(c^2+9h^2\right)}^{5/2}}}={\displaystyle \frac{\frac{{\overline{b}}_{i-2}\left(c^2+2h^2\right)}{\sqrt{c^2+h^2}}+\frac{{\overline{b}}_{i-1}\left(c^2+8h^2\right)}{\sqrt{c^2+4h^2}}+\frac{{\overline{b}}_i\left(c^2+18h^2\right)}{\sqrt{c^2+9h^2}}+\frac{{\overline{b}}_{i+1}\left(c^2+32h^2\right)}{\sqrt{c^2+16h^2}}+\frac{{\overline{b}}_{i+2}\left(c^2+50h^2\right)}{\sqrt{c^2+25h^2}}+\frac{{\overline{b}}_{i+3}\left(c^2+72h^2\right)}{\sqrt{c^2+36h^2}}+c{\overline{b}}_{i-3}}{3c^4}},\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{{\left(c^2+4h^2\right)}^{5/2}}}={\displaystyle \frac{\frac{{\overline{b}}_{i-3}\left(c^2+2h^2\right)}{\sqrt{c^2+h^2}}+\frac{{\overline{b}}_{i-1}\left(c^2+2h^2\right)}{\sqrt{c^2+h^2}}+\frac{{\overline{b}}_i\left(c^2+8h^2\right)}{\sqrt{c^2+4h^2}}+\frac{{\overline{b}}_{i+1}\left(c^2+18h^2\right)}{\sqrt{c^2+9h^2}}+\frac{{\overline{b}}_{i+2}\left(c^2+32h^2\right)}{\sqrt{c^2+16h^2}}+\frac{{\overline{b}}_{i+3}\left(c^2+50h^2\right)}{\sqrt{c^2+25h^2}}+c{\overline{b}}_{i-2}}{3c^4}},\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{{\left(c^2+h^2\right)}^{5/2}}}={\displaystyle \frac{\frac{{\overline{b}}_{i-3}\left(c^2+8h^2\right)}{\sqrt{c^2+4h^2}}+\frac{{\overline{b}}_{i-2}\left(c^2+2h^2\right)}{\sqrt{c^2+h^2}}+\frac{{\overline{b}}_i\left(c^2+2h^2\right)}{\sqrt{c^2+h^2}}+\frac{{\overline{b}}_{i+1}\left(c^2+8h^2\right)}{\sqrt{c^2+4h^2}}+\frac{{\overline{b}}_{i+2}\left(c^2+18h^2\right)}{\sqrt{c^2+9h^2}}+\frac{{\overline{b}}_{i+3}\left(c^2+32h^2\right)}{\sqrt{c^2+16h^2}}+c{\overline{b}}_{i-1}}{3c^4}},\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{c^5}}={\displaystyle \frac{\frac{{\overline{b}}_{i-3}\left(c^2+18h^2\right)}{\sqrt{c^2+9h^2}}+\frac{{\overline{b}}_{i-2}\left(c^2+8h^2\right)}{\sqrt{c^2+4h^2}}+\frac{{\overline{b}}_{i-1}\left(c^2+2h^2\right)}{\sqrt{c^2+h^2}}+\frac{{\overline{b}}_{i+1}\left(c^2+2h^2\right)}{\sqrt{c^2+h^2}}+\frac{{\overline{b}}_{i+2}\left(c^2+8h^2\right)}{\sqrt{c^2+4h^2}}+\frac{{\overline{b}}_{i+3}\left(c^2+18h^2\right)}{\sqrt{c^2+9h^2}}+c{\overline{b}}_i}{3c^4}},\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{{\left(c^2+h^2\right)}^{5/2}}}={\displaystyle \frac{\frac{{\overline{b}}_{i-3}\left(c^2+32h^2\right)}{\sqrt{c^2+16h^2}}+\frac{{\overline{b}}_{i-2}\left(c^2+18h^2\right)}{\sqrt{c^2+9h^2}}+\frac{{\overline{b}}_{i-1}\left(c^2+8h^2\right)}{\sqrt{c^2+4h^2}}+\frac{{\overline{b}}_i\left(c^2+2h^2\right)}{\sqrt{c^2+h^2}}+\frac{{\overline{b}}_{i+2}\left(c^2+2h^2\right)}{\sqrt{c^2+h^2}}+\frac{{\overline{b}}_{i+3}\left(c^2+8h^2\right)}{\sqrt{c^2+4h^2}}+c{\overline{b}}_{i+1}}{3c^4}},\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{{\left(c^2+4h^2\right)}^{5/2}}}={\displaystyle \frac{\frac{{\overline{b}}_{i-3}\left(c^2+50h^2\right)}{\sqrt{c^2+25h^2}}+\frac{{\overline{b}}_{i-2}\left(c^2+32h^2\right)}{\sqrt{c^2+16h^2}}+\frac{{\overline{b}}_{i-1}\left(c^2+18h^2\right)}{\sqrt{c^2+9h^2}}+\frac{{\overline{b}}_i\left(c^2+8h^2\right)}{\sqrt{c^2+4h^2}}+\frac{{\overline{b}}_{i+1}\left(c^2+2h^2\right)}{\sqrt{c^2+h^2}}+\frac{{\overline{b}}_{i+3}\left(c^2+2h^2\right)}{\sqrt{c^2+h^2}}+c{\overline{b}}_{i+2}}{3c^4}},\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{{\left(c^2+9h^2\right)}^{5/2}}}={\displaystyle \frac{\frac{{\overline{b}}_{i-3}\left(c^2+72h^2\right)}{\sqrt{c^2+36h^2}}+\frac{{\overline{b}}_{i-2}\left(c^2+50h^2\right)}{\sqrt{c^2+25h^2}}+\frac{{\overline{b}}_{i-1}\left(c^2+32h^2\right)}{\sqrt{c^2+16h^2}}+\frac{{\overline{b}}_i\left(c^2+18h^2\right)}{\sqrt{c^2+9h^2}}+\frac{{\overline{b}}_{i+1}\left(c^2+8h^2\right)}{\sqrt{c^2+4h^2}}+\frac{{\overline{b}}_{i+2}\left(c^2+2h^2\right)}{\sqrt{c^2+h^2}}+c{\overline{b}}_{i+3}}{3c^4}},\hfill \end{array}$$

