Regression-based variance reduction approach for strong approximation schemes

In this paper we present a novel approach towards variance reduction for discretised diffusion processes. The proposed approach involves specially constructed control variates and allows for a significant reduction in the variance for the terminal functionals. In this way the complexity order of the standard Monte Carlo algorithm ($\varepsilon^{-3}$) can be reduced down to $\varepsilon^{-2}\sqrt{\left|\log(\varepsilon)\right|}$ in case of the Euler scheme with $\varepsilon$ being the precision to be achieved. These theoretical results are illustrated by several numerical examples.Comment: arXiv admin note: text overlap with arXiv:1510.0314

A Variance Reduction Method for Parametrized Stochastic Differential Equations using the Reduced Basis Paradigm

In this work, we develop a reduced-basis approach for the efficient computation of parametrized expected values, for a large number of parameter values, using the control variate method to reduce the variance. Two algorithms are proposed to compute online, through a cheap reduced-basis approximation, the control variates for the computation of a large number of expectations of a functional of a parametrized Ito stochastic process (solution to a parametrized stochastic differential equation). For each algorithm, a reduced basis of control variates is pre-computed offline, following a so-called greedy procedure, which minimizes the variance among a trial sample of the output parametrized expectations. Numerical results in situations relevant to practical applications (calibration of volatility in option pricing, and parameter-driven evolution of a vector field following a Langevin equation from kinetic theory) illustrate the efficiency of the method

Sensitivities for Bermudan Options by Regression Methods

In this article, we propose several pathwise and finite difference-based methods for calculating sensitivities of Bermudan options using regression methods and Monte Carlo simulation. These methods rely on conditional probabilistic representations that allow, in combination with a regression approach, for efficient simultaneous computation of sensitivities at many initial positions. Assuming that the price of a Bermudan option can be evaluated sufficiently accurate, we develop a method for constructing deltas based on least squares. We finally propose a testing procedure for assessing the performance of the developed methods and give a numerical illustration. © 2009 Springer-Verlag.Partially supported by the Deutsche Forschungsgemeinschaft through SFB 649 “Economic Risk” and DFG Research Center Matheon “Mathematics for Key Technologies” in Berlin

Practical Variance Reduction via Regression for Simulating Diffusions

The well-known variance reduction methods—the method of importance sampling and the method of control variates—can be exploited if an approximation of the required solution is known. Here we employ conditional probabilistic representations of solutions together with the regression method to obtain sufficiently inexpensive (although rather rough) estimates of the solution and its derivatives by using the single auxiliary set of approximate trajectories starting from the initial position. These estimates can effectively be used for significant reduction of variance and further accurate evaluation of the required solution. The developed approach is supported by numerical experiments

Regression-based Monte Carlo methods with optimal control variates

In der vorliegenden Dissertation werden regressionsbasierte Monte-Carlo-Verfahren für diskretisierte Diffusionsprozesse vorgestellt. Diese Verfahren beinhalten die Konstruktion von geeigneten Kontrollvariaten, die zu einer signifikanten Reduktion der Varianz führen. Dadurch kann die Komplexität des Standard-Monte-Carlo-Ansatzes (epsilon^{-3} für Schemen erster Ordnung und epsilon^{-2.5} für Schemen zweiter Ordnung) im besten Fall reduziert werden auf eine Ordnung von epsilon^{-2+delta} für ein beliebiges delta aus [0,0.25), wobei epsilon die zu erzielende Genauigkeit bezeichnet. In der Komplexitätsanalyse werden sowohl die Fehler, die auch beim Standard-Monte-Carlo-Ansatz auftreten (Diskretisierungs- und statistischer Fehler), als auch die aus der Schätzung bedingter Erwartungswerte mittels Regression resultierenden Fehler berücksichtigt. Darüber hinaus werden verschiedene Algorithmen hergeleitet, die zwar zu einer ähnlichen theoretischen Komplexität führen, jedoch numerisch gesehen bei der Regressionsschätzung unterschiedlich stabil und genau sind. Die Effektivität dieser Algorithmen wird anhand von numerischen Beispielen veranschaulicht und mit anderen bekannten Methoden verglichen. Zudem werden geeignete Kontrollvariaten für die Bewertung von Bermuda-Optionen sowie amerikanischen Optionen basierend auf einer dualen Monte-Carlo-Methode hergeleitet. Auch hierbei ergibt sich eine signifikante Komplexitätsreduktion, sofern die zugrunde liegenden Funktionen gewisse Glattheitsannahmen erfüllen