A Preconditioned Policy–Krylov Subspace Method for Fractional Partial Integro-Differential HJB Equations in Finance

Abstract

To better simulate the prices of underlying assets and improve the accuracy of pricing financial derivatives, an increasing number of new models are being proposed. Among them, the Lévy process with jumps has received increasing attention because of its capacity to model sudden movements in asset prices. This paper explores the Hamilton–Jacobi–Bellman (HJB) equation with a fractional derivative and an integro-differential operator, which arise in the valuation of American options and stock loans based on the Lévy-α-stable process with jumps model. We design a fast solution strategy that includes the policy iteration method, Krylov subspace method, and banded preconditioner, aiming to solve this equation rapidly. To solve the resulting HJB equation, a finite difference method including an upwind scheme, shifted Grünwald approximation, and trapezoidal method is developed with stability and convergence analysis. Then, an algorithmic framework involving the policy iteration method and the Krylov subspace method is employed. To improve the performance of the above solver, a banded preconditioner is proposed with condition number analysis. Finally, two examples, sugar option pricing and stock loan valuation, are provided to illustrate the effectiveness of the considered model and the efficiency of the proposed preconditioned policy–Krylov subspace method.