(36)

whose solution yields, after symbolic elimination and Taylor expansion in h, the explicit weights

$$\begin{array}{cc}\hfill {\overline{b}}_{i-3}& ={\overline{b}}_{i+3}={\displaystyle \frac{149380759993h^2-704267904c^2}{371589120c^4}}+{\displaystyle \frac{1}{90h^2}},\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill {\overline{b}}_{i-2}& ={\overline{b}}_{i+2}={\displaystyle \frac{704267904c^2-162082276345h^2}{61931520c^4}}-{\displaystyle \frac{3}{20h^2}},\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill {\overline{b}}_{i-1}& ={\overline{b}}_{i+1}={\displaystyle \frac{848515930781h^2-3521339520c^2}{123863040c^4}}+{\displaystyle \frac{3}{2h^2}},\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill {\overline{b}}_i& ={\displaystyle \frac{3521339520c^2-861217447133h^2}{92897280c^4}}-{\displaystyle \frac{49}{18h^2}}.\hfill \end{array}$$

(40)

The structure (37)–(40) mirrors the classical seventh-point FD operator of order six: the leading $1/h^2$ part reproduces the well-known sixth-order polynomial stencil, while the $h^2/c^4$ and $c^2/c^4=1/c^2$ corrections modulate the high-frequency dispersion and provide robust conditioning for moderate c.

Accuracy for second derivatives is characterized by discrete moment constraints. Let

$$M_m^{\left(2\right)}:=\sum _{j=-3}^3{j}^m{\overline{b}}_{i+j}.$$

Taylor expanding $g\left(y_{i+j}\right)$ about $y_i$ and inserting into (28) gives

$${\widehat{g}}^{″}\left(y_i\right)=\sum _{m=0}^{\infty }{\displaystyle \frac{h^m}{m!}}{M}_m^{\left(2\right)}{g}^{\left(m\right)}\left(y_i\right).$$

(41)

Symmetry (29) enforces $M_{2\mathsf{\ell }+1}^{\left(2\right)}=0$ for all $\mathsf{\ell }\ge 0$. Sixth-order exactness of a centered seven-point second derivative requires the even moment conditions

$$M_0^{\left(2\right)}=0,M_2^{\left(2\right)}={\displaystyle \frac{2}{h^2}},M_4^{\left(2\right)}=0,M_6^{\left(2\right)}=0.$$

(42)

The weights (37)–(40) satisfy (42) exactly at the level of their $1/h^2$ parts, while the c-dependent corrections cancel in the sums due to the Toeplitz symmetry of (30)–(36). Consequently, substituting (42) into (41) yields

$${\widehat{g}}^{″}\left(y_i\right)-g^{″}\left(y_i\right)={\displaystyle \frac{h^8}{8!}}{M}_8^{\left(2\right)}{g}^{\left(8\right)}\left(y_i\right)+\mathrm{lower}\text{-}\mathrm{order}c\text{-}\mathrm{dependent}\mathrm{terms}\mathrm{of}\mathrm{total}\mathrm{order}{h}^6,$$

so the leading truncation error is $\mathcal{O}\left(h^6\right)$. A precise expansion is given in (43) below.

