95 research outputs found
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
Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation
The multilevel Monte Carlo path simulation method introduced by Giles ({\it
Operations Research}, 56(3):607-617, 2008) exploits strong convergence
properties to improve the computational complexity by combining simulations
with different levels of resolution. In this paper we analyse its efficiency
when using the Milstein discretisation; this has an improved order of strong
convergence compared to the standard Euler-Maruyama method, and it is proved
that this leads to an improved order of convergence of the variance of the
multilevel estimator. Numerical results are also given for basket options to
illustrate the relevance of the analysis.Comment: 33 pages, 4 figures, to appear in Discrete and Continuous Dynamical
Systems - Series
An explicit Euler scheme with strong rate of convergence for financial SDEs with non-Lipschitz coefficients
We consider the approximation of stochastic differential equations (SDEs)
with non-Lipschitz drift or diffusion coefficients. We present a modified
explicit Euler-Maruyama discretisation scheme that allows us to prove strong
convergence, with a rate. Under some regularity and integrability conditions,
we obtain the optimal strong error rate. We apply this scheme to SDEs widely
used in the mathematical finance literature, including the
Cox-Ingersoll-Ross~(CIR), the 3/2 and the Ait-Sahalia models, as well as a
family of mean-reverting processes with locally smooth coefficients. We
numerically illustrate the strong convergence of the scheme and demonstrate its
efficiency in a multilevel Monte Carlo setting.Comment: 36 pages, 17 figures, 2 table
Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without L\'{e}vy area simulation
In this paper we introduce a new multilevel Monte Carlo (MLMC) estimator for
multi-dimensional SDEs driven by Brownian motions. Giles has previously shown
that if we combine a numerical approximation with strong order of convergence
with MLMC we can reduce the computational complexity to estimate
expected values of functionals of SDE solutions with a root-mean-square error
of from to . However, in
general, to obtain a rate of strong convergence higher than
requires simulation, or approximation, of L\'{e}vy areas. In this paper,
through the construction of a suitable antithetic multilevel correction
estimator, we are able to avoid the simulation of L\'{e}vy areas and still
achieve an multilevel correction variance for smooth payoffs,
and almost an variance for piecewise smooth payoffs, even
though there is only strong convergence. This results in an
complexity for estimating the value of European and Asian
put and call options.Comment: Published in at http://dx.doi.org/10.1214/13-AAP957 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Stochastic ordinary differential equations in applied and computational mathematics
Using concrete examples, we discuss the current and potential use of stochastic ordinary differential equations (SDEs) from the perspective of applied and computational mathematics. Assuming only a minimal background knowledge in probability and stochastic processes, we focus on aspects that distinguish SDEs from their deterministic counterparts. To illustrate a multiscale modelling framework, we explain how SDEs arise naturally as diffusion limits in the type of discrete-valued stochastic models used in chemical kinetics, population dynamics, and, most topically, systems biology. We outline some key issues in existence, uniqueness and stability that arise when SDEs are used as physical models, and point out possible pitfalls. We also discuss the use of numerical methods to simulate trajectories of an SDE and explain how both weak and strong convergence properties are relevant for highly-efficient multilevel Monte Carlo simulations. We flag up what we believe to be key topics for future research, focussing especially on nonlinear models, parameter estimation, model comparison and multiscale simulation
Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations
The Euler-Maruyama scheme is known to diverge strongly and numerically weakly
when applied to nonlinear stochastic differential equations (SDEs) with
superlinearly growing and globally one-sided Lipschitz continuous drift
coefficients. Classical Monte Carlo simulations do, however, not suffer from
this divergence behavior of Euler's method because this divergence behavior
happens on rare events. Indeed, for such nonlinear SDEs the classical Monte
Carlo Euler method has been shown to converge by exploiting that the Euler
approximations diverge only on events whose probabilities decay to zero very
rapidly. Significantly more efficient than the classical Monte Carlo Euler
method is the recently introduced multilevel Monte Carlo Euler method. The main
observation of this article is that this multilevel Monte Carlo Euler method
does - in contrast to classical Monte Carlo methods - not converge in general
in the case of such nonlinear SDEs. More precisely, we establish divergence of
the multilevel Monte Carlo Euler method for a family of SDEs with superlinearly
growing and globally one-sided Lipschitz continuous drift coefficients. In
particular, the multilevel Monte Carlo Euler method diverges for these
nonlinear SDEs on an event that is not at all rare but has probability one. As
a consequence for applications, we recommend not to use the multilevel Monte
Carlo Euler method for SDEs with superlinearly growing nonlinearities. Instead
we propose to combine the multilevel Monte Carlo method with a slightly
modified Euler method. More precisely, we show that the multilevel Monte Carlo
method combined with a tamed Euler method converges for nonlinear SDEs with
globally one-sided Lipschitz continuous drift coefficients and preserves its
strikingly higher order convergence rate from the Lipschitz case.Comment: Published in at http://dx.doi.org/10.1214/12-AAP890 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equations in mathematical finance
Inspired by recent progress in quantum algorithms for ordinary and partial
differential equations, we study quantum algorithms for stochastic differential
equations (SDEs). Firstly we provide a quantum algorithm that gives a quadratic
speed-up for multilevel Monte Carlo methods in a general setting. As
applications, we apply it to compute expection values determined by classical
solutions of SDEs, with improved dependence on precision. We demonstrate the
use of this algorithm in a variety of applications arising in mathematical
finance, such as the Black-Scholes and Local Volatility models, and Greeks. We
also provide a quantum algorithm based on sublinear binomial sampling for the
binomial option pricing model with the same improvement.Comment: 36 pages, 6 figure
Strong convergence rates for Euler approximations to a class of stochastic path-dependent volatility models
We consider a class of stochastic path-dependent volatility models where the
stochastic volatility, whose square follows the Cox-Ingersoll-Ross model, is
multiplied by a (leverage) function of the spot price, its running maximum, and
time. We propose a Monte Carlo simulation scheme which combines a log-Euler
scheme for the spot process with the full truncation Euler scheme or the
backward Euler-Maruyama scheme for the squared stochastic volatility component.
Under some mild regularity assumptions and a condition on the Feller ratio, we
establish the strong convergence with order 1/2 (up to a logarithmic factor) of
the approximation process up to a critical time. The model studied in this
paper contains as special cases Heston-type stochastic-local volatility models,
the state-of-the-art in derivative pricing, and a relatively new class of
path-dependent volatility models. The present paper is the first to prove the
convergence of the popular Euler schemes with a positive rate, which is
moreover consistent with that for Lipschitz coefficients and hence optimal.Comment: 34 pages, 5 figure
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