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Zbl 1155.65109
Sachs, E.W.; Strauss, A.K.
Efficient solution of a partial integro-differential equation in finance. (English)
[J] Appl. Numer. Math. 58, No. 11, 1687-1703 (2008). ISSN 0168-9274

Summary: Jump-diffusion models for the pricing of derivatives lead under certain assumptions to partial integro-differential equations (PIDE). Such a PIDE typically involves a convection term and a non-local integral. We transform the PIDE to eliminate the convection term, discretize it implicitly, and use finite differences on a uniform grid. The resulting dense linear system exhibits so much structure that it can be solved very efficiently by a circulant preconditioned conjugate gradient method. Therefore, this fully implicit scheme requires only on the order of O$(n\log n)$ operations. Second order accuracy is obtained numerically on the whole computational domain for {\it R. C. Merton}'s model [J. Financ. Econ. 3, No.~1--2, 125--144 (1976; Zbl 1131.91344)].

MSC 2000:: *65R20 Integral equations (numerical methods)
45K05 Integro-partial differential equations
91B28 Finance etc.

Keywords: Lévy process; partial integro-differential equations; conjugate gradient method; Toeplitz matrices; preconditioning; numerical examples; jump-diffusion models; pricing of derivatives; finite differences

Citations: Zbl 1131.91344

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