The symbolic elimination leading to (37)–(40) yields the explicit local error expansion

$${\widehat{g}}^{″}\left(y_i\right)-g^{″}\left(y_i\right)=\left({\displaystyle \frac{1}{560}}{g}^{\left(8\right)}\left(y_i\right)\right)h^6-\left({\displaystyle \frac{1834031}{967680}}{\displaystyle \frac{g^{\left(6\right)}\left(y_i\right)}{c^2}}\right.$$

$$\left.+{\displaystyle \frac{198461193}{967680}}{\displaystyle \frac{g^{\left(4\right)}\left(y_i\right)}{c^4}}\right)h^6+\mathcal{O}\left(h^7\right).$$

(43)

Now we can summarize the discussions as the following theorem.

Theorem 2. 

Let $g\in C^8\left(\mathbb{R}\right)$. On the uniform stencil (18), the RBF–FD operator (28) with weights (37)–(40) satisfies

$${\widehat{g}}^{″}\left(y_i\right)-g^{″}\left(y_i\right)=\mathcal{O}\left(h^6\right),h\to 0,$$

with the error expansion (43).

Proof. 

Due to similarity with the proof of Theorem 1, it is omitted.    □

Let $S_1<S_2<\cdots <S_N$ denote the global asset nodes. The first and second RBF–FD differentiation matrices in the S-direction are

$$D_S={\left(b_{i,j}\right)}_{N\times N},D_{SS}={\left({\overline{b}}_{i,j}\right)}_{N\times N},$$

(44)

with sparsity patterns inherited from (18):

$$b_{i,j}=0={\overline{b}}_{i,j}\mathrm{if}|i-j|>3.$$

(45)

For interior rows $4\le i\le N-3$, we set

$$b_{i,i+\mathsf{\ell }}=b_{i+\mathsf{\ell }}(\mathsf{\ell }=-3,-2,-1,0,1,2,3),{\overline{b}}_{i,i+\mathsf{\ell }}={\overline{b}}_{i+\mathsf{\ell }}(\mathsf{\ell }=-3,-2,-1,0,1,2,3),$$

where $\left\{b_{i+\mathsf{\ell }}\right\}$ are the first-derivative weights from (20)–(23) and $\left\{{\overline{b}}_{i+\mathsf{\ell }}\right\}$ are the second-derivative weights (37)–(40). Hence, each row has at most 7 nonzeros. Near boundaries ($i\in \{1,2,3,N-2,N-1,N\}$), one uses one-sided seven-point stencils that remain full rank and preserve sixth order; these are obtained by translating (30)–(36) to the appropriate one-sided sets and repeating the integrated-kernel consistency construction. In the fractional Black–Scholes setting, the degeneracy of the diffusion coefficient ${\sigma }^2{S}^2$ at $S=0$ interacts benignly with the stencil: the $S^2$ factor annihilates the contribution of $D_{SS}$ in the first row, while $D_S$ and the reaction term remain active, which is consistent with the financial boundary condition $u(0,\tau )$ and yields a diagonally dominant first row in the assembled operator.

To make the discussion of conditioning more precise, consider the local interpolation matrix

$$A\left(c\right)={\left({\varphi }_{\star }(y_{i+\mathsf{\ell }}-y_{i+m};c)\right)}_{\mathsf{\ell },m=-3}^3,$$

built on the uniform stencil (18) with either the standard MQ kernel (14), denoted by ${\varphi }_{\star }=\varphi$, or the integrated MQ kernel ${\varphi }_{\star }={\varphi }_2$ from (16). For each fixed $c>0$, the matrix $A\left(c\right)$ is symmetric positive definite, so its spectral condition number in the Euclidean norm is given by

$${\kappa }_2\left(A\left(c\right)\right):={\displaystyle \frac{{\lambda }_{max}\left(A\left(c\right)\right)}{{\lambda }_{min}\left(A\left(c\right)\right)}},$$

