Efficient Scheme for the Economic Heston–Hull–White Problem Using Novel RBF-FD Coefficients Derived from Multiquadric Function Integrals

Abstract

This study presents an efficient method using the local radial basis function finite difference scheme (RBF-FD). The innovative coefficients are derived from the integrals of the multiquadric (MQ) function. Theoretical convergence rates for the coefficients used in function derivative approximation are provided. The proposed scheme utilizes RBF-FD estimations on three-point non-uniform stencils to construct the final approximation on a tensor grid for the 3D Heston–Hull–White (HHW) PDE, which is relevant in economics and mathematical finance. Numerical evidence and comparative analyses validate the results and the proposed scheme.

1. Introduction

1.1. Financial View

Tech stocks are known for their volatility (vol), which can vary significantly over time. To illustrate, consider a scenario where the tech sector experiences increased vol due to uncertainty surrounding regulatory changes. Traditional models that assume constant vol might underestimate the risk associated with options, leading to mispricing [1,2]. In such a circumstance, the Heston model [3] can be given by the following:

$$\begin{array}{ccc}\hfill dA\left(t\right)=& \sqrt{Q\left(t\right)}A\left(t\right)d{B}_1\left(t\right)+\overline{r}A\left(t\right)dt,\hfill & \hfill A\left(t\right)>0,\\ \hfill dQ\left(t\right)=& \gamma \sqrt{Q\left(t\right)}d{B}_2\left(t\right)+\upsilon (\eta -Q\left(t\right))dt,\hfill & \hfill Q\left(t\right)>0.\end{array}$$

(1)

In (1), we considered the variance and stock processes denoted by $Q\left(t\right)$ and $A\left(t\right)$, respectively. The vol of vol is represented by $\gamma >0$, while the risk-free fixed rate of interest is $\overline{r}>0$. The set of stochastic differential equations (SDEs) (1) governing the system is characterized by $d{B}_1\left(t\right)d{B}_2\left(t\right)=\rho dt$, where $B_1\left(t\right)$ and $B_2\left(t\right)$ are two Wiener processes. The adjustment speed for the vol to its mean is determined by $\upsilon >0$, with $\eta >0$ denoting the mean level. To ensure $Q\left(t\right)>0$, the Feller condition must be read: $2\upsilon \eta >{\gamma }^2$.

To clarify the need in practice more, suppose there is a risk manager at an investment bank, and a client requests pricing for European call options on a popular tech stock index, such as the NASDAQ-100. The client wants accurate pricing for options expiring in six months (or one year) to hedge their portfolio against potential market vol. The Heston–Hull–White (HHW) model [4] captures the realistic dynamics of stock prices, incorporating factors like mean reversion and stochastic interest rates. In the tech sector, where sentiment can rapidly shift due to news about technological breakthroughs, regulatory changes, or geopolitical events, accurately modeling these dynamics is crucial. For instance, if a major tech company announces a breakthrough in artificial intelligence (AI) [5], it could trigger a surge in stock prices across the sector. The HHW model’s ability [6] may capture such rapid movements to some extent, and mean reversion helps in pricing options more accurately than simpler models.

Interest rate movements [7], particularly changes in the yield curve, can impact option pricing. In the tech sector, where companies often rely on debt financing for research and development (R&D), interest rate dynamics play a crucial role.

1.2. The Mathematical Model

The HHW model is expressed by the following dynamic [8,9]:

$$\begin{array}{cc}\hfill dA\left(t\right)=& A\left(t\right)C\left(t\right)dt+A\left(t\right)\sqrt{Q\left(t\right)}d{B}_1\left(t\right),\hfill \\ \hfill dQ\left(t\right)=& \upsilon (-Q\left(t\right)+\eta )dt+{\sigma }_1\sqrt{Q\left(t\right)}d{B}_2\left(t\right),\hfill \\ \hfill dC\left(t\right)=& a(-C\left(t\right)+b\left(t\right))dt+{\sigma }_{2}d{B}_3\left(t\right),\hfill \end{array}$$

(2)

where $C\left(t\right)$ represents the process of interest at $0<t\le T$, $B_3\left(t\right)$ is another Brownian motion, and b is a positive given function. In this stochastic evolution dynamic, $\upsilon ,\eta ,{\sigma }_1,{\sigma }_2,a$ are real suitable positive constants.

To illustrate this, assume we are using the HHW model to price European call options on the NASDAQ-100 index with a six-month maturity (T). You input the current stock index level, the estimated vol of the index (which follows a stochastic process in the HHW model), the prevailing risk-free rate of interest, and the strike price of the option into the model. The HHW model then simulates thousands of possible future paths for the index level, each with its associated vol and interest rate dynamics. For each simulated path, it calculates the payoff of the call option at expiration based on the simulated index level. By averaging these payoffs across all simulated paths and discounting them back to the present value employing the corresponding simulated interest rates, the HHW model (2) derives the fair price of the call European option.

1.3. PDE Formulation

The 3D HHW time-dependent partial differential equation (PDE) stands as a significant model in mathematical finance. It characterizes both vol and interest rate dynamics through stochastic processes; see [10] for some pioneer discussions.

Using the dynamics outlined in (2) and having ${\rho }_{12}$, ${\rho }_{13}$, ${\rho }_{23}\in [-1,1]$ as the correlation parameters, we can derive the value of a European option given as [11]

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{\mathsf{\partial }V(x,p,o,\tau )}{\mathsf{\partial }\tau }}=& {\displaystyle \frac{1}{2}}{x}^{2}p{\displaystyle \frac{{\mathsf{\partial }}^{2}V(x,p,o,\tau )}{\mathsf{\partial }{x}^2}}+{\displaystyle \frac{1}{2}}{\sigma }_1^{2}p{\displaystyle \frac{{\mathsf{\partial }}^{2}V(x,p,o,\tau )}{\mathsf{\partial }{p}^2}}\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\sigma }_2^2{\displaystyle \frac{{\mathsf{\partial }}^{2}V(x,p,o,\tau )}{\mathsf{\partial }{o}^2}}+{\rho }_{12}{\sigma }_{1}xp{\displaystyle \frac{{\mathsf{\partial }}^{2}V(x,p,o,\tau )}{\mathsf{\partial }x\mathsf{\partial }p}}\hfill \\ \hfill & +{\rho }_{13}{\sigma }_{2}x\sqrt{p}{\displaystyle \frac{{\mathsf{\partial }}^{2}V(x,p,o,\tau )}{\mathsf{\partial }x\mathsf{\partial }o}}+{\rho }_{23}{\sigma }_1{\sigma }_2\sqrt{p}{\displaystyle \frac{{\mathsf{\partial }}^{2}V(x,p,o,\tau )}{\mathsf{\partial }p\mathsf{\partial }o}}\hfill \\ \hfill & +ox{\displaystyle \frac{\mathsf{\partial }V(x,p,o,\tau )}{\mathsf{\partial }x}}+\upsilon (\eta -p){\displaystyle \frac{\mathsf{\partial }V(x,p,o,\tau )}{\mathsf{\partial }p}}\hfill \\ \hfill & +a(b(T-\tau )-o){\displaystyle \frac{\mathsf{\partial }V(x,p,o,\tau )}{\mathsf{\partial }o}}-oV(x,p,o,\tau ),\hfill \end{array}\end{array}$$

