3,061 research outputs found
Multidimensional Quasi-Monte Carlo Malliavin Greeks
We investigate the use of Malliavin calculus in order to calculate the Greeks
of multidimensional complex path-dependent options by simulation. For this
purpose, we extend the formulas employed by Montero and Kohatsu-Higa to the
multidimensional case. The multidimensional setting shows the convenience of
the Malliavin Calculus approach over different techniques that have been
previously proposed. Indeed, these techniques may be computationally expensive
and do not provide flexibility for variance reduction. In contrast, the
Malliavin approach exhibits a higher flexibility by providing a class of
functions that return the same expected value (the Greek) with different
accuracies. This versatility for variance reduction is not possible without the
use of the generalized integral by part formula of Malliavin Calculus. In the
multidimensional context, we find convenient formulas that permit to improve
the localization technique, introduced in Fourni\'e et al and reduce both the
computational cost and the variance. Moreover, we show that the parameters
employed for variance reduction can be obtained \textit{on the flight} in the
simulation. We illustrate the efficiency of the proposed procedures, coupled
with the enhanced version of Quasi-Monte Carlo simulations as discussed in
Sabino, for the numerical estimation of the Deltas of call, digital Asian-style
and Exotic basket options with a fixed and a floating strike price in a
multidimensional Black-Scholes market.Comment: 22 pages, 6 figure
Smoking Adjoints: fast evaluation of Greeks in Monte Carlo calculations
This paper presents an adjoint method to accelerate the calculation of Greeks by Monte Carlo simulation. The method calculates price sensitivities along each path; but in contrast to a forward pathwise calculation, it works backward recursively using adjoint variables. Along each path, the forward and adjoint implementations produce the same values, but the adjoint method rearranges the calculations to generate potential computational savings. The adjoint method outperforms a forward implementation in calculating the sensitivities of a small number of outputs to a large number of inputs. This applies, for example, in estimating the sensitivities of an interest rate derivatives book to multiple points along an initial forward curve or the sensitivities of an equity derivatives book to multiple points on a volatility surface. We illustrate the application of the method in the setting of the LIBOR market model. Numerical results confirm that the computational advantage of the adjoint method grows in proportion to the number of initial forward rates
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
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
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
Vibrato Monte Carlo and the calculation of greeks
In computational ¯nance Monte Carlo simulation can be used to calculate
the correct prices of ¯nancial options, and to compute the values of the as-
sociated Greeks (the derivatives of the option price with respect to certain
input parameters). The main methods used for the calculation of Greeks
are finite difference, likelihood ratio, and pathwise sensitivity. Each of these
has its limitations and in particular the pathwise sensitivity approach may
not be used for an option whose payoff function is discontinuous. Vibrato
Monte Carlo is a new idea that addresses the limitations of previous methods;
it combines the pathwise sensitivity approach for the SDE path calculation
with the likelihood ratio method for payoff evaluation. This thesis discusses
Vibrato Monte Carlo approximations for a digital option on an asset follow-
ing one-dimensional geometric Brownian motion
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
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