(46)

where ${\lambda }_{max}$ and ${\lambda }_{min}$ denote the largest and smallest eigenvalues of $A\left(c\right)$, respectively. It is well known that for the standard MQ kernel, the quantity ${\kappa }_2\left(A\left(c\right)\right)$ grows rapidly when the shape parameter c is either too small or too large relative to the local spacing h; see, for example, [30,31]. In terms of the dimensionless ratio $\beta =c/h$, one typically observes

$${\kappa }_2\left(A_{\mathrm{MQ}}\left(\beta \right)\right)\gg 1\mathrm{whenever}\beta \ll 1\mathrm{or}\beta \gg 1,$$

(47)

reflecting the well-known trade-off between flat and peaked RBFs.

The use of the integrated MQ kernel ${\varphi }_2$ in (16) prevents the basis functions from becoming excessively sharp at the grid scale and therefore reduces the spectral spread of the local matrix. On the seven-point stencil (18), one may express the entries of the integrated matrix $A_{\mathrm{IMQ}}\left(c\right)$ as discrete averages of the corresponding MQ entries, so that

$$A_{\mathrm{IMQ}}\left(c\right)=B^T{A}_{\mathrm{MQ}}\left(c\right)B,$$

(48)

for a certain $7\times 7$ matrix B whose entries depend only on the normalized locations ${\{(y_{i+\mathsf{\ell }}-y_i)/h\}}_{\mathsf{\ell }=-3}^3$. Relation (48) implies the bound

$${\kappa }_2\left(A_{\mathrm{IMQ}}\left(c\right)\right)\le {\kappa }_2{\left(B\right)}^2{\kappa }_2\left(A_{\mathrm{MQ}}\left(c\right)\right),$$

(49)

so the integration acts as a regularization in the sense that the conditioning of the local system is controlled by the fixed constant ${\kappa }_2\left(B\right)$, while all dependence on the shape parameter c is inherited from $A_{\mathrm{MQ}}\left(c\right)$. Since B is determined solely by the uniform stencil geometry, ${\kappa }_2\left(B\right)$ is bounded independently of c and h, and the integrated MQ kernel cannot introduce additional ill-conditioning beyond that already present in the underlying MQ system. In practice, replacing $\varphi$ by ${\varphi }_2$ smooths the basis functions on each stencil and contracts the spectrum of $A\left(c\right)$, which is consistent with the stabilization effect observed in the truncation error expansions (27) and (43).

To complement the qualitative discussion, one may report representative values of ${\kappa }_2\left(A\left(c\right)\right)$ for a fixed stencil and several values of $\beta =c/h$. In typical ranges of interest for option pricing, for instance, $2\le c/h\le 8$, one observes that

$${\kappa }_2\left(A_{\mathrm{IMQ}}\left(c\right)\right)\ll {\kappa }_2\left(A_{\mathrm{MQ}}\left(c\right)\right),$$

(50)

which quantitatively supports the interpretation of the integrated-kernel construction as a regularization mechanism that yields better conditioned stencil systems and, consequently, more stable differentiation weights for the high-order RBF–FD operators.

4. Full Discretization of the Fractional Black–Scholes PDE

Before presenting the fully discrete scheme, we clarify the interaction between the spatial RBF–FD discretization and the fractional time derivative. The numerical treatment follows a method-of-lines philosophy. First, the spatial derivatives in the fractional Black–Scholes Equation (5) are discretized by the seventh-point RBF–FD operators constructed in Section 2 and Section 3, yielding a system of time–fractional ordinary differential equations of the form

$${}^{C}D_{\tau }^{\alpha }U\left(\tau \right)=LU\left(\tau \right),$$

where $U\left(\tau \right)$ collects the nodal values of the option price and L is the sparse matrix assembled from $D_S$ and $D_{SS}$. At this stage, all spatial nonlocality and high-order accuracy are contained entirely in L, while the fractional memory effect is confined to the Caputo operator acting on $U\left(\tau \right)$.

The fractional time derivative is then discretized independently using the L1 scheme on a uniform temporal grid, which replaces ${}^{C}D_{\tau }^{\alpha }$ by a discrete convolution in time involving past solution values. Consequently, the spatial RBF–FD matrices enter the fully discrete scheme only through matrix–vector products at each time level, while the memory effect is realized through the time-history weights of the L1 approximation. This separation makes the coupling between the spatial meshless discretization and the fractional time-stepping explicit and allows the high-order spatial accuracy to be preserved under fractional temporal evolution.

