A framework for adaptive Monte-Carlo procedures

Adaptive Monte Carlo methods are recent variance reduction techniques. In this work, we propose a mathematical setting which greatly relaxes the assumptions needed by for the adaptive importance sampling techniques presented by Vazquez-Abad and Dufresne, Fu and Su, and Arouna. We establish the convergence and asymptotic normality of the adaptive Monte Carlo estimator under local assumptions which are easily verifiable in practice. We present one way of approximating the optimal importance sampling parameter using a randomly truncated stochastic algorithm. Finally, we apply this technique to some examples of valuation of financial derivatives

Analytic results and weighted Monte Carlo simulations for CDO pricing

We explore the possibilities of importance sampling in the Monte Carlo pricing of a structured credit derivative referred to as Collateralized Debt Obligation (CDO). Modeling a CDO contract is challenging, since it depends on a pool of (typically about 100) assets, Monte Carlo simulations are often the only feasible approach to pricing. Variance reduction techniques are therefore of great importance. This paper presents an exact analytic solution using Laplace-transform and MC importance sampling results for an easily tractable intensity-based model of the CDO, namely the compound Poissonian. Furthermore analytic formulae are derived for the reweighting efficiency. The computational gain is appealing, nevertheless, even in this basic scheme, a phase transition can be found, rendering some parameter regimes out of reach. A model-independent transform approach is also presented for CDO pricing.Comment: 12 pages, 9 figure

Optimised Importance Sampling in Multilevel Monte Carlo

This dissertation explores the remarkable variance reduction effects that can be achieved combining Multilevel Monte Carlo and Importance Sampling. The analysis is conducted within a Black-Scholes framework, focusing on pricing deep out-of-the-money options. Particular attention is addressed to the choice of the Importance Sampling measure and to the optimisation of its parameters. Numerical results show that the combination of the two methods significantly outperforms both techniques if applied separately. \ud \ud Key words: Monte Carlo, Multilevel Monte Carlo, Option Pricing, Importance Sampling, Variance Reductio

Robust adaptive importance sampling for normal random vectors

Adaptive Monte Carlo methods are very efficient techniques designed to tune simulation estimators on-line. In this work, we present an alternative to stochastic approximation to tune the optimal change of measure in the context of importance sampling for normal random vectors. Unlike stochastic approximation, which requires very fine tuning in practice, we propose to use sample average approximation and deterministic optimization techniques to devise a robust and fully automatic variance reduction methodology. The same samples are used in the sample optimization of the importance sampling parameter and in the Monte Carlo computation of the expectation of interest with the optimal measure computed in the previous step. We prove that this highly dependent Monte Carlo estimator is convergent and satisfies a central limit theorem with the optimal limiting variance. Numerical experiments confirm the performance of this estimator: in comparison with the crude Monte Carlo method, the computation time needed to achieve a given precision is divided by a factor between 3 and 15.Comment: Published in at http://dx.doi.org/10.1214/09-AAP595 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Variance Reduction in a Stochastic Volatility Scenario

none2noThis paper investigates the use of a variance reduction, called importance sampling, for Monte Carlo methods in the case of the stochastic volatility model for option pricing introduced by Hobson and Rogers (1998). We briefly recall that a European call option contract gives the right, but not the obligation, to buy a specific amount of a given stock or index at a specified price (strike price) in a specified time (maturity); we show some evidence on the call options on MIB30 Italian Index to verify the performance of the importance sampling in a complete stochastic volatility model. In Monte Carlo method the price of a call option is obtained as the average value of the simulations of a large number of independent, uniform variates (prices) by means of pseudo-random number generators. It is shown, finally, that variance is dramatically reduced meaning that numerical techniques introduced for variance reduction have still a lot to say.openSORINI LAERTE; GUERRA MARIA LETIZIASorini, Laerte; Guerra, MARIA LETIZI

Simulation of diffusions by means of importance sampling paradigm

The aim of this paper is to introduce a new Monte Carlo method based on importance sampling techniques for the simulation of stochastic differential equations. The main idea is to combine random walk on squares or rectangles methods with importance sampling techniques. The first interest of this approach is that the weights can be easily computed from the density of the one-dimensional Brownian motion. Compared to the Euler scheme this method allows one to obtain a more accurate approximation of diffusions when one has to consider complex boundary conditions. The method provides also an interesting alternative to performing variance reduction techniques and simulating rare events.Comment: Published in at http://dx.doi.org/10.1214/09-AAP659 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Credit Risk Monte Carlos Simulation Using Simplified Creditmetrics' Model: the joint use of importance sampling and descriptive sampling

Monte Carlo simulation is implemented in some of the main models for estimating portfolio credit risk, such as CreditMetrics, developed by Gupton, Finger and Bhatia (1997). As in any Monte Carlo application, credit risk simulation according to this model produces imprecise estimates. In order to improve precision, simulation sampling techniques other than traditional Simple Random Sampling become indispensable. Importance Sampling (IS) has already been successfully implemented by Glasserman and Li (2005) on a simplified version of CreditMetrics, in which only default risk is considered. This paper tries to improve even more the precision gains obtained by IS over the same simplified CreditMetrics' model. For this purpose, IS is here combined with Descriptive Sampling (DS), another simulation technique which has proved to be a powerful variance reduction procedure. IS combined with DS was successful in obtaining more precise results for credit risk estimates than its standard form.

Variance Reduction Techniques in Monte Carlo Methods

Monte Carlo methods are simulation algorithms to estimate a numerical quantity in a statistical model of a real system. These algorithms are executed by computer programs. Variance reduction techniques (VRT) are needed, even though computer speed has been increasing dramatically, ever since the introduction of computers. This increased computer power has stimulated simulation analysts to develop ever more realistic models, so that the net result has not been faster execution of simulation experiments; e.g., some modern simulation models need hours or days for a single ’run’ (one replication of one scenario or combination of simulation input values). Moreover there are some simulation models that represent rare events which have extremely small probabilities of occurrence), so even modern computer would take ’for ever’ (centuries) to execute a single run - were it not that special VRT can reduce theses excessively long runtimes to practical magnitudes.common random numbers;antithetic random numbers;importance sampling;control variates;conditioning;stratied sampling;splitting;quasi Monte Carlo

Towards interactive global illumination effects via sequential Monte Carlo adaptation

Journal ArticleThis paper presents a novel method that effectively combines both control variates and importance sampling in a sequential Monte Carlo context while handling general single-bounce global illumination effects. The radiance estimates computed during the rendering process are cached in an adaptive per-pixel structure that defines dynamic predicate functions for both variance reduction techniques and guarantees well-behaved PDFs, yielding continually increasing efficiencies thanks to a marginal computational overhead