  In this paper we introduce a new multilevel Monte Carlo (MLMC) estimator for
multi-dimensional SDEs driven by Brownian motions. 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 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. In this paper,
through the construction of a suitable antithetic multilevel correction
estimator, we are able to avoid the simulation of L\'{e}vy areas and still
achieve an $O(\Delta t^2)$ multilevel correction variance for smooth payoffs,
and almost an $O(\Delta t^{3/2})$ variance 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 European and Asian
put and call options.