(3)

where x, p, and o represent the price of the asset, the spontaneous variance, and the interest rate, respectively. $\tau$ shows the forward movement along time for the PDE problem. The non-smooth initial condition, (payoff) for the call scenario, may be written as follows:

$$V(x,p,o,0)={(x-E,0)}^{+},$$

(4)

wherein E is the strike price; refer also to the discussion in [12] for discussion on initial and boundary conditions on another similar interest rate model. The boundary conditions for $x,p,o$ are provided as outlined below [13]:

$$\begin{array}{ccc}\hfill & V_x(x,p,o,\tau )=1,\hfill & \hfill x=x_{max},\end{array}$$

(5)

$$\begin{array}{ccc}\hfill & V(x,p,o,\tau )=0,\hfill & \hfill x=0,\end{array}$$

(6)

$$\begin{array}{ccc}\hfill & V(x,p,o,\tau )=x,\hfill & \hfill p=p_{max},\end{array}$$

(7)

$$\begin{array}{ccc}\hfill & V_o(x,p,o,\tau )=0,\hfill & \hfill o=-o_{max},\end{array}$$

(8)

$$\begin{array}{ccc}\hfill & V_o(x,p,o,\tau )=0,\hfill & \hfill o=o_{max}.\end{array}$$

(9)

We recall that in the case where $p=0$, the PDE (3) becomes degenerate, and no side conditions need to be imposed (see for more [14]).

1.4. Methodology and Objectives

The scheme of the radial basis function in the format of finite difference (RBF-FD) extends the classic FD scheme for accommodating scattered node layouts and has successfully been applied for various important option pricing problems in recent literature [15,16]. This approach leverages RBFs to estimate an operator in a local sense [17,18], which is linear and of the differentiation type within a specified neighborhood, as discussed in [19]. This neighborhood, known as the stencil in discrete contexts, is generally expressed by the n closest knots. To provide the definition, an RBF [20] is the following symmetric map

$$f\left(r\right):{\mathbb{R}}^M\to \mathbb{R},$$

in the dimension M, where $\mathbf{x}$ and ${\mathbf{x}}_k$ are the evaluation and center nodes, respectively, while $r=\parallel \mathbf{x}-{\mathbf{x}}_k{\parallel }_2$, where ${\parallel ·\parallel }_2$ represents the 2-norm.

While other kernels like the Gaussian, inverse multiquadric, Matern [21], Wendland, and others could be used, we concentrate solely on the multiquadric (MQ) kernel because its integrals can be more easily managed, allowing us to derive analytical formulas for solving the 3D HHW PDE problem (3) under (4) and (5).

Some comments are in order:

• Our primary objective here is to introduce a novel numerical solver for simulating (3) using the RBF-FD local approximations, resulting in sparse matrices.
• This is particularly crucial since (3) represents a 3D variable-coefficient problem, involving three mixed derivatives. Consequently, designing numerical methods for this purpose requires careful consideration.
• Toward this goal, the RBF-FD approximations are tailored to be applicable on non-uniform meshes, emphasizing regions of high significance.

The motivation behind this study is to introduce an efficient numerical solver using the local RBF-FD for the 3D HHW PDE, which is prevalent in mathematical finance. By deriving innovative coefficients from the integrals of the MQ function and employing three-point non-uniform stencils, the proposed scheme ensures precise approximations on a tensor grid. The emphasis on non-uniform meshes in regions of high significance highlights the robustness and applicability of the proposed RBF-FD solver.

1.5. Structure

The remainder of this paper is arranged in the following order. Section 2 details the major contribution of the work and focuses on using the integrals of the MQ function and then the construction of RBF-FD weights for a novel RBF, demonstrating convergence on non-uniform three-point meshes. These approximations can also be adapted and simplified for uniform meshes. In Section 3, we discuss some techniques for generating a non-uniform set of discretization knots along the spatial variables, focusing on the crucial regions of (3). Our proposed RBF-FD solver, with a focus on financially significant areas where the payoff exhibits nonsmoothness, is presented in Section 4. Section 5 is dedicated to discussing the practical utility and applicability of the proposed scheme, which is accompanied by numerical tests and simulations. Lastly, a concluding remark along some points for future works is provided in the last section.

2. Constructing the Coefficients via MQ Function

Let $X=\{{\mathbf{x}}_1,{\mathbf{x}}_2,\dots ,{\mathbf{x}}_N\}$ be a partition of distinguished points in $D\subset {\mathbb{R}}^M$, including the center ${\mathbf{x}}_c$ and $n-1$ nearest nodes forming a stencil for $n\ll N$. The approximation of the RBF-FD for u is given now in what follows [22] (${\delta }_i$ stands for the coefficients):

$$\mathcal{L}u\left(\mathbf{x}\right){|}_{{\mathbf{x}}_c\in X}\simeq \sum _{i=1}^n{\delta }_i{u}_i,$$

(10)

wherein $\mathcal{L}$ is a differential linear operator and $u\left({\mathbf{x}}_i\right)=u_i$. We state that the MQ infinitely-smooth RBF [23], is expressed by

$$f\left(r\right)=\sqrt{{ϵ}^2+r^2},$$

(11)

where $ϵ>0$ denotes the shape parameter. Extensive discussions along with some algorithms for computing an efficient shape parameter have recently been furnished in [24].

Our objectives in this section are as follows. Firstly, we aim to improve the applicability of (10) by incorporating integrals of RBFs [25]. This involves integrating the RBFs to an appropriate order (see also [26,27]) and then using these integrals as the basis for generating the weights. Secondly, we aim to achieve a theoretical convergence speed by employing three-knot unstructured stencils. The goal is to prove convergence orders for deriving weights using the RBFs’ integrals and then apply these in solving the 3D HHW PDE problem later in Section 4.

Further research is warranted to explore the incorporation of integrals of alternative RBFs such as Gaussian, Matern, and inverse-multiquadric kernels for solving HHW PDE. Currently, there is a gap in the literature regarding these approaches, necessitating a literature review to analyze how these alternative kernels have been utilized and their respective performance outcomes.

