×

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.

MSC:

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.)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] DOI: 10.1002/fut.21647
[2] DOI: 10.1006/jcph.2002.7176 · Zbl 1015.65018
[3] DOI: 10.1086/260062 · Zbl 1092.91524
[4] Boyarchenko S., Non-Gaussian Merton–Black–Scholes Theory, Vol. 9 (2002)
[5] DOI: 10.1111/1540-6261.00544
[6] DOI: 10.1086/338705
[7] DOI: 10.1016/j.physa.2006.08.071
[8] DOI: 10.1137/S0036144594276474 · Zbl 0863.65013
[9] DOI: 10.1137/0910009 · Zbl 0666.65030
[10] DOI: 10.3934/jimo.2015.11.241 · Zbl 1305.91239
[11] DOI: 10.1201/9780203485217 · Zbl 1052.91043
[12] DOI: 10.1007/s11766-012-2516-5 · Zbl 1265.62035
[13] DOI: 10.1002/9781118673447
[14] DOI: 10.1007/s00211-009-0252-4 · Zbl 1185.91176
[15] DOI: 10.1111/j.1467-9965.2008.00338.x · Zbl 1141.91438
[16] DOI: 10.1080/14697688.2010.538074 · Zbl 1279.91183
[17] Itkin A., Algorithmic Financ. 3 pp 233– (2014)
[18] DOI: 10.1080/00207160.2015.1071360 · Zbl 1335.91095
[19] DOI: 10.1007/s10614-011-9269-8 · Zbl 1254.91747
[20] Jin X., Preconditioning Techniques for Toeplitz Systems (2010)
[21] DOI: 10.1016/j.jcp.2013.02.025 · Zbl 1297.65095
[22] DOI: 10.1016/j.nonrwa.2008.10.066 · Zbl 1180.91308
[23] DOI: 10.1016/j.cam.2004.01.033 · Zbl 1126.76346
[24] DOI: 10.1016/j.apnum.2005.02.008 · Zbl 1086.65087
[25] DOI: 10.1002/num.21948 · Zbl 1329.91141
[26] Ng M.K., Iterative Methods for Toeplitz Systems (2004) · Zbl 1059.65031
[27] Petrella G., J. Comput. Financ. 8 pp 1– (2004)
[28] Podlubny I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Vol. 198 (1998) · Zbl 0924.34008
[29] DOI: 10.1137/1.9780898718003
[30] DOI: 10.1016/j.jcp.2005.08.008 · Zbl 1089.65089
[31] Varga R.S., Geršgorin and His Circles (2010)
[32] DOI: 10.1016/j.cam.2006.04.034 · Zbl 1137.91477
[33] DOI: 10.1016/j.advwatres.2010.11.003
[34] DOI: 10.1137/130922495 · Zbl 1296.91288
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.