# 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

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topMatache, 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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