Let us examine the functions derived by integrating the kernel (11). The expressions for the 1st and 2nd integrals of (11) in 1D, based solely on the Euclidean distance, are

$$\begin{array}{cc}\hfill & f^1\left(r\right)=\int f\left(r\right)dr={\displaystyle \frac{{ϵ}^2\left(ϵr{\left(\frac{{ϵ}^2+r^2}{{ϵ}^2}\right)}^{3/2}+{ϵ}^2{sinh}^{-1}\left(\frac{r}{ϵ}\right)+r^2{sinh}^{-1}\left(\frac{r}{ϵ}\right)\right)}{2\left({ϵ}^2+r^2\right)}},\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill f^2\left(r\right)=& \int \int f\left(r\right)drdr={\displaystyle \frac{1}{3}}ϵ\left(-\left({ϵ}^2\left(\sqrt{{\displaystyle \frac{r^2}{{ϵ}^2}}+1}-1\right)\right)\right.\hfill \\ \hfill & \left.-r^2\sqrt{{\displaystyle \frac{r^2}{{ϵ}^2}}+1}+{\displaystyle \frac{3ϵr\left(ϵr{\left(\frac{{ϵ}^2+r^2}{{ϵ}^2}\right)}^{3/2}+{ϵ}^2{sinh}^{-1}\left(\frac{r}{ϵ}\right)+r^2{sinh}^{-1}\left(\frac{r}{ϵ}\right)\right)}{2\left({ϵ}^2+r^2\right)}}\right).\hfill \end{array}$$

(13)

We now derive the coefficients for the RBF-FD method in one dimension regarding the 2nd integral of the MQ function, particularly (13). First, we assume a set of knots $x_1,x_2,\dots ,x_N$ in the working interval. Then, we consider the following [28]:

$$\{x_{i-1},x_i,x_{i+1}\}=\{x_i-h,x_i,x_i+wh\},h>0,$$

(14)

wherein w controls the stretch of the three-closest knots and $x_{i+1}=x_i+wh$.

2.1. First-Derivative Coefficients

We use the formulation (10) when $n=3$ as follows [29]:

$$\begin{array}{cc}\hfill u^{\prime }\left(x_i\right)\simeq & {\widehat{u}}^{\prime }\left(x_i\right)={\delta }_{i-1}u\left(x_{i-1}\right)+{\delta }_{i}u\left(x_i\right)+{\delta }_{i+1}u\left(x_{i+1}\right),2\le i\le N-1,\hfill \end{array}$$

(15)

wherein ${\widehat{u}}^{\prime }$ is the approximate to the 1st differentiation. The relation (13) is employed to substitute u at the knots [29], resulting in

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{3}}\left({ϵ}^3{\delta }_i-{\left({ϵ}^2+h^2\right)}^{3/2}{\delta }_i+{\displaystyle \frac{3{ϵ}^2\left(ϵh{\left(\frac{{ϵ}^2+h^2}{{ϵ}^2}\right)}^{3/2}+{ϵ}^2{sinh}^{-1}\left(\frac{h}{ϵ}\right)+h^2{sinh}^{-1}\left(\frac{h}{ϵ}\right)\right)\left(h{\delta }_i-1\right)}{2\left({ϵ}^2+h^2\right)}}\right.\hfill \\ \hfill & \left.+{ϵ}^3{\delta }_{i+1}-{\delta }_{i+1}{\left({ϵ}^2+h^2{(w+1)}^2\right)}^{3/2}+3ϵh^2{(w+1)}^2{\delta }_{i+1}{\left({\displaystyle \frac{{(hw+h)}^2}{{ϵ}^2}}+1\right)}^{3/2}{\varrho }_1\right)=0,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{3}}\left({\delta }_{i-1}\left({ϵ}^3-{\left({ϵ}^2+h^2\right)}^{3/2}+{\displaystyle \frac{3{ϵ}^{2}h\left(ϵh{\left(\frac{{ϵ}^2+h^2}{{ϵ}^2}\right)}^{3/2}+{ϵ}^2{sinh}^{-1}\left(\frac{h}{ϵ}\right)+h^2{sinh}^{-1}\left(\frac{h}{ϵ}\right)\right)}{2\left({ϵ}^2+h^2\right)}}\right)\right.\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & \left.{\delta }_{i+1}+\left({ϵ}^3-{\left({ϵ}^2+h^2{w}^2\right)}^{3/2}+3ϵh^2{w}^2{\left({\displaystyle \frac{h^2{w}^2}{{ϵ}^2}}+1\right)}^{3/2}{\varrho }_2\right)\right)=0,\hfill \\ \hfill & {\displaystyle \frac{1}{3}}\left({\delta }_{i-1}\left({ϵ}^3-{\left({ϵ}^2+h^2{(w+1)}^2\right)}^{3/2}+3ϵh^2{(w+1)}^2{\left({\displaystyle \frac{{(hw+h)}^2}{{ϵ}^2}}+1\right)}^{3/2}{\varrho }_3\right)\right.\hfill \\ \hfill & \left.+{ϵ}^3{\delta }_i-{\delta }_i{\left({ϵ}^2+h^2{w}^2\right)}^{3/2}+3ϵhw{\left({\displaystyle \frac{h^2{w}^2}{{ϵ}^2}}+1\right)}^{3/2}\left({\displaystyle \frac{1}{2\left(\frac{h^2{w}^2}{{ϵ}^2}+1\right)}}+{\varrho }_4\right)\left(hw{\delta }_i+1\right)\right)=0,\hfill \end{array}$$

(18)

where

$$\begin{array}{cc}\hfill & {\varrho }_1=\left({\displaystyle \frac{1}{2\left(\frac{{(hw+h)}^2}{{ϵ}^2}+1\right)}}+{\displaystyle \frac{ϵ{sinh}^{-1}\left(\frac{hw+h}{ϵ}\right)}{2(hw+h){\left(\frac{{(hw+h)}^2}{{ϵ}^2}+1\right)}^{3/2}}}\right),\hfill \\ \hfill & {\varrho }_2=\left({\displaystyle \frac{1}{2\left(\frac{h^2{w}^2}{{ϵ}^2}+1\right)}}+{\displaystyle \frac{ϵ{sinh}^{-1}\left(\frac{hw}{ϵ}\right)}{2hw{\left(\frac{h^2{w}^2}{{ϵ}^2}+1\right)}^{3/2}}}\right),\hfill \\ \hfill & {\varrho }_3=\left({\displaystyle \frac{1}{2\left(\frac{{(hw+h)}^2}{{ϵ}^2}+1\right)}}+{\displaystyle \frac{ϵ{sinh}^{-1}\left(\frac{hw+h}{ϵ}\right)}{2(hw+h){\left(\frac{{(hw+h)}^2}{{ϵ}^2}+1\right)}^{3/2}}}\right),\hfill \\ \hfill & {\varrho }_4={\displaystyle \frac{ϵ{sinh}^{-1}\left(\frac{hw}{ϵ}\right)}{2hw{\left(\frac{h^2{w}^2}{{ϵ}^2}+1\right)}^{3/2}}}.\hfill \end{array}$$

Solving (16)–(18) by imposing several Taylor expansions around zero to simplify as much as possible leads to

$$\begin{array}{cc}\hfill & {\delta }_{i-1}={\displaystyle \frac{hw}{12{ϵ}^2}}-{\displaystyle \frac{w}{hw+h}},\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & {\delta }_i={\displaystyle \frac{w-1}{hw}}-{\displaystyle \frac{h(w-1)}{12{ϵ}^2}},\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & {\delta }_{i+1}={\displaystyle \frac{1}{h{w}^2+hw}}-{\displaystyle \frac{h}{12{ϵ}^2}}.\hfill \end{array}$$

(21)

Theorem 1.

