We design and analyze a second-order implicit-explicit (IMEX) finite-difference scheme for the liquidity-switching indifference-pricing model. In this model, the market alternates between a liquid regime, where continuous trading is available, and an illiquid regime, where portfolio rebalancing is suspended. The buyer’s exponential-utility indifference price is characterized by a coupled semilinear parabolic-ODE system with nonlinear exponential coupling. The proposed scheme treats the stiff diffusion term implicitly by the second-order backward differentiation formula (BDF2) and the nonlinear coupling explicitly by second-order Adams-Bashforth (AB2) extrapolation. Therefore, each time step requires only one tridiagonal solve for the parabolic component, followed by a pointwise update for the ODE component. For American options, the early-exercise constraint is enforced through an Ikonen-Toivanen operator-splitting projection, which avoids any nonlinear complementarity solve at each time level. A complete stability and convergence analysis is presented. The BDF2 coefficient matrix is an irreducible M-matrix, which ensures inverse positivity of the implicit diffusion solve; for the American option, nonnegativity is enforced by the obstacle projection. For European options, the scheme is conditionally L2 stable and converges with order O(k2+h2) under standard smoothness assumptions. For American options, second-order convergence holds in the continuation region under the stated regularity assumptions, while the projection enforces the obstacle constraint exactly at the discrete level. Numerical experiments for a European call and an American put confirm second-order convergence in both liquidity regimes, exhibit pronounced regime-dependent Gamma amplification near the strike and transition region, and demonstrate stable computation of the Greeks and parameter sensitivities.

This paper presents the stability and error analysis for the IMEX-BDF-OS techniques, which aim to solve the linear complementarity problem (LCP) for pricing American options in a Markovian regime-switching jump-diffusion (RSJD) economy. Multiple regime-switching models have been fitted with the OS schemes with favourable outcomes. The complementarity requirements and the differential equation are separated and studied independently, making it easier to apply to LCP. Nevertheless, no stability or error analysis is provided for the American option under the RSJD model by these operator-splitting procedures despite their popularity. Based on the IMEX-BDF1 and IMEX-BDF2 approaches, we estimated the error for these techniques and offered a priori stability estimates for operator splitting (OS) strategies. We numerically tested the operator splitting techniques and showed how IMEX-BDF1 and IMEX-BDF2 worked for the test issues, highlighting how well they converged.

Runge–Kutta (RK) methods are widely used techniques for solving a class of initial value problems. In this article, we introduce an adaptive multiquadratic (MQ) radial basis function (RBF)-based method to develop enhanced explicit RK methods. These methods achieve a higher order of convergence than the corresponding classical RK methods. To improve the local convergence of the numerical solution, we optimize the free parameters (shape functions) involved in the RBFs by forcing the local truncation errors to vanish. We also present a convergence and stability analysis of the proposed methods. To demonstrate the advantages of these methods in terms of accuracy and convergence, we consider several numerical examples and compare the performance of our methods with that of the classical RK methods. The Tables and Figures presented in this article clearly validate the superiority of the proposed methods.

This article delves into the construction of a new RBF-FD implicit-explicit difference scheme for solving a moving boundary partial integro-differential equation system governing regime-switching jump-diffusion for Asian option pricing. The RBF-FD scheme for spatial discretization is paired with the IMEX schemes for temporal discretization. The stability of the time semi-discretization scheme is also proved theoretically. Numerical examples are provided to illustrate the theoretical findings and highlight the efficacy of the proposed scheme in terms of convergence and accuracy.

The operator splitting method has been effectively applied to jump-diffusion models, and it is also easy to implement because the differential and complementarity restrictions are decoupled and solved separately. Despite their ubiquity, these operator-splitting approaches for jump-diffusion models have no stability and error analysis. In this direction, we performed a priori stability analysis for the implicit–explicit backward difference operator splitting techniques (IMEX-BDF-OS). After the stability analysis, we established the error estimates for IMEX-BDF1-OS and IMEX-BDF2-OS techniques. To validate the theoretical results, numerical evidence of the pricing of American options under Kou’s and Merton’s jump-diffusion models has been shown.