For parabolic problems, the discrete spatial operator should be negative semidefinite on interior rows [26]. The weights (37)–(40) satisfy

$${\overline{b}}_i<0,{\overline{b}}_{i\pm 1}>0,{\overline{b}}_{i\pm 2}<0\mathrm{for}\mathrm{small}c,{\overline{b}}_{i\pm 3}>0\mathrm{for}\mathrm{small}c,$$

with the exact sign pattern depending slightly on $c/h$. In practice, for moderate $c/h$ the row sum vanishes and the diagonal entry dominates the off-diagonal sum in magnitude, which implies the semi-discrete operator $L:={\displaystyle \frac{1}{2}}{\sigma }^2\mathrm{diag}{\left(S\right)}^2{D}_{SS}+(r-q)\mathrm{diag}\left(S\right)D_S-rI$ is an M-matrix up to the skew part from $D_S$.

In the semi-discrete time–fractional Black–Scholes system

$${}^{C}D_{\tau }^{\alpha }u\left(\tau \right)={\displaystyle \frac{1}{2}}{\sigma }^2{S}^2{u}_{SS}\left(\tau \right)+(r-q)S{u}_S\left(\tau \right)-ru\left(\tau \right),$$

the spatial discretization with the present weights yields

$${}^{C}D_{\tau }^{\alpha }U\left(\tau \right)={\displaystyle \frac{1}{2}}{\sigma }^2\mathrm{diag}{\left(S\right)}^2{D}_{SS}U\left(\tau \right)+(r-q)\mathrm{diag}\left(S\right)D_{S}U\left(\tau \right)-rU\left(\tau \right),$$

where $U\left(\tau \right)={(u(S_1,\tau ),\dots ,u(S_N,\tau ))}^T$. The matrices $D_S$ and $D_{SS}$ can now be filled, and with boundary rows populated by the corresponding one-sided seven-point weights. The resulting matrices have exactly 7 nonzeros per interior row and at most 7 per boundary row, which keeps the total storage at $\mathcal{O}\left(7N\right)$.

In view of the discussion in Section 1, the time–fractional operator in (5) is taken in the Caputo sense and is denoted by ${}^{C}D_{\tau }^{\alpha }$. For payoffs with limited regularity, the modified RL operator ${}^{\mathrm{mRL}}D_{\tau }^{\alpha }$ defined in (8) coincides with ${}^{C}D_{\tau }^{\alpha }$ whenever $u(·,\tau )$ is differentiable in $\tau$ and the initial value $u(·,0)$ is incorporated explicitly, see the identity following (8). Thus, the formulations based on ${}^{C}D_{\tau }^{\alpha }$ and on ${}^{\mathrm{mRL}}D_{\tau }^{\alpha }$ are equivalent for the present pricing problem, and it is sufficient to discretize the Caputo derivative.

Assume that the temporal axis is divided into n discrete instants. This discretization is realized by a uniform mesh of the form

$$0={\tau }_0<{\tau }_1<\cdots <{\tau }_n=T,$$

(51)

covering the interval $[0,T]$, where each subinterval possesses the constant length $\delta$. According to the construction introduced in [36], the classical L1 procedure furnishes an estimation for the Caputo fractional derivative of the function b. Specifically, one has

$${}^{C}D_{\tau }^{\alpha }b(s_i,{\tau }_n)={\delta }^{-\alpha }\sum _{j=0}^n{l}_{j,n}b(s_i,{\tau }_{n-j})+\mathcal{O}\left({\delta }^{2-\alpha }\right).$$

(52)

The coefficients $l_{j,n}$ appearing above may be evaluated using the relation

$$\left(l_{j,n}\right)\mathsf{\Gamma }(2-\alpha )=\left\{\begin{array}{cc}-j^{1-\alpha }+{(j-1)}^{1-\alpha },\hfill & j=n,\hfill \\ 1,\hfill & j=0,\hfill \\ {(j-1)}^{1-\alpha }-2j^{1-\alpha }+{(j+1)}^{1-\alpha },\hfill & 1\le j\le n-1.\hfill \end{array}\right.$$

(53)

Whenever the nodes in (51) are equally spaced, a reformulation of the L1 discretization is attainable, leading to

$$\begin{array}{c}\hfill {}^{C}D_{\tau }^{\alpha }U(s_i,{\tau }_{k+1})={\displaystyle \frac{{\delta }^{-\alpha }}{\mathsf{\Gamma }(2-\alpha )}}\sum _{j=0}^k{\displaystyle \frac{l_{i,j+1}-l_{i,j}}{\delta }}\left(-j^{1-\alpha }+{(j+1)}^{1-\alpha }\right),\end{array}$$

(54)

which is the standard first-order approximation of the Caputo derivative and is consistent with the governing Equation (5).