Consider the function $f^2\left(r\right)$ as defined in (13). Formula (15), which estimates the first derivative of a suitably differentiable function $u\left(x\right)$ evaluated at the nodes given by (14), demonstrates the 2nd speed of convergence.

Proof. 

To verify this convergence speed, we must use the weights derived analytically in Equations (19)–(21) and incorporate them into Formula (15) as outlined in (19)–(21). By expanding the resulting expressions in a Taylor series around zero and considering terms up to the second order, we derive the following local truncation error (LTE):

$${\epsilon }_1\left(x_i\right)={\displaystyle \frac{1}{6}}w\left(-{\displaystyle \frac{u^{\prime }\left(x_i\right)}{{ϵ}^2}}+u^{\left(3\right)}\left(x_i\right)\right)h^2+\mathcal{O}\left(h^3\right),$$

(22)

where ${\epsilon }_1\left(x_i\right)={\widehat{u}}^{\prime }\left(x_i\right)-u^{\prime }\left(x_i\right)$. The error in Equation (22) confirms that the approximation reaches a quadratic convergence rate, which is applicable to both uniform and non-uniform grids. This finishes the proof. □

Corollary 1.

Based on the findings in Theorem 1, as $w=1$, an equidistant stencil is created. This leads to simplifications of the coefficients and equations of error, as shown below:

$${\delta }_{i-1}={\displaystyle \frac{h}{12{ϵ}^2}}-{\displaystyle \frac{1}{2h}},{\delta }_i=0,{\delta }_{i+1}=-{\delta }_{i-1},$$

(23)

and

$${\epsilon }_1\left(x_i\right)={\displaystyle \frac{1}{6}}\left(-{\displaystyle \frac{u^{\prime }\left(x_i\right)}{{ϵ}^2}}+u^{\left(3\right)}\left(x_i\right)\right)h^2+\mathcal{O}\left(h^3\right).$$

2.2. Second-Derivative Coefficients

Addressing the issue of the second differentiation, we now utilize (14) and proceed with the following formulation,

$$u^{″}\left(x_i\right)\simeq {\widehat{u}}^{″}\left(x_i\right)={\pi }_{i-1}u\left(x_{i-1}\right)+{\pi }_{i}u\left(x_i\right)+{\pi }_{i+1}u\left(x_{i+1}\right),2\le i\le N-1,$$

(24)

wherein ${\widehat{u}}^{″}$ represents the approximate of the RBF-FD for the 2nd differentiation. Similarly, we move forward by constructing the corresponding linear system at the stencil’s knots

$$\begin{array}{cc}\hfill & \sqrt{{ϵ}^2+h^2}={\displaystyle \frac{1}{3}}ϵ\left({\pi }_i\left(-{\displaystyle \frac{{\left({ϵ}^2+h^2\right)}^{3/2}}{ϵ}}+{\varrho }_5+{ϵ}^2\right)+\right.\hfill \\ \hfill & \left.{\pi }_{i+1}+\left(-{\displaystyle \frac{{\left({ϵ}^2+h^2{(w+1)}^2\right)}^{3/2}}{ϵ}}+{\varrho }_6+{ϵ}^2\right)\right),\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{3}}\left(-3ϵ+{\pi }_{i-1}\left({ϵ}^3-{\left({ϵ}^2+h^2\right)}^{3/2}+{\varrho }_7\right)\right.\hfill \\ \hfill & \left.{\pi }_{i+1}+\left({ϵ}^3-{\left({ϵ}^2+h^2{w}^2\right)}^{3/2}+3ϵh^2{w}^2{\left({\displaystyle \frac{h^2{w}^2}{{ϵ}^2}}+1\right)}^{3/2}{\varrho }_8\right)\right)=0,\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & \sqrt{{ϵ}^2+h^2{w}^2}=\hfill \\ \hfill & {\displaystyle \frac{1}{3}}ϵ\left({\pi }_{i-1}\left(-{\displaystyle \frac{{\left({ϵ}^2+h^2{(w+1)}^2\right)}^{3/2}}{ϵ}}+3h^2{(w+1)}^2{\left({\displaystyle \frac{{(hw+h)}^2}{{ϵ}^2}}+1\right)}^{3/2}{\varrho }_9+{ϵ}^2\right)\right.\hfill \\ \hfill & \left.{\pi }_i+\left(-{\displaystyle \frac{{\left({ϵ}^2+h^2{w}^2\right)}^{3/2}}{ϵ}}+3h^2{w}^2{\left({\displaystyle \frac{h^2{w}^2}{{ϵ}^2}}+1\right)}^{3/2}{\varrho }_{10}+{ϵ}^2\right)\right).\hfill \end{array}$$

(27)

where

$$\begin{array}{cc}\hfill & {\varrho }_5={\displaystyle \frac{3ϵh\left(ϵh{\left(\frac{{ϵ}^2+h^2}{{ϵ}^2}\right)}^{3/2}+{ϵ}^2{sinh}^{-1}\left(\frac{h}{ϵ}\right)+h^2{sinh}^{-1}\left(\frac{h}{ϵ}\right)\right)}{2\left({ϵ}^2+h^2\right)}},\hfill \\ \hfill & {\varrho }_6=3h^2{(w+1)}^2{\left({\displaystyle \frac{{(hw+h)}^2}{{ϵ}^2}}+1\right)}^{3/2}\left({\displaystyle \frac{1}{2\left(\frac{{(hw+h)}^2}{{ϵ}^2}+1\right)}}+{\displaystyle \frac{ϵ{sinh}^{-1}\left(\frac{hw+h}{ϵ}\right)}{2(hw+h){\left(\frac{{(hw+h)}^2}{{ϵ}^2}+1\right)}^{3/2}}}\right),\hfill \\ \hfill & {\varrho }_7={\displaystyle \frac{3{ϵ}^{2}h\left(ϵh{\left(\frac{{ϵ}^2+h^2}{{ϵ}^2}\right)}^{3/2}+{ϵ}^2{sinh}^{-1}\left(\frac{h}{ϵ}\right)+h^2{sinh}^{-1}\left(\frac{h}{ϵ}\right)\right)}{2\left({ϵ}^2+h^2\right)}},\hfill \\ \hfill & {\varrho }_8=\left({\displaystyle \frac{1}{2\left(\frac{h^2{w}^2}{{ϵ}^2}+1\right)}}+{\displaystyle \frac{ϵ{sinh}^{-1}\left(\frac{hw}{ϵ}\right)}{2hw{\left(\frac{h^2{w}^2}{{ϵ}^2}+1\right)}^{3/2}}}\right),\hfill \\ \hfill & {\varrho }_9=\left({\displaystyle \frac{1}{2\left(\frac{{(hw+h)}^2}{{ϵ}^2}+1\right)}}+{\displaystyle \frac{ϵ{sinh}^{-1}\left(\frac{hw+h}{ϵ}\right)}{2(hw+h){\left(\frac{{(hw+h)}^2}{{ϵ}^2}+1\right)}^{3/2}}}\right),\hfill \\ \hfill & {\varrho }_{10}=\left({\displaystyle \frac{1}{2\left(\frac{h^2{w}^2}{{ϵ}^2}+1\right)}}+{\displaystyle \frac{ϵ{sinh}^{-1}\left(\frac{hw}{ϵ}\right)}{2hw{\left(\frac{h^2{w}^2}{{ϵ}^2}+1\right)}^{3/2}}}\right).\hfill \end{array}$$

