10,812 research outputs found
Pricing and Risk Management with High-Dimensional Quasi Monte Carlo and Global Sensitivity Analysis
We review and apply Quasi Monte Carlo (QMC) and Global Sensitivity Analysis
(GSA) techniques to pricing and risk management (greeks) of representative
financial instruments of increasing complexity. We compare QMC vs standard
Monte Carlo (MC) results in great detail, using high-dimensional Sobol' low
discrepancy sequences, different discretization methods, and specific analyses
of convergence, performance, speed up, stability, and error optimization for
finite differences greeks. We find that our QMC outperforms MC in most cases,
including the highest-dimensional simulations and greeks calculations, showing
faster and more stable convergence to exact or almost exact results. Using GSA,
we are able to fully explain our findings in terms of reduced effective
dimension of our QMC simulation, allowed in most cases, but not always, by
Brownian bridge discretization. We conclude that, beyond pricing, QMC is a very
promising technique also for computing risk figures, greeks in particular, as
it allows to reduce the computational effort of high-dimensional Monte Carlo
simulations typical of modern risk management.Comment: 43 pages, 21 figures, 6 table
Greedy vector quantization
We investigate the greedy version of the -optimal vector quantization
problem for an -valued random vector . We show the
existence of a sequence such that minimizes
(-mean quantization error at level induced by
). We show that this sequence produces -rate
optimal -tuples ( the -mean
quantization error at level induced by goes to at rate
). Greedy optimal sequences also satisfy, under natural
additional assumptions, the distortion mismatch property: the -tuples
remain rate optimal with respect to the -norms, .
Finally, we propose optimization methods to compute greedy sequences, adapted
from usual Lloyd's I and Competitive Learning Vector Quantization procedures,
either in their deterministic (implementable when ) or stochastic
versions.Comment: 31 pages, 4 figures, few typos corrected (now an extended version of
an eponym paper to appear in Journal of Approximation
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