In this manuscript, we proposed the stability and error analysis for the backward difference operator splitting (BDF-OS) methods to solve the linear complementarity problem (LCP) for pricing the American option under the Black–Scholes framework. The OS schemes have been successfully applied to a variety of Black–Scholes models. It is easy to apply on LCP because the complementarity conditions and the differential equation are segregated and examined separately. We provided an error estimate for these methods and the priori stability estimates for operator splitting strategies based on the BDF1 and BDF2 approaches. We performed numerical experiments and illustrated the order and efficiency of the BDF1 and BDF2 approaches for the test problems to emphasize the convergence behavior of the proposed methods. We have also verified the numerical results with the existing methods in the literature.

In this work, we introduce two accurate and efficient finite difference methods based on radial basis functions (RBF-FD) for pricing European and American options under liquidity shocks. The problem is formulated as the semi-linear complementarity and PDE systems for American and European options, respectively. In the context of temporal semi-discretization, we provide two backward difference formulas of order one and two (BDF1 & BDF2). Furthermore, we discuss the stability and convergence properties of the proposed methods. For the American option, to solve the semi-linear system of complementarity problems (LCPs), we combine the RBF-FD approaches with an operator splitting (OS) method. To illustrate the efficiency and accuracy of the suggested methods, we provide numerical examples for both European and American call options and verify them with the existing work in the literature. In numerical discussion, we show the Greeks (Delta & Gamma) plots for the American options.

In this paper, a new meshless symplectic and multi-symplectic scheme is developed for solving the coupled nonlinear Schrödinger system. The key feature of the method is the utilization of local radial basis function (RBF) approximation (LRBFA) method in spatial discretization. The LRBFA only requires solving a small linear system, thus it can circumvent the ill-conditioned problem and shape-parameter-sensitivity of global RBF method. The proposed method offers a high order of convergence and conserves the invariants well. Various numerical examples are presented to verify that the proposed method is valid for both uniform knots and random knots.

In this manuscript, we introduced the radial basis function based three implicit-explicit (IMEX) finite difference techniques for pricing European and American options in an extended Markovian regime-switching jump-diffusion (RSJD) economy. A partial integrodifferential equation (PIDE) yields the values of the European option, which is one of the financial options, and a linear complementary problem (LCP) yields the prices of the American option. To solve the LCP for American option pricing, we combine the suggested techniques with the operator splitting methods. The suggested methods are designed to prevent the use of any fixed-point repetition approaches at each economic stage and time increment. We analyzed the stability of the proposed time discretization methods. We performed numerical experiments and illustrated the second-order convergence and efficiency of the three IMEX numerical techniques (BDF2, CNAB, CNLF) under the extended RSJD model.

In this manuscript, we presented some efficient and accurate radial basis function-based finite difference (RBF-FD) implicit–explicit (IMEX) numerical techniques for pricing the option when the underlying asset follows the jump-diffusion process with local volatility. For the time semi-discretization, we present three numerical techniques Crank–Nikolson Leap-Frog (CNLF), Crank–Nikolson AdamBashforth (CNAB), and Backward difference formula of order two (BDF2), incorporated with the radial basis function based finite difference (RBF-FD) method. The stabilities of time semi-discretized schemes are also proved. The computational methods developed for the European option are extended for the American option. We amalgamate the RBF-FD implicit–explicit methods with an operator splitting (OS) method for solving the linear complementarity problem (LCP) with variable parameters that determines the price of an American option. In order to demonstrate the effectiveness and precision of the current techniques, numerical data for European and American put options under the Merton and Kou models are presented.

The main goal of this manuscript is to develop an RBF-based meshfree method to solve the multi-term time-fractional nonlinear two-dimensional diffusion-wave equation numerically. We discussed the present scheme’s stability analysis and theoretically proved that the scheme is convergent. Time fractional derivatives are defined in Caputo’s sense. Numerical examples on the regular and irregular domains with uniform and non-uniform points are given to validate the ability and accuracy of the developed scheme. The present results show that the proposed method is efficient and reliable for modeling and simulating the considered problems.

