Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation

By M. Giles, Lukasz Szpruch. 2012

129 citations recorded by the source corpus.

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.

Subfields: Finance, Numerical Analysis

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In Finance

  1. Lukasz Szpruch
  2. Nicolas Chopin
  3. Chris J. Oates
  4. Mark A. Girolami
  5. M. Giles
  6. Steffen Dereich
  7. Raphael Kruse
  8. Lukasz Szpruch

In Numerical Analysis

  1. Chris J. Oates
  2. Mark A. Girolami
  3. Lukasz Szpruch
  4. Nicolas Chopin
  5. M. Giles
  6. Qiang Liu
  7. Dilin Wang
  8. Jason D. Lee

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