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We consider the problem of estimating a nested structure of two expectationstaking the form $U_0 = E[\max\{U_1(Y), \pi(Y)\}]$, where $U_1(Y) = E[X\ |\ Y]$.Terms of this form arise in financial risk estimation and option pricing. When$U_1(Y)$ requires approximation, but exact samples of $X$ and $Y$ areavailable, an antithetic multilevel Monte Carlo (MLMC) approach has beenwell-studied in the literature. Under general conditions, the antithetic MLMCestimator obtains a root mean squared error $\varepsilon$ with order$\varepsilon^{-2}$ cost. If, additionally, $X$ and $Y$ require approximatesampling, careful balancing of the various aspects of approximation is requiredto avoid a significant computational burden. Under strong convergence criteriaon approximations to $X$ and $Y$, randomised multilevel Monte Carlo techniquescan be used to construct unbiased Monte Carlo estimates of $U_1$, which can bepaired with an antithetic MLMC estimate of $U_0$ to recover order$\varepsilon^{-2}$ computational cost. In this work, we instead consider biasedmultilevel approximations of $U_1(Y)$, which require less strict assumptions onthe approximate samples of $X$. Extensions to the method consider anapproximate and antithetic sampling of $Y$. Analysis shows the resultingestimator has order $\varepsilon^{-2}$ asymptotic cost under the conditionsrequired by randomised MLMC and order $\varepsilon^{-2}|\log\varepsilon|^3$cost under more general assumptions.

Nested Multilevel Monte Carlo with Biased and Antithetic Sampling