The paper aims to put forth a radial basis function-based meshless approach for the numerical solution of the time-fractional nonlinear mixed diffusion and diffusion-wave equation. The time-fractional derivative is defined in Caputo's sense and discretized by the finite difference method. The spatial discretization is done using a radial basis function-based local meshless method. Stability of time semi-discretization is rigorously set up. Proposed method's efficiency is validated with different numerical examples on an irregular domain with uniform and nonuniform points. Numerical results obtained demonstrate the ability and accuracy of the present method.

In the current work, a radial basis function based local meshless method is taken into consideration to solve the multi-term time fractional nonlinear diffusion equation. We mentioned the proof of unconditional stability and also theoretically discussed the convergence of the proposed numerical scheme. Some numerical problems are given to show the exactness and efficiency of the developed scheme. The present result indicates that the proposed numerical scheme is very accurate and efficient for modeling and simulating the considered problems.

In the present work, we investigate the numerical solution of time-fractional telegraph equation by a local meshless method. The fractional-order derivative is defined in the Caputo's sense. The time semi-discretization was carried out using finite difference method followed by radial basis function-based spatial discretization. The theoretical convergence analysis and stability analysis of time semi-discrete scheme are also proved. Several test problems with regular and irregular domains with uniform and non-uniform points are considered. To demonstrate the accuracy and efficiency of the proposed method, we compared the analytical and numerical solution of the proposed problem.

In the present manuscript, we present an RBF based meshless method to investigate the time-fractional Tricomi-type equation, which has been arising in transonic flow. The unconditional stability of the proposed numerical scheme is discussed and theoretically proved. The time semi discretization has been done by using the finite difference method and for space discretization, we proposed an RBF based local collocation method. Some test problems are considered for regular as well as an irregular domain with uniform and non-uniform points to show the feasibility and efficiency of the proposed method.

In this paper, we have developed an radial basis function (RBF) based meshless method to solve the time-fractional mixed diffusion and diffusion-wave equation which involves two fractional Caputo derivatives of order $$\alpha \in (0,1)$$ and $$\beta \in (1,2)$$ . The unconditional stability of the proposed numerical scheme is discussed and proved theoretically. The time semi discretization has been done by using the finite difference method and for space discretization, we proposed an RBF-based local collocation method. Some test problems are considered for regular as well as an irregular domain with uniform and non-uniform points to validate the efficiency and accuracy of the method.

We present a radial basis function-based local collocation method for solving time fractional nonlinear diffusion wave equation.The main beauty of the local collocation method is that only the nodes located in the subdomain, surrounding the local collocation point, need to be considered when we are calculating the numerical solution at this point. We also prove the unconditional stability and convergence of the proposed scheme. Some numerical experiments are carried out and numerical results are compared with an analytical solution to confirm the efficiency and reliability of the proposed method.

In this article, we presented a method for option pricing problem under regime-switching jump-diffusion models. We have proposed a numerical method for solving a partial integro-differential equation (PIDE) for pricing European option and for solving linear complementarity problem (LCP), to evaluate the price of American options. We use implicit explicit method for time semi discretization, followed by radial basis function based finite difference (RBF-FD) method for spatial discretization to solve PIDE. The proposed method is further extended to solve the LCP by coupling it with operator splitting method. Numerical simulation is done for European and American option to demonstrate efficiency and accuracy of the proposed method.

In this manuscript, we present a radial basis function based local collocation method for solving time fractional diffusion-wave equation. The advantage of the local collocation method is only the discretization points located in each sub-domain, surrounding the collocation point, need to be considered. It also avoids the ill-conditioning problem arises due to the use of global collocation method. The theoretical stability and convergence of the proposed time semi-discrete scheme are also discussed. Numerical experiments for one dimension and two dimension problems are carried out. It is shown that the present method is efficient and numerical results have nice agreements with theoretical result.

The aim of this article is to develop and analyze a finite element method, combined with implicit-explicit (IMEX) time semidiscretizations, for pricing American options under Merton's and Kou's jump-diffusion models. Under realistic regularity assumptions on the data, some error estimates are established. The theoretical findings and the efficiency of the proposed methods are demonstrated by several numerical experiments.