The resolution for (25)–(27) yields as follows:

$$\begin{array}{cc}\hfill & {\pi }_{i-1}={\displaystyle \frac{720{ϵ}^4+60{ϵ}^2{h}^2\left(w^2-w-1\right)+h^4\left(-17w^4-68w^3+63w^2+22w+11\right)}{360{ϵ}^4{h}^2(w+1)}},\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & {\pi }_i={\displaystyle \frac{-720{ϵ}^4-60{ϵ}^2{h}^2\left(w^2-3w+1\right)+h^4\left(17w^4+75w^2+17\right)}{360{ϵ}^4{h}^{2}w}},\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & {\pi }_{i+1}={\displaystyle \frac{720{ϵ}^4-60{ϵ}^2{h}^2\left(w^2+w-1\right)+h^4\left(11w^4+22w^3+63w^2-68w-17\right)}{360{ϵ}^4{h}^{2}w(w+1)}}.\hfill \end{array}$$

(30)

Theorem 2.

Let $f^2\left(r\right)$ be as defined in (13). Then, the second derivative of a sufficiently smooth function $u\left(x\right)$, as given by (24) and evaluated using (28) through (30) at the points specified in (14), demonstrates linear convergence.

Proof. 

The demonstration follows a method analogous to the one used in proving Theorem 1. By developing Taylor’s expansion for h on 0 to the first order for the coefficients and then replacing these into (24), we can articulate

$${\epsilon }_2\left(x_i\right)={\displaystyle \frac{(w-1)\left({ϵ}^2{u}^{\left(3\right)}\left(x_i\right)-u^{\prime }\left(x_i\right)\right)}{3{ϵ}^2}}h+\mathcal{O}\left(h^2\right).$$

(31)

We considered the definition ${\epsilon }_2\left(x_i\right)={\widehat{u}}^{″}\left(x_i\right)-u^{″}\left(x_i\right)$ in (31). The proof is complete. □

Corollary 2.

Based on the findings provided in Theorem 2, when $w=1$, a uniform stencil is formed. As a result, the analytical weights are simplified, leading to a corresponding simplification in the error equation, which is expressed as

$$\begin{array}{cc}\hfill & {\pi }_{i-1}={\displaystyle \frac{11h^2}{720{ϵ}^4}}-{\displaystyle \frac{1}{12{ϵ}^2}}+{\displaystyle \frac{1}{h^2}},\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & {\pi }_i={\displaystyle \frac{109h^2}{360{ϵ}^4}}+{\displaystyle \frac{1}{6{ϵ}^2}}-{\displaystyle \frac{2}{h^2}},\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & {\pi }_{i+1}={\pi }_{i-1},\hfill \end{array}$$

(34)

and

$${\epsilon }_2\left(x_i\right)={\displaystyle \frac{\left({ϵ}^4{u}^{\left(4\right)}\left(x_i\right)-{ϵ}^2{u}^{\prime \prime }\left(x_i\right)+4u\left(x_i\right)\right)}{12{ϵ}^4}}{h}^2+\mathcal{O}\left(h^3\right).$$

With this type of stencil, the convergence order improves to two.

The coefficients derived in this research are innovative, as they originate from the integrals of the MQ function. The weights under this approach enhance the accuracy and stability in approximating function derivatives since they are built upon the integral of the RBFs. This gives us more generality as well as more flexibility when approximating a suitable function, as also pointed out in [30].

For RBFs, Gaussian (smooth, but sensitive to shape parameter), Matern (less smooth than Gaussian, a balanced choice between smoothness and non-sensitivity to shape parameters) and inverse-multiquadric (bounded, more stable) kernels are more popular than multiquadric kernels (unbounded growth, sensitive to shape parameters, not smooth as Gaussian). I understand that authors choose the MQ kernel because its integral is easier to manage, thus making the technicalities of this manuscript possible. However, I recommend that the authors conduct a literature review to compare how alternative kernels have been utilized and their performance outcomes in solving HHW PDEs.

3. Set of Discretization Knots

The HHW 3D PDE (3) is defined in

$$(x,p,o,\tau )\in [0,+\infty )\times [0,+\infty )\times (-\infty ,+\infty )\times (0,T].$$

A key advantage of (3) is its ability to accommodate negative interest rates, unlike the HCIR model where the rate of interest must always remain positive, as noted by [11]. To price under the HHW PDE, it is essential to first truncate the domain in the following format:

$$(x,p,o,\tau )\in [0,x_{max}]\times [0,p_{max}]\times [-o_{max},o_{max}]\times (0,T],$$

(35)

where $x_{max},p_{max},o_{max}$ are positive scalars.

