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Fast Laplace transform methods for the PDE system of Parisian and Parasian option pricing

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Science China mathematics 2022年第6期65卷 1229-1246页

作者： jingtang ma Zhiqiang Zhou School of Economic mathematics Southwestern University of Finance and EconomicsChengdu 611130China School of mathematics and Finance Xiangnan UniversityChenzhou 423000China

This paper develops a fast Laplace transform method for solving the complex PDE system arising from Parisian and Parasian option *** value functions of the options are governed by a system of partial differential equations(PDEs)of two and three *** the Laplace transform to the PDEs with respect to the calendar time to maturity leads to a coupled system consisting of an ordinary differential equation(ODE)and a 2-dimensional partial differential equation(2d-PDE).The solution to this ODE is found analytically on a specific parabola contour that is used in the fast Laplace inversion,whereas the solution to the 2d-PDE is approximated by solving 1-dimensional integro-differential *** Laplace inversion is realized by the fast contour integral *** results confirm that the Laplace transform methods have the exponential convergence rates and are more efficient than the implicit finite difference methods,Monte Carlo methods and moving window methods.

关键词： Parisian option Parasian option coupled PDE Laplace transform method convergence rate

CONVERGENCE RATES OF MOVING MESH RANNACHER METHODS FOR PDES OF ASIAN OPTIONS PRICING

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Journal of Computational mathematics 2016年第3期34卷 240-261页

作者： jingtang ma Zhiqiang Zhou gchool of Economic mathematics Southwestern University of Finance and Economics Chengdu 611130 China

This paper studies the convergence rates of a moving mesh implicit finite difference method with interpolation for partial differential equations （PDEs） with moving boundary arising in Asian option pricing. The moving mesh scheme is based on Rnnacher timestepping approach whose idea is running the implicit Euler schemes in the initial few steps and continuing with Crank-Nicolson schemes. With graded meshes for time direction and moving meshes for space direction, the fully discretized scheme is constructed using quadratic interpolation between two consecutive time level for the PDEs with moving boundary. The second-order convergence rates in both time and space are proved and numerical examples are carried out to confirm the theoretical results.

关键词： Asian option pricing Moving mesh methods Crank-Nicolson schemes Ran-nacher time-stepping schemes Convergence analysis.

Lattice Boltzmann methods for solving partial differential equations of exotic option pricing

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Frontiers of mathematics in China 2016年第1期11卷 237-254页

作者： Zhiqiang ZHOU jingtang ma School of Economic mathematics Southwestern University of Finance and EconomicsChengdu 611130 China

This paper establishes a lattice Boltzmann method （LBM） with two amending functions for solving partial differential equations （PDEs） arising in Asian and lookback options pricing. The time evolution of stock prices can be regarded as the movement of randomizing particles in different directions, and the discrete scheme of LBM can be interpreted as the binomial models. With the Chapman-Enskog multi-scale expansion, the PDEs are recovered correctly from the continuous Boltzmann equation and the computational complexity is O（N）, where N is the number of space nodes. Compared to the traditional LBM, the coefficients of equilibrium distribution and amending functions are taken as polynomials instead of constants. The stability of LBM is studied via numerical examples and numerical comparisons show that the LBM is as accurate as the existing numerical methods for pricing the exotic options and takes much less CPU time.

关键词： Exotic option pricing lattice Boltzmann method Chapman-Enskogmulti-scale expansion stability computational complexity

ON A MOVING MESH METHOD FOR SOLVING PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS

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Journal of Computational mathematics 2009年第6期27卷 713-728页

作者： jingtang ma Yingjun Jiang Kaili Xiang School of Economic mathematics Southwestern University of Finance and Economics Chengdu 610074 China Department of mathematics and Scientific Computing Changsha University of Science and Technology Changsha 410076 China

This paper develops and analyzes a moving mesh finite difference method for solving partial integro-differential equations. First, the time-dependent mapping of the coordinate transformation is approximated by a a piecewise linear function in time. Then, piecewise quadratic polynomial in space and an efficient method to discretize the memory term of the equation is designed using the moving mesh approach. In each time slice, a simple piecewise constant approximation of the integrand is used, and thus a quadrature is constructed for the memory term. The central finite difference scheme for space and the backward Euler scheme for time are used. The paper proves that the accumulation of the quadrature error is uniformly bounded and that the convergence of the method is second order in space and first order in time. Numerical experiments are carried out to confirm the theoretical predictions.

关键词： Partial integro-differential equations Moving mesh methods Stability and convergence.

