1,257 research outputs found
Monte Carlo Greeks for financial products via approximative transition densities
In this paper we introduce efficient Monte Carlo estimators for the valuation
of high-dimensional derivatives and their sensitivities (''Greeks''). These
estimators are based on an analytical, usually approximative representation of
the underlying density. We study approximative densities obtained by the WKB
method. The results are applied in the context of a Libor market model.Comment: 24 page
Monte Carlo Greeks for financial products via approximative Greenian kernels
In this paper we introduce efficient Monte Carlo estimators for the valuation of high-dimensional derivatives and their sensitivities (''Greeks''). These estimators are based on an analytical, usually approximative representation of the underlying density. We study approximative densities obtained by the WKB method. The results are applied in the context of a Libor market model
Efficient Monte Carlo for high excursions of Gaussian random fields
Our focus is on the design and analysis of efficient Monte Carlo methods for
computing tail probabilities for the suprema of Gaussian random fields, along
with conditional expectations of functionals of the fields given the existence
of excursions above high levels, b. Na\"{i}ve Monte Carlo takes an exponential,
in b, computational cost to estimate these probabilities and conditional
expectations for a prescribed relative accuracy. In contrast, our Monte Carlo
procedures achieve, at worst, polynomial complexity in b, assuming only that
the mean and covariance functions are H\"{o}lder continuous. We also explain
how to fine tune the construction of our procedures in the presence of
additional regularity, such as homogeneity and smoothness, in order to further
improve the efficiency.Comment: Published in at http://dx.doi.org/10.1214/11-AAP792 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On the Convergence of the Laplace Approximation and Noise-Level-Robustness of Laplace-based Monte Carlo Methods for Bayesian Inverse Problems
The Bayesian approach to inverse problems provides a rigorous framework for
the incorporation and quantification of uncertainties in measurements,
parameters and models. We are interested in designing numerical methods which
are robust w.r.t. the size of the observational noise, i.e., methods which
behave well in case of concentrated posterior measures. The concentration of
the posterior is a highly desirable situation in practice, since it relates to
informative or large data. However, it can pose a computational challenge for
numerical methods based on the prior or reference measure. We propose to employ
the Laplace approximation of the posterior as the base measure for numerical
integration in this context. The Laplace approximation is a Gaussian measure
centered at the maximum a-posteriori estimate and with covariance matrix
depending on the logposterior density. We discuss convergence results of the
Laplace approximation in terms of the Hellinger distance and analyze the
efficiency of Monte Carlo methods based on it. In particular, we show that
Laplace-based importance sampling and Laplace-based quasi-Monte-Carlo methods
are robust w.r.t. the concentration of the posterior for large classes of
posterior distributions and integrands whereas prior-based importance sampling
and plain quasi-Monte Carlo are not. Numerical experiments are presented to
illustrate the theoretical findings.Comment: 50 pages, 11 figure
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