In this paper, we construct and analyze a Crank-Nicolson fitted finite volume scheme for pricing European options under regime-switching Kou's jump diffusion model which is governed by a system of partial integro-differential equations (PIDEs). We show that this scheme is consistent, stable and monotone as the mesh sizes in space and time approach zero, hence it ensures the convergence to the solution of continuous problem. Finally, numerical experiments are performed to demonstrate the efficiency, accuracy and robustness of the proposed method.

[Purposes]As for the valuation of American options under regime-switching Kou’s jump-diffusion processes. [Methods]Based on the linear complementarity problem discretized by Crank Nicolson fitted finite volume method, an efficient modulus matrix splitting iterative method is introduced to solve it. [Findings]The convergence theorem of the algorithms are given under the H+-matrix property of the discrete matrix. [Conclusions]Numerical experiments verify the effectiveness, robustness and convergence of the new method, and the computational efficiency of themodulus-based matrix splitting iteration methods is superior to the projected successive overrelaxation iteration method.

In this paper we introduce a new numerical method for the linear complementarity problems (LCPs) arising from two-asset Black–Scholes and Heston's stochastic volatility American options pricing. Based on barycenter dual mesh, a class of finite volume method (FVM) is proposed for the spatial discretization, coupled with the backward Euler and Crank–Nicolson schemes are employed for time stepping of the partial differential equations (PDEs). Then, for the resulting time-dependent LCPs are solved by using an efficient modulus-based successive overrelaxation (MSOR) iteration method. Numerical experiments are carried out to verify the efficiency and usefulness of the proposed method.

本文采用VaR模型的三种常用方法参数法、历史模拟法、蒙特卡洛模拟法分别对投资组合的风险价值进行计算.通过选取2018年10月8日到2019年3月1日的二十只股票数据,以此期间的股票基金指数成分股作为投资组合,利用MATLAB编程计算,估算出在一定持有期和给定置信度下的VaR值.数值结果表明,三种方法效果差别不大,参数法较好.

首先,考虑一种求解美式Kou型跳扩散期权模型的Crank-Nicolson拟合有限体积方法,并给出收敛性分析;其次,针对非线性代数系统设计一个迭代算法,并证明其收敛性;最后,用数值实验验证了新方法的收敛性、稳健性和有效性.

主要研究了一类状态转换下美式跳扩散期权定价模型的修正Crank-Nicolson拟合有限体积法并且给出收敛性分析.文章所构造的新方法是对[Gan X T,Yin J F,Li R,Fitted finite volume method for pricing American options under regime-switching jump-diffusion models based on penalty method.Adv.Appl.Math.Mech., 2020, 12(3): 748-773]中时间方向上Crank-Nicolson格式的改进.同时,还对求解非线性系统迭代方法的收敛性证明进行了补充.最后,数值实验验证了新方法的有效性.

本文主要研究状态转换下欧式Merton跳扩散期权定价模型的拟合有限体积方法.针对该定价模型中的偏积分-微分方程,空间方向采用拟合有限体积方法离散,时间方向构造Crank-Nicolson格式.理论证明了数值格式的一致性、稳定性和单调性,因此收敛至原连续问题的解.数值实验验证了新方法的稳健性,有效性和收敛性.

Unlike the European options pricing,the closed-form solution generally does not exist due to the early exercise feature of American options.Hence,numerical ap-proximation methods are normally employed to solve them.Presented in this paper is a new numerical method to price American bond options.To numerically solve the resulting partial differential complementarity problem(PDCP),we develop a class of finite volume method for the spatial discretization,coupled with the stable fully implicit time stepping scheme of the partial differential equation(PDE).Then,the resulting linear complementarity problems(LCPs)are solved by using an effi-cient iterative method,the modulus-based matrix splitting iteration method,where the H+-matrix property of the system matrix guarantees its convergence.Numer-ical experiments are implemented to verify the accuracy,efficiency and robustness of the new method.

This paper develops and analyses a novel numerical scheme to price European options under regime switching model which is governed by a system of partial differential equations (PDEs).To numerically solve these PDEs,we introduce a fitted finite volume method for the spatial discretization,coupled with the Crank-Nicolson time stepping scheme.We show that this scheme is consistent,stable and monotone,and hence the convergence of the numerical solution to the viscosity solution of the continuous problem is guaranteed.Numerical experiments are presented to demonstrate the accuracy,efficiency and robustness of the new numerical method.