In the context of option pricing for put and call contracts, the boundary prescriptions are typically stated as [37]

$$U(0,\tau )=K{e}^{-r\tau },\underset{S\to \infty }{lim}U(S,\tau )=0,$$

(55)

for the put case, whereas for the call variant, the boundary behavior becomes

$$U(0,\tau )=0,\underset{S\to \infty }{lim}U(S,\tau )=S_{max}{e}^{-q\tau }-K{e}^{-r\tau },$$

(56)

where $S_{max}$ is chosen sufficiently large and positive.

To formulate the spatial–temporal discretization, the Kronecker product ⊗ will be employed on the truncated spatial interval $[S_{min},S_{max}]=[0,S_{max}]$. By gathering the relevant discretized operators, one may write

$$M={\displaystyle \frac{1}{2}}{\sigma }^2{S}^2(D_{SS}\otimes I_{\tau })+(r-q)S(D_S\otimes I_{\tau })-r{I}_W,$$

(57)

with

$$S=diag(S_1,S_2,\dots ,S_N)\otimes I_{\tau },I_W=I_S\otimes I_{\tau }.$$

Here the identity unit $I_W$ is of size $W\times W$, where $W=Nn$, and $I_S\in {\mathbb{R}}^{N\times N}$, $I_{\tau }\in {\mathbb{R}}^{n\times n}$.

Implementation of the boundary constraints (55) and (56) can be accomplished by enforcing them directly on the first and last rows of the matrix M, corresponding to the spatial endpoints of the truncated domain.

The discrete operator defined by (54) can be assembled into a lower triangular matrix $D_{\tau }$. Consequently, a full semi-discrete analogue of (5) is obtained in the compact linear system

$$(I_S\otimes D_{\tau })U=MU,$$

(58)

where the vector of unknowns $U\in {\mathbb{R}}^W$ is arranged under the Kronecker structure as

$$U={(u_{1,1},u_{1,2},\dots ,u_{N,n})}^{\ast }.$$

5. Computational Results

This section gives a numerical study for (5) with spatial operators discretized by the seven-point RBF–FD matrices derived in Section 2 and Section 3. Three solvers have been compared on identical hardware and software stacks: the proposed method (abbreviated by PM) that uses the analytically integrated MQ weights of order six in space; a FD baseline (FD2), second order in space on uniform stencils but first order in time, following [21]; and a standard RBF–FD approach on uniform meshes using conventional MQ weights without the integrated construction (SM2), following [37]. Unless stated otherwise, the computational domain is $S\in [0,S_{max}]$ with $S_{max}=3K$, the MQ parameter is set to $c=4h$ with h the local spacing.

All numerical experiments reported in Section 5 were carried out in Mathematica 14.0 [38,39] running on a standard desktop workstation equipped with an Intel^® Core™ i7-12700 processor (12 cores, base frequency 2.1 GHz), 32 GB RAM, and Windows 11 64-bit operating system. The sparse linear algebra routines (including LinearSolve[] with Krylov subspace acceleration) were used with default internal settings. To ensure reproducibility, all computations were executed in a single-threaded mode unless stated otherwise, and no GPU acceleration or external libraries were employed.

All matrix assembly routines for the RBF–FD differentiation operators, including the construction of the seven-point local stencils, analytic weight evaluation, and sparsification of the global operators, were written in native Mathematica code. Time-stepping for the Caputo derivative used custom routines based on the L1 weights (52)–(54). No external numerical libraries or GPU-accelerated modules were used. The reproducibility of all computations follows from the fact that every component—assembly, time discretization, and sparse solves—relies entirely on deterministic built-in Mathematica functionality.

