Multilevel Monte Carlo For Exponential Lévy Models

Abstract: 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.