A new numerical method for pricing American options under regime-switching model is developed. The original problem is first approximated by a set of nonlinear partial differential equations. After that a novel fitted finite volume method for the spatial discretisation of the nonlinear penalised system of partial differential equations is coupled with the Crank-Nicolson time stepping scheme. It is shown that the discretisation scheme is consistent, stable, monotone and hence convergent. In order to solve nonlinear algebraic systems, we apply an iterative algorithm and show its convergence. Numerical experiments demonstrate the convergence, efficiency and robustness of the numerical method.

This paper develops and analyses a Crank–Nicolson fitted finite volume method to price American options on a zero-coupon bond under the Cox–Ingersoll–Ross (CIR) model governed by a partial differential complementarity problem (PDCP). Based on a penalty approach, the PDCP results in a nonlinear partial differential equation (PDE). We then apply a fitted finite volume method for the spatial discretization along with a Crank–Nicolson time-stepping scheme for the PDE, which results in a nonlinear algebraic equation. We show that this scheme is consistent, stable, and monotone, and hence, the convergence of the numerical solution to the viscosity solution of the continuous problem is guaranteed. To solve the system of nonlinear equations effectively, an iterative algorithm is established and its convergence is proved. Numerical experiments are presented to demonstrate the accuracy, efficiency, and robustness of the new numerical method.

In this paper we develop a novel numerical method for pricing American options under regime-switching jump-diffusion models which are governed by a system of partial integro-differential complementarity problems (PIDCPs). Based on a penalty approach, the PIDCPs results in a set of coupled nonlinear partial integro-differential equations (PIDEs). To numerically solve these nonlinear penalized PIDEs, we introduce a fitted finite volume method for the spatial discretization, coupled with the backward Euler and Crank-Nicolson time stepping schemes. We show that these schemes are consistent, stable and monotone, hence it ensures the convergence to the solution of continuous problem. To solve the discretized nonlinear system effectively, an iterative method is designed. Numerical experiments are presented to demonstrate the accuracy, efficiency and robustness of the new numerical method.

We develop a novel numerical method for pricing American options under Kou’s jump-diffusion model which governed by a partial integro-differential complementarity problem (PIDCP). By using a penalty approach, the PIDCP results in a nonlinear partial integro-differential equation (PIDE). To numerically solve this nonlinear penalized PIDE, a fitted finite volume method is introduced for the spatial discretization and the Backward Euler and Crank–Nicolson schemes for the time discretization. We show that these schemes are consistent, stable and monotone, hence convergence to the solution of continuous problem. Numerical experiments are performed to verify the effectiveness of this new method.

考虑有限体积法定价欧式的Merton型跳扩散期权模型.基于线性有限元空间,构造了向后Euler和Crank-Nicolson两种全离散有限体积格式,且离散矩阵均为M-矩阵.针对方程中的积分项,采用一类高效的线性插值技术进行逼近.数值实验验证了本文方法的有效性和稳健性.

In view of the numerical solution of option pricing model under uncertain volatility , w e constructed a fully implicit finite volume scheme of the nonlinear HJB (Hamilton‐Jacobi‐Bellman ) equation ,and proved the stability , existence and uniqueness of the scheme.The robustness and effectiveness of the proposed method were verified by numerical experiments .

考虑有限体积法求解Kou跳扩散期权定价模型.基于线性有限元空间,构造了向后Euler和Crank-Nicolson两种全离散有限体积格式,并结合简单高效的递推公式逼近方程中的积分项.理论分析表明所得的离散矩阵为M-矩阵.数值实验验证了方法的有效性.

针对美式债券期权定价模型的数值解法,构造全隐式的有限差分格式,并给出格式的稳定性证明.采用模系矩阵分裂迭代法求解离散得到的线性互补问题,并与投影超松弛迭代法进行比较.数值实验验证了新方法的有效性和稳健性.

针对低比特JPEG图像因量化过程中产生的量化噪声问题,提出一种核范数JPEG解码算法.首先基于自然图像的低秩性得到一个带无穷范数约束问题的低秩矩阵恢复模型,其次将约束凸优化问题转换为无约束优化问题,降低其计算难度.最后,利用经典的原对偶算法结合块匹配方法处理低秩矩阵模型,得到后处理JPEG解码图像.实验结果表明,该文算法比基于总变分后处理方法在去除量化噪声方面具有优越性.

In this paper,finite element analysis for a class of neutral delay parabolic equation is considered.Based on a linear finite element space,one semi-discrete and two full discrete finite element schemes are constructed for the neutral delay parabolic equation.Then,the L2 norm error estimates of the semi-discrete and two full discrete schemes are obtained by the projection operator.Numerical experiments proves the validity of the theoretical analysis.

An implicit double discretization method is developed for pricing European and American options under Merton's jump-diffusion model.Stability of the method is discussed.Numerical experiments show that the proposed method is effective and robust,and has advantages over the explicit scheme.