# Antithetic multilevel Monte Carlo estimation of financial options

Lukas Szpruch, Mike Giles

Research output: Chapter in Book/Report/Conference proceedingConference contribution

## Abstract

In this paper we develop antithetic multilevel Monte Carlo
(MLMC) estimators for multidimensional SDEs driven by Brownian motion.
Giles has previously shown that if we combine a numerical approximation
with strong order of convergence $O(\Delta t)$ with MLMC we can reduce
the computational complexity to estimate expected values of Lipschitz
functionals of SDE solutions with a root-mean-square error of $\epsilon$
from $O(\epsilon^{-3})$ to $O(\epsilon^{-2})$. However, in general, to obtain
a rate of strong convergence higher than $O(\Delta t^{1/2})$ requires
simulation, or approximation, of L{\'e}vy areas.
Recently, Giles and Szpruch \cite{giles2012antithetic} constructed
an antithetic multilevel estimator that avoids the simulation of
L{\'e}vy areas and still achieves an MLMC correction variance which
is $O(\Delta t^2)$ for smooth payoffs and almost $O(\Delta t^{3/2})$ for
piecewise smooth payoffs, even though there is only $O(\Delta t^{1/2})$
strong convergence. This results in an $O(\epsilon^{-2})$ complexity
for estimating the value of financial European and Asian put and call
options.
In this paper, we extend these results to more complex payoffs based
on the path minimum. To achieve this, an approximation of the L{\'e}vy
areas is needed, resulting in $O(\Delta t^{3/4})$ strong convergence.
By modifying the antithetic MLMC estimator we are able to obtain
$O(\epsilon^{-2}\log(\epsilon)^2)$ complexity for estimating financial
barrier and lookback options.
Original language English Monte Carlo and Quasi-Monte Carlo Methods 2012 Springer-Verlag GmbH 367-384 978-3-642-41095-6 978-3-642-41094-9 https://doi.org/10.1007/978-3-642-41095-6_16 Published - 2013

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