To continue, let us assume ${\left\{x_i\right\}}_{i=1}^{n_1}$ represents a partition for $x\in [x_{min},x_{max}]$. A widely recognized and effective graded mesh can then be expressed for $i=1,2,\dots ,n_1$ by [31]:

$$x_i=\Psi \left({\lambda }_i\right),$$

(36)

where $n_1\gg 3$ and

$${\lambda }_{max}={\lambda }_{n_1}>\cdots >{\lambda }_2>{\lambda }_1={\lambda }_{min},$$

(37)

are $n_1$ uniform knots, and it is defined that

$$\begin{array}{cc}\hfill & {\lambda }_{min}={sinh}^{-1}\left({\displaystyle \frac{x_{min}-x_{\mathrm{left}}}{d_1}}\right),\hfill \\ \hfill & {\lambda }_{\mathrm{int}}={\displaystyle \frac{x_{\mathrm{right}}-x_{\mathrm{left}}}{d_1}},\hfill \\ \hfill & {\lambda }_{max}={\lambda }_{\mathrm{int}}+{sinh}^{-1}\left({\displaystyle \frac{x_{max}-x_{\mathrm{right}}}{d_1}}\right).\hfill \end{array}$$

(38)

We also consider $x_{min}=0$ and $x_{max}=14E$. The parameter $d_1>0$ determines the node density around $x=E$. Furthermore, the following relation is defined:

$$\Psi \left(\lambda \right)=\left\{\begin{array}{cc}{x}_{\mathrm{left}}+d_{1}sinh\left(\lambda \right),\hfill & {\lambda }_{min}\le \lambda <0,\hfill \\ x_{\mathrm{left}}+d_1\lambda ,\hfill & 0\le \lambda \le {\lambda }_{\mathrm{int}},\hfill \\ x_{\mathrm{right}}+d_{1}sinh(\lambda -{\lambda }_{\mathrm{int}}),\hfill & {\lambda }_{\mathrm{int}}<\lambda \le {\lambda }_{max}.\hfill \end{array}\right.$$

(39)

Here, $d_1={\displaystyle \frac{E}{20}}$ is a suitable selection in (38), while $x_{\mathrm{left}}=max\{e^{-0.0025T},$$0.5\}\times E$, $x_{\mathrm{right}}=E$, and $[x_{\mathrm{left}},x_{\mathrm{right}}]\subset [0,x_{max}]$.

After having the set of discretization non-uniform knots for the variable x, now we produce in a similar format for the the independent variable p, that is ${\left\{p_j\right\}}_{j=1}^{n_2}$, as follows [31]:

$$p_j=sinh\left({\varsigma }_j\right)d_2,j=1,2,\dots ,n_2,$$

(40)

where $d_2>0$ regulates the density around $p=0$. In this study, we used $d_2={\displaystyle \frac{p_{max}}{500}}$, with $p_{max}=10$. Additionally, for $1\le j\le n_2$, ${\varsigma }_j$ represents equidistant points provided by

$$\Delta \varsigma ={sinh}^{-1}\left({\displaystyle \frac{p_{max}}{d_2}}\right)(1/(n_2-1)),{\varsigma }_j=(j-1)\Delta \varsigma .$$

(41)

Lastly, the non-uniform knots for the dimension o are defined as

$$o_k=d_{3}sinh\left({\zeta }_k\right),1\le k\le n_3,$$

(42)

where $o_{max}=1$, $\Delta \zeta ={\displaystyle \frac{1}{n_3-1}}{sinh}^{-1}\left({\displaystyle \frac{o_{max}}{d_3}}\right)$, ${\zeta }_k=(\Delta \zeta )(k-1)$, and $d_3={\displaystyle \frac{o_{max}}{500}}$ is a scalar; see also [32].

4. Discretization of the HHW PDE (3)

To develop our scheme in an algorithmic way, we initially employ the semi-discretization for the time-dependent problem (3) [33,34], wherein all space variables are discretized to derive a set of ordinary differential equations (ODEs), which is linear. Consequently, some matrices of differentiation (DMs) should be filled that incorporate the coefficients for the RBF-FD processes on non-uniform stencils, as in Section 2, in the following way

$$H_x=\left\{\begin{array}{ccc}{\delta }_{i,j}\mathrm{via}\left(16\right)\hfill & \hfill & i-j=1,\hfill \\ {\delta }_{i,j}\mathrm{via}\left(17\right)\hfill & \hfill & i-j=0,\hfill \\ {\delta }_{i,j}\mathrm{via}\left(18\right)\hfill & \hfill & j-i=1,\hfill \\ 0\hfill & \hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(43)

and

$$H_{xx}=\left\{\begin{array}{ccc}{\pi }_{i,j}\mathrm{via}\left(28\right)\hfill & \hfill & i-j=1,\hfill \\ {\pi }_{i,j}\mathrm{via}\left(29\right)\hfill & \hfill & i-j=0,\hfill \\ {\pi }_{i,j}\mathrm{via}\left(30\right)\hfill & \hfill & j-i=1,\hfill \\ 0\hfill & \hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(44)

To discuss further, we computed the weights using three nodes from the stencil. For each iteration, three interior stencil nodes are considered, and their respective weights are calculated. These calculated weights, along with those associated to the first and last nodes, are assembled into the DM. The derived coefficients only vary if the three adjacent stencil nodes or their spacings (h or w) are altered.

For boundary points, we can only construct two-knot sided approximants of the RBF-FD. These approximations are similarly employed to populate the DMs in the first and last rows, preserving the triangular structure of $H_x$ and $H_{xx}$.

For simplicity, and given that the function $b(T-\tau )$ exhibits no fluctuations, we approximate it as a scalar by expanding it to zeroth order, i.e., $b(T-\tau )\simeq \beta$. This approach eliminates the time-dependent coefficient function, simplifying our numerical solver by avoiding a time-varying matrix. If ⊗ stands for the Kronecker product and $I=I_x\otimes I_p\otimes I_o$ is an identity matrix of the size $N_1\times N_1$, where $N_1=n_1\times n_2\times n_3$, $I_x$ is the $n_1\times n_1$ identity matrix for x, and similarly for $I_p$ and $I_o$. This allows us to reformulate the entire scheme in the following matrix format:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill G=& {\displaystyle \frac{1}{2}}{\mathcal{A}}^2\mathcal{Q}(H_{xx}\otimes I_p\otimes I_o)+{\displaystyle \frac{1}{2}}{\sigma }_1^2\mathcal{Q}(I_x\otimes H_{pp}\otimes I_o)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\sigma }_2^2(I_x\otimes I_p\otimes H_{oo})+{\rho }_{12}{\sigma }_1\mathcal{A}\mathcal{Q}(H_x\otimes H_p\otimes I_o)\hfill \\ \hfill & +{\rho }_{13}{\sigma }_2\mathcal{A}{\left(\mathcal{Q}\right)}^{{\displaystyle \frac{1}{2}}}(H_x\otimes I_p\otimes H_o)+{\rho }_{23}{\sigma }_1{\sigma }_2{\left(\mathcal{Q}\right)}^{{\displaystyle \frac{1}{2}}}(I_x\otimes H_p\otimes H_o)\hfill \\ \hfill & +\mathcal{C}\mathcal{A}(H_x\otimes I_p\otimes I_o)+\upsilon (aI-\mathcal{Q})(I_x\otimes H_p\otimes I_o)\hfill \\ \hfill & +a(\beta I-\mathcal{C})(I_x\otimes I_p\otimes H_o)-oI.\hfill \end{array}\end{array}$$

(45)

