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

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.

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

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

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

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 techniques.Then the nonlinear fractional Ric-cati equation for the characteristic function of the asset log-price is derived.The existence,uniqueness and regularity of the solution to the nonlinear fractional Riccati equation are proved and the equation is solved by the Adams methods.Finally the Fourier-cosine methods are combined with the Adams methods to price the options.

有限期美式封顶股票抵押贷款的定价

本文研究有限期的美式封顶式股票抵押贷款的定价问题.股票抵押贷款是一种用股票作为抵押品的贷款产品,它们的定价问题是一种最优停时问题.带封顶的股票抵押贷款通过设定股票价格的上限,将'上限'功能纳入股票抵押贷款中,这样贷款人就可以避免由于股价上涨而造成巨大损失.本文利用随机分析方法推导出最优执行边界函数的积分方程,从而得到有限期的美式封顶式股票抵押贷款价格的解析公式.本文还进一步研究一类随机利率模型下美式封顶式股票抵押贷款的定价问题,通过数值算例分析最优执行边界的性质.

DEV模型和SAHARA效用下DC型养老金的最优投资策略

本文研究不完全金融市场中缴费确定型养老金的最优投资问题,并假设金融市场由一种无风险资产和一种价格过程服从动态方差弹性模型的风险资产组成.在对称渐近双曲绝对风险厌恶效用下,养老金参与者旨在最大化其终端时刻财富的期望效用.本文推导Hamilton-Jacobi-Bellman方程,给出验证结果和值函数的上界和下界,然后应用对偶控制Monte Carlo算法计算最优策略.数值算例验证了该方法的准确性.

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

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.

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

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.

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

This paper develops a fast Laplace transform method for solving the complex PDE system arising from Parisian and Parasian option pricing.The value functions of the options are governed by a system of partial differential equations(PDEs)of two and three dimensions.Applying 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 equations.The Laplace inversion is realized by the fast contour integral methods.Numerical 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.

ERROR ANALYSIS FOR A FAST NUMERICAL METHOD TO A BOUNDARY INTEGRAL EQUATION OF THE FIRST KIND

For two-dimensional boundary integral equations of the first kind with logarithmic kernels, the use of the conventional boundary element methods gives linear systems with dense matrix. In a recent work [J. Comput. Math., 22 （2004）, pp. 287-298], it is demonstrated that the dense matrix can be replaced by a sparse one if appropriate graded meshes are used in the quadrature rules. The numerical experiments also indicate that the proposed numerical methods require less computational time than the conventional ones while the formal rate of convergence can be preserved. The purpose of this work is to establish a stability and convergence theory for this fast numerical method. The stability analysis depends on a decomposition of the coefficient matrix for the collocation equation. The formal orders of convergence observed in the numerical experiments are proved rigorously.

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

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 two.Moving finite element methods are proposed to solve the system of fractional diffusion equations and the convergence rates of the methods are proved.Numerical examples are carried out to confirm the theoretical findings.Some applications in anomalous diffusive Lotka-Volterra and Michaelis-Menten-Holling predator-preymodels are studied.