Conference Paper

# A rate result for simulation optimization with conditional value-at-risk constraints.

DOI: 10.1109/WSC.2008.4736121 Conference: Proceedings of the 2008 Winter Simulation Conference, Global Gateway to Discovery, WSC 2008, InterContinental Hotel, Miami, Florida, USA, December 7-10, 2008

Source: DBLP

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**ABSTRACT:**This paper develops e#cient methods for computing portfolio value-at-risk (VAR) when the underlying risk factors have a heavy-tailed distribution. In modeling heavy tails, we focus on multivariate t distributions and some extensions thereof. We develop two methods for VAR calculation that exploit a quadratic approximation to the portfolio loss, such as the delta-gamma approximation. In the first method, we derive the characteristic function of the quadratic approximation and then use numerical transform inversion to approximate the portfolio loss distribution. Because the quadratic approximation may not always yield accurate VAR estimates, we also develop a low variance Monte Carlo method. This method uses the quadratic approximation to guide the selection of an e#ective importance sampling distribution that samples risk factors so that large losses occur more often. Variance is further reduced by combining the importance sampling with stratified sampling. Numerical results on a variety of test portfolios indicate that large variance reductions are typically obtained. Both methods developed in this paper overcome di#culties associated with VAR calculation with heavy-tailed risk factors. The Monte Carlo method also extends to the problem of estimating the conditional excess, sometimes known as the conditional VAR. Key Words: value-at-risk, delta-gamma approximation, Monte Carlo, simulation, variance reduction, importance sampling, stratified sampling, conditional excess, conditional value-at-risk 1Mathematical Finance 08/2001; · 1.25 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this chapter we review some of the recent developments for efficient estimation of rare-events, most of which involve application of importance sampling techniques to achieve variance reduction. The zero-variance importance sampling measure is well known and in many cases has a simple representation. Though not implementable, it proves useful in selecting good and implementable importance sampling changes of measure that are in some sense close to it and thus provides a unifying framework for such selections. Specifically, we consider rare events associated with: 1) multi-dimensional light-tailed random walks, 2) with certain events involving heavy-tailed random variables and 3) queues and queueing networks. In addition, we review the recent literature on development of adaptive importance sampling techniques to quickly estimate common performance measures associated with finite-state Markov chains. We also discuss the application of rare-event simulation techniques to problems in financial engineering. The discussion in this chapter is non-measure theoretic and kept sufficiently simple so that the key ideas are accessible to beginners. References are provided for more advanced treatments.Transactions of The Society for Modeling and Simulation International - SIMULATION. - [Show abstract] [Hide abstract]

**ABSTRACT:**Fundamental properties of conditional value-at-risk (CVaR), as a measure of risk with significant advantages over value-at-risk (VaR), are derived for loss distributions in finance that can involve discreetness. Such distributions are of particular importance in applications because of the prevalence of models based on scenarios and finite sampling. CVaR is able to quantify dangers beyond VaR and moreover it is coherent. It provides optimization short-cuts which, through linear programming techniques, make practical many large-scale calculations that could otherwise be out of reach. The numerical efficiency and stability of such calculations, shown in several case studies, are illustrated further with an example of index tracking.Journal of Banking & Finance 01/2002; · 1.29 Impact Factor

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