933 research outputs found
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
Pricing American Options by Exercise Rate Optimization
We present a novel method for the numerical pricing of American options based
on Monte Carlo simulation and the optimization of exercise strategies. Previous
solutions to this problem either explicitly or implicitly determine so-called
optimal exercise regions, which consist of points in time and space at which a
given option is exercised. In contrast, our method determines the exercise
rates of randomized exercise strategies. We show that the supremum of the
corresponding stochastic optimization problem provides the correct option
price. By integrating analytically over the random exercise decision, we obtain
an objective function that is differentiable with respect to perturbations of
the exercise rate even for finitely many sample paths. The global optimum of
this function can be approached gradually when starting from a constant
exercise rate.
Numerical experiments on vanilla put options in the multivariate
Black-Scholes model and a preliminary theoretical analysis underline the
efficiency of our method, both with respect to the number of
time-discretization steps and the required number of degrees of freedom in the
parametrization of the exercise rates. Finally, we demonstrate the flexibility
of our method through numerical experiments on max call options in the
classical Black-Scholes model, and vanilla put options in both the Heston model
and the non-Markovian rough Bergomi model
Computing mean first exit times for stochastic processes using multi-level Monte Carlo
The multi-level approach developed by Giles (2008) can be used to estimate mean first exit times for stochastic differential equations, which are of interest in finance, physics and chemical kinetics. Multi-level improves the computational expense of standard Monte Carlo in this setting by an order of magnitude. More precisely, for a target accuracy of TOL, so that the root mean square error of the estimator is O(TOL), the O(TOL-4) cost of standard Monte Carlo can be reduced to O(TOL-3|log(TOL)|1/2) with a multi-level scheme. This result was established in Higham, Mao, Roj, Song, and Yin (2013), and illustrated on some scalar examples. Here, we briefly overview the algorithm and present some new computational results in higher dimensions
Chebyshev Interpolation for Parametric Option Pricing
Function approximation with Chebyshev polynomials is a well-established and thoroughly investigated method within the field of numerical analysis. The method enjoys attractive convergence properties and its implementation is straightforward. We propose to apply tensorized Chebyshev interpolation to computing Parametric Option Prices (POP). This allows us to exploit the recurrent nature of the pricing problem in an efficient, reliable and general way. For a large variety of option types and affine asset models we prove that the convergence rate of the method is exponential if there is a single varying parameter and of any arbitrary polynomial order in the multivariate case. Numerical experiments confirm these findings and show that the method achieves a significant gain in efficiency
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
Decision-making under uncertainty: using MLMC for efficient estimation of EVPPI
In this paper we develop a very efficient approach to the Monte Carlo
estimation of the expected value of partial perfect information (EVPPI) that
measures the average benefit of knowing the value of a subset of uncertain
parameters involved in a decision model. The calculation of EVPPI is inherently
a nested expectation problem, with an outer expectation with respect to one
random variable and an inner conditional expectation with respect to the
other random variable . We tackle this problem by using a Multilevel Monte
Carlo (MLMC) method (Giles 2008) in which the number of inner samples for
increases geometrically with level, so that the accuracy of estimating the
inner conditional expectation improves and the cost also increases with level.
We construct an antithetic MLMC estimator and provide sufficient assumptions on
a decision model under which the antithetic property of the estimator is well
exploited, and consequently a root-mean-square accuracy of can be
achieved at a cost of . Numerical results confirm the
considerable computational savings compared to the standard, nested Monte Carlo
method for some simple testcases and a more realistic medical application
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