Fast deterministic pricing of options on Lévy driven assets

Ana-Maria Matache; Tobias von Petersdorff; Christoph Schwab

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 38, Issue: 1, page 37-71
  • ISSN: 0764-583X

Abstract

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Arbitrage-free prices u of European contracts on risky assets whose log-returns are modelled by Lévy processes satisfy a parabolic partial integro-differential equation (PIDE) ∂ t u + 𝒜 [ u ] = 0 . This PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the θ-scheme in time and a wavelet Galerkin method with N degrees of freedom in log-price space. The dense matrix for 𝒜 can be replaced by a sparse matrix in the wavelet basis, and the linear systems in each implicit time step are solved approximatively with GMRES in linear complexity. The total work of the algorithm for M time steps is bounded by O(MN(log(N))2) operations and O(Nlog(N)) memory. The deterministic algorithm gives optimal convergence rates (up to logarithmic terms) for the computed solution in the same complexity as finite difference approximations of the standard Black–Scholes equation. Computational examples for various Lévy price processes are presented.

How to cite

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Matache, Ana-Maria, von Petersdorff, Tobias, and Schwab, Christoph. "Fast deterministic pricing of options on Lévy driven assets." ESAIM: Mathematical Modelling and Numerical Analysis 38.1 (2010): 37-71. <http://eudml.org/doc/194208>.

@article{Matache2010,
abstract = { Arbitrage-free prices u of European contracts on risky assets whose log-returns are modelled by Lévy processes satisfy a parabolic partial integro-differential equation (PIDE) $\partial_t u + \{\mathcal\{A\}\}[u] = 0$. This PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the θ-scheme in time and a wavelet Galerkin method with N degrees of freedom in log-price space. The dense matrix for $\{\mathcal\{A\}\}$ can be replaced by a sparse matrix in the wavelet basis, and the linear systems in each implicit time step are solved approximatively with GMRES in linear complexity. The total work of the algorithm for M time steps is bounded by O(MN(log(N))2) operations and O(Nlog(N)) memory. The deterministic algorithm gives optimal convergence rates (up to logarithmic terms) for the computed solution in the same complexity as finite difference approximations of the standard Black–Scholes equation. Computational examples for various Lévy price processes are presented. },
author = {Matache, Ana-Maria, von Petersdorff, Tobias, Schwab, Christoph},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Parabolic partial integro-differential equations; Lévy processes; Markov processes; Galerkin finite element method; wavelet; matrix compression; GMRES.; arbitrage-free prices; European contracts on risky assets; parabolic partial integro-differential equation; Galerkin method; Lévy price processes},
language = {eng},
month = {3},
number = {1},
pages = {37-71},
publisher = {EDP Sciences},
title = {Fast deterministic pricing of options on Lévy driven assets},
url = {http://eudml.org/doc/194208},
volume = {38},
year = {2010},
}

TY - JOUR
AU - Matache, Ana-Maria
AU - von Petersdorff, Tobias
AU - Schwab, Christoph
TI - Fast deterministic pricing of options on Lévy driven assets
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 38
IS - 1
SP - 37
EP - 71
AB - Arbitrage-free prices u of European contracts on risky assets whose log-returns are modelled by Lévy processes satisfy a parabolic partial integro-differential equation (PIDE) $\partial_t u + {\mathcal{A}}[u] = 0$. This PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the θ-scheme in time and a wavelet Galerkin method with N degrees of freedom in log-price space. The dense matrix for ${\mathcal{A}}$ can be replaced by a sparse matrix in the wavelet basis, and the linear systems in each implicit time step are solved approximatively with GMRES in linear complexity. The total work of the algorithm for M time steps is bounded by O(MN(log(N))2) operations and O(Nlog(N)) memory. The deterministic algorithm gives optimal convergence rates (up to logarithmic terms) for the computed solution in the same complexity as finite difference approximations of the standard Black–Scholes equation. Computational examples for various Lévy price processes are presented.
LA - eng
KW - Parabolic partial integro-differential equations; Lévy processes; Markov processes; Galerkin finite element method; wavelet; matrix compression; GMRES.; arbitrage-free prices; European contracts on risky assets; parabolic partial integro-differential equation; Galerkin method; Lévy price processes
UR - http://eudml.org/doc/194208
ER -

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