290 research outputs found
Conditional Quasi-Monte Carlo with Constrained Active Subspaces
Conditional Monte Carlo or pre-integration is a useful tool for reducing
variance and improving regularity of integrands when applying Monte Carlo and
quasi-Monte Carlo (QMC) methods. To choose the variable to pre-integrate with,
one need to consider both the variable importance and the tractability of the
conditional expectation. For integrals over a Gaussian distribution, one can
pre-integrate over any linear combination of variables. Liu and Owen (2022)
propose to choose the linear combination based on an active subspace
decomposition of the integrand. However, pre-integrating over such selected
direction might be intractable. In this work, we address this issue by finding
the active subspaces subject to the constraints such that pre-integration can
be easily carried out. The proposed method is applied to some examples in
derivative pricing under stochastic volatility models and is shown to
outperform previous methods
Sequential Monte Carlo Methods for Option Pricing
In the following paper we provide a review and development of sequential
Monte Carlo (SMC) methods for option pricing. SMC are a class of Monte
Carlo-based algorithms, that are designed to approximate expectations w.r.t a
sequence of related probability measures. These approaches have been used,
successfully, for a wide class of applications in engineering, statistics,
physics and operations research. SMC methods are highly suited to many option
pricing problems and sensitivity/Greek calculations due to the nature of the
sequential simulation. However, it is seldom the case that such ideas are
explicitly used in the option pricing literature. This article provides an
up-to date review of SMC methods, which are appropriate for option pricing. In
addition, it is illustrated how a number of existing approaches for option
pricing can be enhanced via SMC. Specifically, when pricing the arithmetic
Asian option w.r.t a complex stochastic volatility model, it is shown that SMC
methods provide additional strategies to improve estimation.Comment: 37 Pages, 2 Figure
Quasi-Monte Carlo methods for calculating derivatives sensitivities on the GPU
The calculation of option Greeks is vital for risk management. Traditional
pathwise and finite-difference methods work poorly for higher-order Greeks and
options with discontinuous payoff functions. The Quasi-Monte Carlo-based
conditional pathwise method (QMC-CPW) for options Greeks allows the payoff
function of options to be effectively smoothed, allowing for increased
efficiency when calculating sensitivities. Also demonstrated in literature is
the increased computational speed gained by applying GPUs to highly
parallelisable finance problems such as calculating Greeks. We pair QMC-CPW
with simulation on the GPU using the CUDA platform. We estimate the delta, vega
and gamma Greeks of three exotic options: arithmetic Asian, binary Asian, and
lookback. Not only are the benefits of QMC-CPW shown through variance reduction
factors of up to , but the increased computational speed
through usage of the GPU is shown as we achieve speedups over sequential CPU
implementations of more than x for our most accurate method.Comment: 26 pages, 12 figure
Kernel estimation of Greek weights by parameter randomization
A Greek weight associated to a parameterized random variable is
a random variable such that
for any function
. The importance of the set of Greek weights for the purpose of Monte
Carlo simulations has been highlighted in the recent literature. Our main
concern in this paper is to devise methods which produce the optimal weight,
which is well known to be given by the score, in a general context where the
density of is not explicitly known. To do this, we randomize the
parameter by introducing an a priori distribution, and we use
classical kernel estimation techniques in order to estimate the score function.
By an integration by parts argument on the limit of this first kernel
estimator, we define an alternative simpler kernel-based estimator which turns
out to be closely related to the partial gradient of the kernel-based estimator
of . Similarly to the finite differences
technique, and unlike the so-called Malliavin method, our estimators are
biased, but their implementation does not require any advanced mathematical
calculation. We provide an asymptotic analysis of the mean squared error of
these estimators, as well as their asymptotic distributions. For a
discontinuous payoff function, the kernel estimator outperforms the classical
finite differences one in terms of the asymptotic rate of convergence. This
result is confirmed by our numerical experiments.Comment: Published in at http://dx.doi.org/10.1214/105051607000000186 the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Multilevel Monte Carlo methods for applications in finance
Since Giles introduced the multilevel Monte Carlo path simulation method
[18], there has been rapid development of the technique for a variety of
applications in computational finance. This paper surveys the progress so far,
highlights the key features in achieving a high rate of multilevel variance
convergence, and suggests directions for future research.Comment: arXiv admin note: text overlap with arXiv:1202.6283; and with
arXiv:1106.4730 by other author
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