Extrapolation Variable Step-size Implicit Runge-Kutta Numerical Algorithm For European And American Options Under The Jump-diffusion Model

The traditional option pricing model assumes that the underlying asset price obeys Brow-nian motion with drift and does not consider market volatility.On this basis,some scholars have proposed a more realistic jump-diffusion option pricing model.The model is a partial integro-difFerential equation(PIDE)with a non-local integral term.This thesis mainly studies the extrapolated variable-step implicit explicit Runge-Kutta(IMEX RK)method for European option pricing and American option pricing problems with jumps.For the European option pricing problem,in the time direction,we proposed the extrap-olation variable step-sizes IMEX RK method,that is,the function u in the non-local integral operator is approximated by the extrapolated method to achieve the explicit discrete effects,the rest of the operators are implicitly discrete.The stability of the time semi-discrete scheme and the global error bound are analyzed.In the spatial direction,we use the finite difference method to discretize the differential operator,and the non-local integral operator is approximated by the composite trapezoidal rule.The system of linear equations with tridiagonal coefficient matri-ces can reduce the computational complexity and improve the solution efficiency.Numerical experiments verify the feasibility of the extrapolated variable-step IMEX RK method.The characteristic of American options is that the option can be exercised at any time during the validity period of the option,and its pricing model satisfies a linear complementarity problem(LCP).We choose the penalty function method to convert the LCP into a nonlinear partial integro-difFerential equation for solving.The discretization method is the same as the discretization format of European option.In the time direction,the extrapolated variable stepsizes IMEX RK method is also used to discretize the nonlinear PIDE,and the extrapolated is used to approximate the integral term and penalty term.The American option pricing problem is equally valid.