Dual Control Methods for a Mixed Control Problem with Optimal Stopping Arising in Optimal Consumption and Investment

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Numerical mathematics(Theory,Methods and Applications) 2022年第3期15卷 641-661页

作者： jingtang ma Jie Xing Shan Yang School of mathematics Southwestern University of Finance and EconomicsChengdu 611130China Guiyang Institute for Big Data and Finance Guizhou University of Finance and EconomicsGuiyang 550025China

This paper studies a problem of optimal investment and consumption with early retirement option under constant elasticity variation(CEV)model with finite *** risky assets are involved in the model with one following geometric Brownian motion and the other a CEV *** problem is a kind of two dimensional mixed control and optimal stopping problems with finite *** existence and continuity of the optimal retirement threshold surfaces are proved and the working and retirement regions are characterized ***-squares Monte-Carlo methods are developed to solve this mixed control and optimal stopping *** algorithms are well implemented and the optimal retirement threshold surfaces,optimal investment strategies and the optimal consumptions are drawn via examples.

关键词： Optimal investment and consumption stochastic control with optimal stopping nonlinear free boundary problems least-squares Monte-Carlo methods

Moving Finite Element Methods for a System of Semi-Linear Fractional Diffusion Equations

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Advances in Applied mathematics and Mechanics 2016年第6期8卷 911-931页

作者： jingtang ma Zhiqiang Zhou School of Economic mathematics Southwestern University of Finance and EconomicsChengdu 611130China

This paper studies a system of semi-linear fractional diffusion equations which arise in competitive predator-preymodels by replacing the second-order derivatives in the spatial variables with fractional derivatives of order less than *** finite element methods are proposed to solve the system of fractional diffusion equations and the convergence rates of the methods are *** examples are carried out to confirm the theoretical *** applications in anomalous diffusive Lotka-Volterra and Michaelis-Menten-Holling predator-preymodels are studied.

关键词： Finite element methods fractional differential equations predator-preymodels

Moving finite element methods for time fractional partial differential equations

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Science China mathematics 2013年第6期56卷 1287-1300页

作者： JIANG YingJun ma jingtang Department of mathematics and Scientific Computing Changsha University of Science and Technology School of Economic mathematics Southwestern University of Finance and Economics

With the aim of simulating the blow-up solutions, a moving finite element method, based on nonuni- form meshes both in time and in space, is proposed in this paper to solve time fractional partial differential equations （FPDEs）. The unconditional stability and convergence rates of 2 -a for time and r for space are proved when the method is used for the linear time FPDEs with a-th order time derivatives. Numerical exam-ples are provided to support the theoretical findings, and the blow-up solutions for the nonlinear FPDEs are simulated by the method.

关键词： fractional partial differential equations moving finite element methods blow-up solutions

Analysis of a moving collocation method for one-dimensional partial differential equations

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Science China mathematics 2012年第4期55卷 827-840页

作者： ma jingtang Huang WeiZhang Russell Robert D. School of Ecomomic mathematics Southwestern University of Finance and EconomicsChengdu 611130China Department of mathematics University of KansasLawrenceKS 66045USA Department of mathematics Simon Fraser UniversityBurnabyBC V5A 1S6Canada

A moving collocation method has been shown to be very effcient for the adaptive solution of second- and fourth-order time-dependent partial differential equations and forms the basis for the two robust codes MOVCOL and MOVCOL4. In this paper, the relations between the method and the traditional collocation and finite volume methods are investigated. It is shown that the moving collocation method inherits desirable properties of both methods： the ease of implementation and high-order convergence of the traditional collocation method and the mass conservation of the finite volume method. Convergence of the method in the maximum norm is proven for general linear two-point boundary value problems. Numerical results are given to demonstrate the convergence order of the method.

关键词： collocation method finite volume method Hermite basis function conservation convergence moving mesh

Rough Heston Models with Variable Vol-of-Vol and Option Pricing

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Annals of Applied mathematics 2023年第2期39卷 206-238页

作者： Hui Liang jingtang ma Zhengguang Shi School of Science Harbin Institute of TechnologyShenzhenGuangdong 518055China School of mathematics Southwestern University of Finance and EconomicsChengduSichuan 611130China School of mathematical Sciences Sichuan Normal UniversityChengduSichuan 610066China

In this paper,a rough Heston model with variable volatility of volatility(vol-of-vol)is derived by modifying the generalized nonlinear Hawkes process and extending the scaling *** the nonlinear fractional Ric-cati equation for the characteristic function of the asset log-price is *** existence,uniqueness and regularity of the solution to the nonlinear fractional Riccati equation are proved and the equation is solved by the Adams *** the Fourier-cosine methods are combined with the Adams methods to price the options.

关键词： Rough Heston model option pricing Hawkes process fractional differential equations Fourier-cosine methods