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Efficient solution of a partial integro-differential equation in finance. (English) Zbl 1155.65109
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$\left(nlogn\right)$ operations. Second order accuracy is obtained numerically on the whole computational domain for R. C. Merton’s model [J. Financ. Econ. 3, No. 1–2, 125–144 (1976; Zbl 1131.91344)].
##### MSC:
 65R20 Integral equations (numerical methods) 45K05 Integro-partial differential equations 91B28 Finance etc. (MSC2000)