  We apply multilevel Monte Carlo for option pricing problems using exponential
L\'{e}vy models with a uniform timestep discretisation to monitor the running
maximum required for lookback and barrier options. The numerical results
demonstrate the computational efficiency of this approach. We derive estimates
of the convergence rate for the error introduced by the discrete monitoring of
the running supremum of a broad class of L\'{e}vy processes. We use these to
obtain upper bounds on the multilevel Monte Carlo variance convergence rate for
the Variance Gamma, NIG and $\alpha$-stable processes used in the numerical
experiments. We also show numerical results and analysis of a trapezoidal
approximation for Asian options.

Computational Finance