A Newton–Krylov method with a tridiagonal preconditioner for American option pricing under jump–diffusion model with transaction costs

In financial markets, trading activities are often subject to transaction costs, which are ignored under the assumptions of the traditional Black–Scholes model. In addition, the Black–Scholes model cannot capture the discontinuous jumps commonly observed in asset prices. To address these limitations, jump–diffusion models have been developed for option pricing, and recent studies have further incorporated transaction costs into such models. However, such models involve integro–differential operators and nonlinear terms, which pose challenges to existing numerical methods. This paper aims to develop a fast algorithm with theoretical guarantees for the American option pricing based on jump–diffusion models with transaction costs, which is essentially a nonlinear complementarity problem (NCP) involving integro–differential operators. To develop the fast solver, the NCP is transformed into a constant-coefficient partial integro–differential equation with two nonlinear terms, and a numerical scheme is proposed to discretize it, whose stability and positivity-preserving properties are analyzed. Then, a nested Newton–Krylov iteration framework is designed to solve the nonlinear scheme and the associated linear systems. To accelerate the convergence rate of the proposed method, a tridiagonal preconditioner is introduced. It is proven that the eigenvalues of the preconditioned matrix are clustered around one and its condition number is bounded. Numerical experiments, including an empirical example, are given to demonstrate the efficiency and effectiveness of the proposed fast solution strategy.