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

TL;DR

This paper enhances MLMC methods for barrier option pricing in non-Lipschitz models like CIR and CEV by using a Brownian interpolation of the drift implicit Euler scheme, supported by theoretical density formulas and numerical validation.

Contribution

It introduces a novel MLMC efficiency improvement for non-Lipschitz diffusion models using Lamperti transformation and Brownian interpolation techniques.

Findings

MLMC efficiency is improved for CIR and CEV models.

Semi-explicit density formulas for minima and maxima are derived.

Numerical tests confirm theoretical results.

Abstract

Recently, Giles et al. [14] proved that the efficiency of the Multilevel Monte Carlo (MLMC) method for evaluating Down-and-Out barrier options for a diffusion process $(X_t)_{t\in[0,T]}$ with globally Lipschitz coefficients, can be improved by combining a Brownian bridge technique and a conditional Monte Carlo method provided that the running minimum $\inf_{t\in[0,T]}X_t$ has a bounded density in the vicinity of the barrier. In the present work, thanks to the Lamperti transformation technique and using 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 the evaluation of barrier options for models with non-Lipschitz diffusion coefficients under certain moment constraints. We study two example models: the Cox-Ingersoll-Ross (CIR) and the Constant of Elasticity of Variance (CEV) processes for which we show that the conditions of our theoretical framework are satisfied under certain restrictions on the models parameters. In particular, we develop semi-explicit formulas for the densities of the running minimum and running maximum of both CIR and CEV processes which are of independent interest. Finally, numerical tests are processed to illustrate our results.