The matrices $H_p$, $H_o$, $H_{pp}$, and $H_{oo}$ in (45) have been filled via a similar strategy as in (43) and (44); see also the discussions in [11]. Additionally, the sparse diagonal matrices $\mathcal{A}$, $\mathcal{Q}$ and $\mathcal{C}$ are filled in through the following relations:

$$\mathcal{A}=\mathrm{diag}\left(x_1,x_2,\cdots ,x_{n_1}\right)\otimes I_p\otimes I_o.$$

(46)

$$\mathcal{Q}=I_x\otimes \mathrm{diag}\left(p_1,p_2,\cdots ,p_{n_2}\right)\otimes I_o,$$

(47)

$$\mathcal{C}=I_x\otimes I_p\otimes \mathrm{diag}\left(o_1,o_2,\cdots ,o_{n_3}\right),$$

(48)

By incorporating all the weights into appropriate matrices, the PDE can be transformed into a set of ODEs using the method of lines methodology in order to resolve (3) by

$$v^{\prime }\left(\tau \right)=Gv\left(\tau \right),$$

(49)

where $v\left(\tau \right)=\underset{N_1\mathrm{entries}}{\underbrace{{\left(v_{1,1,1}\left(\tau \right),v_{1,1,2}\left(\tau \right),\dots ,v_{n_1,n_2,n_3-1}\left(\tau \right),v_{n_1,n_2,n_3}\left(\tau \right)\right)}^{*}}}.$

After considering the boundary conditions (5), the system of ODEs (49) is derived in the following format:

$$\begin{array}{c}{v}^{\prime }\left(\tau \right)=\overline{G}v\left(\tau \right)=Y(\tau ,v\left(\tau \right)),\hfill \end{array}$$

(50)

where the matrix $\overline{G}$ includes the conditions on the boundaries.

The system (50) can be addressed using various time-stepping solvers [35]. For this purpose, consider selecting $\vartheta +1$ evenly spaced time knots with the size $\xi ={\displaystyle \frac{T}{\vartheta }}>0$. The time points are defined as ${\tau }_{\iota +1}=\xi +{\tau }_{\iota }$ for $0\le \iota \le \vartheta$, starting with ${\mathbf{v}}^0=v\left(0\right)$. Given that ${\mathbf{v}}^{\iota }$ approximates $\mathbf{v}\left({\tau }_{\iota }\right)$, we can obtain our final time-integration scheme. The four-stage Runge–Kutta (RK) explicit method, as detailed in [36,37], is provided by

$${\mathbf{v}}^{\iota +1}={\mathbf{v}}^{\iota }+{\displaystyle \frac{\xi }{6}}\left(P_1+2P_2+2P_3+P_4\right),$$

(51)

and

$$\begin{array}{cc}\hfill P_1& =Y\left({\tau }_{\iota },{\mathbf{v}}^{\iota }\right),\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill P_2& =Y\left({\tau }_{\iota }+{\displaystyle \frac{\xi }{2}},{\mathbf{v}}^{\iota }+{\displaystyle \frac{\xi }{2}}{P}_1\right),\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill P_3& =Y\left({\tau }_{\iota }+{\displaystyle \frac{\xi }{2}},{\mathbf{v}}^{\iota }+{\displaystyle \frac{\xi }{2}}{P}_2\right),\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill P_4& =Y\left({\tau }_{\iota }+\xi ,{\mathbf{v}}^{\iota }+\xi P_3\right).\hfill \end{array}$$

(55)

Since the time-stepping solver is similar to what we have previously employed in the work [11], we do not employ a rigorous stability analysis for this time marcher. In fact, the stability bound discussed in [11] can similarly be obtained but using the final system matrix $\overline{G}$ in (50).

5. Financial Numerical Observations

This section aims to evaluate the performance of different solvers for addressing (3) within the same numerical domain, given the parameters $p_0=0.04$, $E=100\$$, $T=1$, and $o_0=10\%$. The methods under comparison are outlined below:

• The standard FD scheme of the 2nd spatial discretization order with equidistant points and the explicitly linearly convergent Euler’s solver, which is denoted as FD2 [11,13].
• The method introduced in [31], applied on similar non-uniform stencils and referred to as HM2, which is short for Haentjens’s method. HM2 is selected due to its fundamental importance and high efficiency in tackling (3).
• The contributed effective scheme based on in Section 2, Section 3 and Section 4, which is known as IRBF2.

In this work, we describe the process of selecting the stencil configuration and collocation points. First, we generate a sufficient number of points along each spatial variable using the non-uniform generators described in Section 3. Specifically, we generate $n_1$, $n_2$, and $n_3$ points in one dimension. For each set of one-dimensional discretization points, we use three-point stencils, except at the boundaries. This means that for each set of discretization knots, from the second point to the penultimate point, we calculate the weights and insert them into the corresponding rows of the DM. Subsequently, as detailed in Section 4, we construct the DMs, which ultimately form the final system presented in (50).

Here, the methods are implemented in Mathematica 13.3 [38,39]. The elapsed computational times are measured in seconds, and the relative absolute error is calculated via

$$\mathrm{Err}=\left|{\displaystyle \frac{u_{\mathrm{ref}}-u_{\mathrm{num}}}{u_{\mathrm{ref}}}}\right|,$$

(56)

wherein $u_{\mathrm{ref}}$ [13] and $u_{\mathrm{num}}$ are the referenced and numerical solutions, respectively.

It is again noted that $b(·)$ can be given in the following format:

$$b(T-\tau )=c_1-c_{2}exp(-c_3\tau )\simeq \beta ,\tau \ge 0,$$

(57)

where $c_1$, $c_2$, $c_3$ are constants.

Example 1

([13]). We pursue an option pricing using the following conditions: $a=0.20$, $\eta =0.12$, $\upsilon =3.0$, ${\sigma }_1=0.80$, ${\sigma }_2=0.03$, ${\rho }_{23}=0.4$, ${\rho }_{13}=0.2$, ${\rho }_{12}=0.6$, $c_1=0.05$, $c_2=0$, $c_3=0$, where the reference value is $u_{\mathrm{ref}}(E,p_0,o_0,T)\simeq 16.176$.

Example 2

([13]). In this test, we use a different set of parameters to compare the performance of various schemes: $a=0.16$, $\upsilon =0.5$, ${\sigma }_1=0.90$, ${\sigma }_2=0.03$, ${\rho }_{23}=0.1$, ${\rho }_{13}=0.2$, ${\rho }_{12}=-0.5$, $\eta =0.8$, $c_1=0.055$, $c_2=0$, $c_3=0$, with the reference value being $u_{\mathrm{ref}}(E,p_0,o_0,T)\simeq 20.994$.

Due to the fact that the truncated domain along x is larger compared to the other two space variables, it is beneficial to use more discretization points, especially on non-uniform stencils along x.

The numerical results are compiled in

Table 1. Convergence histories for various schemes under Example 1.

