| Issue |
ESAIM: M2AN
Volume 38, Number 1, January-February 2004
|
|
|---|---|---|
| Page(s) | 37 - 71 | |
| DOI | https://doi.org/10.1051/m2an:2004003 | |
| Published online | 15 February 2004 | |
Fast deterministic pricing of options on Lévy driven assets
1
RiskLab and
Seminar for Applied Mathematics,
ETH-Zentrum, 8092 Zürich, Switzerland.
2
Department of Mathematics,
University of Maryland, College Park, MD 20742, USA.
3
Seminar for Applied Mathematics,
ETH-Zentrum, 8092 Zürich, Switzerland. schwab@sam.math.ethz.ch.
Received:
27
February
2003
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)
![$\partial_t u + {\mathcal{A}}[u] = 0$](/articles/m2an/abs/2004/01/m2an0309/m2an0309_tex_eq1.png)
.
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 
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.
Mathematics Subject Classification: 65N30 / 60J75
Key words: Parabolic partial integro-differential equations / Lévy processes / Markov processes / Galerkin finite element method / wavelet / matrix compression / GMRES.
© EDP Sciences, SMAI, 2004
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