A robust algorithm and convergence analysis for static replications of nonlinear payoffs

In this paper we propose a new robust algorithm to find the optimal static replicating portfolios for general nonlinear payoff functions and give the estimate of the rate of convergence that is absent in the literature. We choose the static replication by minimizing the error bound between the nonlinear payoff function and the linear spline approximation and derive the equidistribution equation for selecting the optimal strike prices. The numerical tests for variance swaps and swaptions and also for the static quadratic replication and the model with counterparty risk show that the proposed algorithm is simple, fast and accurate. The paper has generalized and improved the results of the static replication and approximation in the literature.