Study On Numerical Methods For Volterra Integral Equations With Pantograph Delays And The Pricing Of American Option Model

The integral equation can be used to solve the mathematics, natural science, engi-neering technology field and financial field, many practical problems etc, Then the integral equation is to transform in order to find a solution to solve practical problems. The op-timal exercise boundary B(t) of American options under the Block-Scholes model can be transformed into nonlinear Volterra integral equation of the second kind. In this paper, first, the convergence analysis of the Volterra integral equation of second kind with weakly singular kernel and pantograph delays is provided. Then we solve the American put op-tion under Block-Scholes model by a projection and contraction method, The final topic is devoted to the American put option pricing problem governed by the Black-Scholes equa-tion.Applying the PML technique and finite difference method coupled with Newton’s method to solve the Black-Scholes equation. There are three chapters in this paper.In the first Chapter, we make a detailed introduction of the history and current status of the Volterra integral equation and American option pricing. We carry on the induction and the summary to the existing numerical methods for solving the Volterra integral equation and the American option pricing.In the second Chapter, the convergence analysis of the Volterra integral equation of second kind with weakly singular kernel and pantograph delays is provided and numerical examples are presented to complement the theoretical convergence results.The main conclusions of this chapter are as follows:In this chapter, we consider the delay integral equations with a weakly singular kernelwhere 0<q<1,0<μ<1, y(t) is the unknown function, g,K are given functions and K(t, qt)≠0 for t ∈I.We use some function transformations and variable transformations to transform the equation into a new Volterra integral equation: wherewhere (x) is the standard Lagrange interpolation polynomial with (N+1) Jacobi Gauss, Jacobi Gauss-Radau, or Jacobi Gauss-Lobatto points{xi}i=0N. Then, the collocation spec-tral numerical scheme is given byFirst, we give the error estimate in the L∞-norm.Theorem 1 1 be the exact solution to equation (2) and uN be the spectral collocation solutions of (4). If O<μ<1/2, we have the following error estimate provided that N is sufficiently large, where where K is defined by (3), and C is a constant that does not depend on N.Next we provide the error estimate in weighted L2-norm. Theorem 2 2 Let u be the exact solution of (2) and uN be the corresponding approx-imation obtained by using the spectral collocation scheme (4)-If 0<μ<2/1and u∈ Hm(—1,1)∩Hω-μO,*m(—1,1), we have the error estimateIn the third Chapter, we solve the American put option under Block-Scholes model by a projection and contraction method. here G(S)= [K—5)+, B(t)is unknown optimal exercise boundary, KX is the perpetual American put option andThe algorithm of the projection and contraction method is as follows:Finally we solve American put option under the Black-Scholes model: here G(S)=(K-S)+, B(t) is unknown optimal exercise boundary. Applying finite dif-ference method coupled with Newton’s method to solve the Black-Scholes equation, we can get the numerical approximations of the option price and the optimal exercise bound-ary simultaneously. The algorithm in this paper has timeliness, and can guarantee the numerical accuracy, which is an efficient method of quickly pricing the financial products. Numerical experiments verify the efficiency of this method.