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Analysing multi-level Monte Carlo for options with non-globally Lipschitz payoff

Giles, Michael B. and Higham, Desmond J. and Mao, Xuerong (2009) Analysing multi-level Monte Carlo for options with non-globally Lipschitz payoff. Finance and Stochastics, 13 (3). pp. 403-413. ISSN 0949-2984

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Abstract

Giles (Multilevel Monte Carlo path simulation Operations Research, 2008; 56:607-617) introduced a multi-level Monte Carlo method for approximating the expected value of a function of a stochastic differential equation solution. A key application is to compute the expected payff of a financial option. This new method improves on the computational complexity of standard Monte Carlo. Giles analysed globally Lipschitz payoffs, but also found good performance in practice for non-globally Lipschitz cases. In this work, we show that the multi-level Monte Carlo method can be rigorously justifed for non-globally Lipschitz payoffs. In particular, we consider digital, lookback and barrier options. This requires non-standard strong convergence analysis of the Euler-Maruyama method.