Circulant preconditioning technique for barrier options pricing under fractional diffusion models. (English) Zbl 1337.91130

Summary: In recent years, considerable literature has proposed the more general class of exponential Lévy processes as the underlying model for prices of financial quantities, which thus better explain many important empirical facts of financial markets. Finite moment log stable, Carr-Geman-Madan-Yor and KoBoL models are chosen from those above-mentioned models as the dynamics of underlying equity prices in this paper. With such models pricing barrier options, one kind of financial derivatives is transformed to solve specific fractional partial differential equations (FPDEs). This study focuses on numerically solving these FPDEs via the fully implicit scheme, with the shifted Grünwald approximation. The circulant preconditioned generalized minimal residual method which converges very fast with theoretical proof is incorporated for solving resultant linear systems. Numerical examples are given to demonstrate the effectiveness of the proposed preconditioner and show the accuracy of our method compared with that done by the Fourier cosine expansion method as a benchmark.


91G60 Numerical methods (including Monte Carlo methods)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
60G22 Fractional processes, including fractional Brownian motion
60G51 Processes with independent increments; Lévy processes
65F08 Preconditioners for iterative methods
65F10 Iterative numerical methods for linear systems
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M22 Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs
91G20 Derivative securities (option pricing, hedging, etc.)
Full Text: DOI


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