Using spectral element method to solve variational inequalities with applications in finance

作者：Moradipour M; Yousefi S A^*

来源：Chaos Solitons & Fractals, 2015, 81: 208-217.

DOI：10.1016/j.chaos.2015.09.006

• American options
• Spectral element methods
• Variational inequalities
• Linear complementarity problems

Under the Black-Scholes model, the value of an American option solves a time dependent variational inequality problem (VIP). In this paper, first we discretize the variational inequality of American option in temporal direction by applying the Rannacher time stepping and achieve a sequence of elliptic variational inequalities. Second we discretize the spatial domain of variational inequalities by using spectral element methods with high order Lagrangian polynomials introduced on Gauss-Legendre-Lobatto points. Also by computing integrals by the Gauss-Legendre-Lobatto quadrature rule we derive a sequence of the linear complementarity problems (LCPs) having a positive definite sparse coefficient matrix. To find the unique solutions of the LCPs, we use the projected successive over-relaxation (PSOR) algorithm. Furthermore we present some existence and uniqueness theorems for the variational inequalities and LCPs. Finally, theoretical results are verified on the relevant numerical examples.

• 出版日期2015-12