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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 tothe Lamperti transformation technique and using a Brownian interpolation of thedrift implicit Euler scheme of Alfonsi [2], we show that the efficiency of theMLMC can be also improved for the evaluation of barrier options for models withnon-Lipschitz diffusion coefficients under certain moment constraints. We studytwo example models: the Cox-Ingersoll-Ross (CIR) and the Constant of Elasticityof Variance (CEV) processes for which we show that the conditions of ourtheoretical framework are satisfied under certain restrictions on the modelsparameters. In particular, we develop semi-explicit formulas for the densitiesof the running minimum and running maximum of both CIR and CEV processes whichare of independent interest. Finally, numerical tests are processed toillustrate our results.

Interpolated Drift Implicit Euler MLMC Method for Barrier Option Pricing and application to CIR and CEV Models