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Recently, Giles et al. [14] proved that the efficiency of the MultilevelMonte Carlo (MLMC) method for evaluating Down-and-Out barrier options for adiffusion process $(X_t)_{t\in[0,T]}$ with globally Lipschitz coefficients, canbe improved by combining a Brownian bridge technique and a conditional MonteCarlo method provided that the running minimum $\inf_{t\in[0,T]}X_t$ has abounded density in the vicinity of the barrier.  In the present work, thanks to the Lamperti transformation technique andusing a Brownian interpolation of the drift implicit Euler scheme of Alfonsi[2], we show that the efficiency of the MLMC can be also improved for theevaluation of barrier options for models with non-Lipschitz diffusioncoefficients under certain moment constraints. We study two example models: theCox-Ingersoll-Ross (CIR) and the Constant of Elasticity of Variance (CEV)processes for which we show that the conditions of our theoretical frameworkare satisfied under certain restrictions on the models parameters. Inparticular, we develop semi-explicit formulas for the densities of the runningminimum and running maximum of both CIR and CEV processes which are ofindependent interest. Finally, numerical tests are processed to illustrate ourresults.

The interpolated drift implicit Euler scheme Multilevel Monte Carlo method for pricing Barrier options and applications to the CIR and CEV models