## 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.

(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 |
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Title of host publication | Monte Carlo and Quasi-Monte Carlo Methods 2012 |

Publisher | Springer-Verlag GmbH |

Pages | 367-384 |

ISBN (Electronic) | 978-3-642-41095-6 |

ISBN (Print) | 978-3-642-41094-9 |

DOIs | |

Publication status | Published - 2013 |