To cleanly separate spatial from temporal effects, the time discretization for PM has used the classical $L1$ formula on a uniform grid ${\tau }_j=j\delta$, $j=0,\dots ,n$, for the Caputo derivative. With $U^j\in {\mathbb{R}}^N$ the vector of nodal values at time level j, the fully discrete scheme reads

$${\displaystyle \frac{1}{\mathsf{\Gamma }(2-\alpha ){\delta }^{\alpha }}}\left(U^j-\sum _{k=0}^{j-1}{a}_k{U}^{j-1-k}\right)={\displaystyle \frac{1}{2}}{\sigma }^2\mathrm{diag}{\left(S\right)}^2{D}_{SS}{U}^j+(r-q)\mathrm{diag}\left(S\right)D_S{U}^j-r{U}^j,j\ge 1,$$

(59)

with $a_k={(k+1)}^{1-\alpha }-k^{1-\alpha }$, and $U^0$ given by the payoff. Formula (59) is first order in time and, in combination with the sixth-order space, yields a global error dominated by $\mathcal{O}\left(\delta \right)+\mathcal{O}\left(h^6\right)$. To balance the two contributions, we have used $\delta =\kappa h^6$ in the convergence experiments (constant $\kappa$ across all methods), so that spatial rates are visible in the error decay.

Error metrics have been recorded as follows. Let $U^n$ be the vector of numerical prices at maturity $\tau =T$ and $U^{\mathrm{ref}}$ a reference vector. The absolute error at the strike has been reported as

$$\mathrm{AE}=|u_{\mathrm{approx}}(K,T)-u_{\mathrm{ref}}(K,T)|.$$

(60)

In

Table 1. Comparison of solvers for Example 1 at the strike $S=K$; $\mathrm{AE}=|u_{\mathrm{approx}}(K,T)-18.1883|$.

N	u FD2	$\mathbf{AE}$ FD2	Time FD2	u SM2	$\mathbf{AE}$ SM2	Time SM2	u PM	$\mathbf{AE}$ PM	Time PM
20	17.88	2.9 × ${10}^{-1}$	0.02	17.69	4.9 × ${10}^{-1}$	0.02	18.058	1.3 × ${10}^{-1}$	0.02
40	17.91	2.7 × ${10}^{-1}$	0.15	18.00	1.8 × ${10}^{-1}$	0.17	18.098	9.0 × ${10}^{-2}$	0.19
80	18.03	1.4 × ${10}^{-1}$	3.91	18.21	2.4 × ${10}^{-2}$	3.71	18.165	2.3 × ${10}^{-2}$	3.97
120	18.05	1.3 × ${10}^{-1}$	19.55	18.19	1.6 × ${10}^{-3}$	19.85	18.187	1.3 × ${10}^{-3}$	20.57

and

Table 2. Results for Example 2. $\mathrm{AE}=|u_{\mathrm{approx}}(K,T)-24.6299|$.

N	u FD2	$\mathbf{AE}$ FD2	Time FD2	u SM2	$\mathbf{AE}$ SM2	Time SM2	u PM	$\mathbf{AE}$ PM	Time PM
21	24.31	3.1 × ${10}^{-1}$	0.11	24.11	5.1 × ${10}^{-1}$	0.19	24.236	3.9 × ${10}^{-1}$	0.20
41	24.46	1.6 × ${10}^{-1}$	0.40	24.35	2.7 × ${10}^{-1}$	0.52	24.598	3.1 × ${10}^{-2}$	0.75
81	24.54	8.4 × ${10}^{-2}$	6.39	24.56	6.4 × ${10}^{-2}$	7.65	24.622	7.9 × ${10}^{-3}$	8.25
161	24.58	4.9 × ${10}^{-2}$	85.97	24.63	6.1 × ${10}^{-3}$	88.25	24.627	2.9 × ${10}^{-3}$	90.25

we have reported $\mathrm{AE}$ at $S=K$ and CPU times. The reference prices $u(K,T)$ in Examples 1 and 2 have been taken from [37] and independently verified by PM on a very fine grid (h reduced by a factor $2^5$ with $\delta =\kappa h^6$), which produced agreement to at least four significant digits.

Example 1. 

European call with $K=\$100$, valuation date 11 November 2025, maturity 11 November 2026, parameters $r=5\%$, $q=0$, $\sigma =40\%$, fractional order $\alpha =0.8$, and $u(K,T)\simeq 18.1883$ from [37].

Table 1 summarizes the absolute strike error $\mathrm{AE}$ and CPU time for increasing numbers of spatial nodes. The proposed PM achieves visibly smaller errors than FD2 and SM2 at comparable costs, especially once $N\ge 40$. For $N=120$, PM attains a good accuracy at the strike. We remark that stability has remained robust for all three solvers; however, PM has produced better accuracy-to-time ratios due to the analytic weights that reduce dispersion and improve the conditioning of the sparse operator.

Example 2. 

European put with $K=\$100$, valuation date 25 March 2025, maturity 25 March 2026, parameters $r=5\%$, $q=20\%$, $\sigma =30\%$, and $\alpha =0.7$. The reference strike price is $u(K,T)\simeq 24.6299$ [37].

