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

Michael B. Giles; Desmond J. Higham; Xuerong Mao

doi:10.1007/s00780-009-0092-1

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

Michael B. Giles, Desmond J. Higham, Xuerong Mao

Mathematics And Statistics

Research output: Contribution to journal › Article › peer-review

46 Citations (Scopus)

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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.

Original language	English
Pages (from-to)	403-413
Number of pages	10
Journal	Finance and Stochastics
Volume	13
Issue number	3
DOIs	https://doi.org/10.1007/s00780-009-0092-1
Publication status	Published - Sep 2009

Keywords

barrier option
complexity
digital option
Euler-Maruyama
lookback option
path
dependent option
statistical error
strong error
weak error

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10.1007/s00780-009-0092-1

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title = "Analysing multi-level Monte Carlo for options with non-globally Lipschitz payoff",

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.",

keywords = "barrier option, complexity, digital option, Euler-Maruyama, lookback option, path, dependent option, statistical error, strong error, weak error",

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AB - 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.

KW - barrier option

KW - complexity

KW - digital option

KW - Euler-Maruyama

KW - lookback option

KW - path

KW - dependent option

KW - statistical error

KW - strong error

KW - weak error

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DO - 10.1007/s00780-009-0092-1

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