Solver	${\mathit{n}}_1$	${\mathit{n}}_2$	${\mathit{n}}_3$	${\mathit{N}}_1$	$\mathit{\xi }$	${\mathit{u}}_{\mathbf{num}}$	Er	Time
FD2
	10	8	6	480	0.002	25.492	5.7E-1	0.38
	14	10	10	1400	0.001	11.098	3.1E-1	0.65
	18	12	12	2592	0.0005	17.203	6.3E-2	1.02
	24	14	14	4704	0.00025	18.731	1.5E-1	2.11
	28	16	16	7168	0.0002	13.329	1.7E-1	6.14
	45	22	22	21,780	0.00005	14.636	9.4E-2	69.25
HM2
	10	8	6	480	0.001	14.472	1.0E-1	0.50
	14	10	10	1400	0.0005	15.300	5.3E-2	0.89
	18	12	12	2592	0.00025	15.615	3.4E-2	2.03
	24	14	14	4704	0.0001	15.806	2.2E-2	8.31
	28	16	16	7168	0.0001	15.871	1.8E-2	11.06
	50	22	22	24,200	0.000025	16.006	5.9E-3	185.43
IRBF2
	10	8	6	480	0.004	15.234	5.8E-2	0.61
	14	10	10	1400	0.0025	15.869	1.8E-2	1.03
	18	12	12	2592	0.002	16.027	9.2E-3	2069
	24	14	14	4704	0.0005	16.287	6.8E-3	6.27
	28	16	16	7168	0.0004	16.242	4.0E-3	10.81
	50	22	22	24,200	0.0002	16.211	2.1E-3	118.20

and

Table 2. Convergence histories for various schemes under Example 2.

Solver	${\mathit{n}}_1$	${\mathit{n}}_2$	${\mathit{n}}_3$	${\mathit{N}}_1$	$\mathit{\xi }$	${\mathit{u}}_{\mathbf{num}}$	Er	Time
FD2
	20	10	10	2000	0.00025	22.022	4.9E-2	1.04
	24	12	12	3456	0.0002	21.436	2.1E-2	3.46
	26	14	14	5096	0.0001	19.678	6.1E-2	8.81
	28	16	16	7168	0.0001	17.376	1.7E-1	11.05
	30	18	18	9720	0.00005	17.404	1.7E-1	34.79
	36	20	20	14,400	0.000025	20.510	2.2E-2	104.32
	38	22	22	18,392	0.000025	20.275	3.3E-2	160.13
	42	22	22	20,328	0.00002	18.370	1.2E-1	225.78
HM2
	20	10	10	2000	0.00025	20.631	1.6E-2	1.12
	24	12	12	3456	0.0002	20.709	1.2E-2	3.58
	26	14	14	5096	0.0001	20.729	1.1E-2	9.02
	28	16	16	7168	0.0001	20.748	1.0E-2	12.22
	30	18	18	9720	0.00005	20.767	9.9E-3	36.55
	36	20	20	14,400	0.000025	20.810	7.8E-3	107.24
	38	22	22	18,392	0.000025	20.818	7.5E-3	166.42
	42	22	22	20,328	0.00002	20.833	6.7E-3	238.67
IRBF2
	20	10	10	2000	0.004	19.765	5.8E-2	2.02
	24	12	12	3456	0.002	19.920	5.1E-2	4.01
	26	14	14	5096	0.001	20.258	3.5E-2	7.22
	28	16	16	7168	0.0005	20.645	1.6E-2	13.25
	30	18	18	9720	0.0004	20.786	9.9E-3	36.81
	36	20	20	14,400	0.0002	20.881	5.3E-3	101.28
	38	22	22	18,392	0.0001	20.921	3.4E-3	162.40
	42	22	22	20,328	0.00005	20.958	1.7E-3	236.07

. These tables demonstrate that boosting the number of discretization knots enhances accuracy for all the schemes. However, the graded meshes and efficient structure of the proposed solver facilitate achieving higher accuracies more quickly. The computation time for determining the coefficients of the RBF-FD method is included. The reported times represent the total computational duration from initial input consideration to final interpolation and results output. The shape parameter is always chosen to be three times higher than the maximum spacing for each set of nodes in each spatial direction.

One might question why the equal number of space and temporal steps is not used in various schemes in Table 1 and Table 2. The rationale is that we employ more nodes along x and p due to their larger computational domains—larger domains necessitate more nodes. Additionally, smaller time sizes are employed for FD2 and HM2 because their time-stepping solvers have lower stability regions. The scheme are compared via their elapsed times as well as evaluating (56). In Table 1 and Table 2, $N_1=n_1\times n_2\times n_3$ are the total numbers of points we have taken.

The simulation results demonstrate that the proposed method, despite requiring a bit longer computation time, achieves higher accuracy. This improved accuracy, evidenced by lower relative absolute errors, is primarily due to the implementation of the IRBF2 solver with a good $ϵ$. The reduced computational time is majorally attributed to the use of smaller temporal step sizes, which are made possible by the larger stability regions of the (51) method. Based on the computational observations presented here, we state that our fast scheme (IRBF2) opens new avenues for effective option pricing.

6. Concluding Remarks

Sometimes, there is a sudden shift in monetary policy that leads to an unexpected increase in short-term interest rates. This could affect tech companies’ borrowing costs and, consequently, their stock prices. The HHW model’s incorporation of interest rate dynamics enables more accurate pricing by accounting for these impacts on option values.

In this paper, we introduce an effective numerical scheme utilizing the local RBF-FD. The novelty of our work lies in the derivation of coefficients from the integrals of the MQ function, which are employed to approximate function derivatives. We provide theoretical convergence rates for these coefficients, underscoring their robustness and accuracy. Our proposed scheme leverages RBF-FD approximations on three-point non-uniform stencils to construct a final approximation on a tensor grid specifically tailored for the 3D HHW PDE, which is a model frequently encountered in mathematical finance.

The numerical evidence and comparative analysis presented herein validate the efficacy and reliability of our method. The primary contribution of this research is the development of a novel numerical solver for simulating the 3D PDE with variable coefficients and mixed derivatives, leading to the formulation of sparse block matrices. This is particularly significant given the complexities inherent in solving such high-dimensional problems. Our approach ensures that the RBF-FD approximations are applicable on non-uniform meshes, allowing for refined accuracy in regions of high significance.

In conclusion, the results demonstrate that our RBF-FD based scheme offers a powerful and flexible tool for addressing complex PDEs in mathematical finance, paving the way for further advancements in numerical methods for high-dimensional problems. Future work will focus on extending this methodology to other types of PDEs and exploring its potential applications in various fields. Additionally, forthcoming research could involve constructing higher-order approximations for the 2nd differentiation using more knots in the stencil, which may improve efficiency when accompanied by a Krylov-type solver [40] as a one-step method for time stepping.