Table 2 reports the results for Example 2. Both FD2 and PM have remained stable across all N, with PM again providing the smallest errors for moderate and large N. For $N=161$ the strike error of PM is about one order of magnitude smaller than that of FD2, with virtually identical CPU times. Visual profiles of the option surface $u(S,T)$ at $N=n=81$ confirm that PM exhibits less numerical dispersion near the strike kink, an expected consequence of the integrated MQ-variant construction and the sixth-order moment cancellations of the second-derivative stencil.

Two additional observations are relevant for fractional pricing. First, the $L1$ history weights in (59) are positive and monotonically decreasing, implying a discrete maximum principle for the homogeneous part of the evolution when coupled with a negative semidefinite $D_{SS}$. The sign pattern of (37)–(40) and the vanishing row sums ensure that the semi-discrete operator has a nonpositive symbol, which contributes to the observed stability. Second, the memory cost of (59) grows like $\mathcal{O}\left(nN\right)$; in our experiments, n has been chosen proportional to $h^{-6}$ to balance errors, which keeps runtimes competitive due to the sparse seven-band structure of the spatial matrices. Also note that to clearly observe the sixth-order accuracy in the numerical work out for such a PDE problem, a smoothing on the initial condition is required. Since, due to the kink at the initial payoff condition, the real higher order might not be seen properly.

To complement the CPU-time comparison in Section 5, we briefly discuss the memory requirements of the three numerical solvers (FD2, SM2, and PM). Let N denote the number of spatial nodes and n the number of time levels. For each method, the dominant memory consumption arises from storing (i) the spatial differentiation matrices, (ii) the time-history vectors associated with the L1 discretization, and (iii) intermediate Krylov subspace iterates used in LinearSolve[]. For the FD2 method, the spatial discretization produces banded matrices with a bandwidth of 3, which require

$${\mathcal{M}}_{\mathrm{FD}2}\sim \mathcal{O}\left(3N\right)$$

memory units. For the SM2, each interior row contains $m=3$ nonzero entries, and it is similar to FD2, as

$${\mathcal{M}}_{\mathrm{SM}2}\sim \mathcal{O}\left(3N\right).$$

For the PM, the spatial matrices also contain exactly 7 nonzeros per interior row, but no extended-precision regularization is needed because the integrated MQ kernel produces well-conditioned local matrices. Therefore, the overall memory footprint behaves as

$${\mathcal{M}}_{\mathrm{PM}}\sim \mathcal{O}\left(7N\right),$$

with smaller constant factors than SM2 due to the absence of stabilization stages. The L1 time discretization requires, for all three methods, storing the fractional-history vectors

$$\mathcal{O}\left(nN\right)\mathrm{units},$$

which dominate the asymptotic memory usage when n is large. A representative measurement on a mesh with $N=161$ and $n=161$ shows that the peak memory footprint (for storing matrices, solution snapshots, and solver work arrays) is approximately:

$$\mathrm{FD}2:140\mathrm{MB},\mathrm{SM}2:190\mathrm{MB},\mathrm{PM}:155\mathrm{MB}.$$

Finally, the choice $c=4h$ has been consistent with the error expansions (43) and with the first-derivative expansion (27). This scaling keeps the c-dependent terms at the same asymptotic order as the $g^{\left(8\right)}$ contribution and has yielded near-optimal constants in practice. A mild dependence of the optimal $c/h$ ratio on $\alpha$ has been observed: smaller $\alpha$ tends to benefit from slightly larger $c/h$ to further suppress high-frequency dispersion in $D_{SS}$ as the memory kernel in (59) becomes heavier.

6. Conclusions

This work has developed a sixth-order meshless discretization for the spatial operator in the temporal–fractional Black–Scholes equation. Closed-form differentiation weights have been constructed from analytically integrated inverse MQ-variant kernels, producing sparse and symmetric RBF–FD matrices on seven-point stencils. The resulting operators have preserved polynomial reproduction up to degree five and have delivered provable sixth-order truncation error for both first- and second-order derivatives, including one-sided stencils near boundaries. Numerical tests for European options have verified that the scheme has achieved high accuracy and strong stability when coupled with fractional time-stepping, outperforming classical FDs and standard RBF–FD implementations.

Future research will address several promising directions. The present methodology will be extended to American options through the incorporation of early-exercise features and free-boundary tracking. This could be done for future work in this direction.