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We investigate the problem of computing a nested expectation of the form $mathbb{P}[mathbb{E}[X|Y] !geq!0]!=!mathbb{E}[textrm{H}(mathbb{E}[X|Y])]$ where $textrm{H}$ is the Heaviside function. This nested expectation appears, for example, when estimating the probability of a large loss from a financial portfolio. We present a method that combines the idea of using Multilevel Monte Carlo (MLMC) for nested expectations with the idea of adaptively selecting the number of samples in the approximation of the inner expectation, as proposed by (Broadie et al., 2011). We propose and analyse an algorithm that adaptively selects the number of inner samples on each MLMC level and prove that the resulting MLMC method with adaptive sampling has an $mathcal{O}left( varepsilon^{-2}|logvarepsilon|^2 right)$ complexity to achieve a root mean-squared error $varepsilon$. The theoretical analysis is verified by numerical experiments on a simple model problem. We also present a stochastic root-finding algorithm that, combined with our adaptive methods, can be used to compute other risk measures such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR), with the latter being achieved with $mathcal{O}left(varepsilon^{-2}right)$ complexity.
We consider the problem of estimating the probability of a large loss from a financial portfolio, where the future loss is expressed as a conditional expectation. Since the conditional expectation is intractable in most cases, one may resort to neste
In this paper we develop a novel methodology for estimation of risk capital allocation. The methodology is rooted in the theory of risk measures. We work within a general, but tractable class of law-invariant coherent risk measures, with a particular
Computing risk measures of a financial portfolio comprising thousands of options is a challenging problem because (a) it involves a nested expectation requiring multiple evaluations of the loss of the financial portfolio for different risk scenarios
We derive representations of local risk-minimization of call and put options for Barndorff-Nielsen and Shephard models: jump type stochastic volatility models whose squared volatility process is given by a non-Gaussian rnstein-Uhlenbeck process. The
In this article we consider static Bayesian parameter estimation for partially observed diffusions that are discretely observed. We work under the assumption that one must resort to discretizing the underlying diffusion process, for instance using th