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2018 | OriginalPaper | Buchkapitel

8. Monte Carlo Methods

verfasst von : David Jamieson Bolder

Erschienen in: Credit-Risk Modelling

Verlag: Springer International Publishing

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Abstract

Stochastic-simulation, or Monte-Carlo, methods are used extensively in the area of credit-risk modelling. This technique has, in fact, been employed inveterately in previous chapters. Care and caution are always advisable when employing a complex numerical technique. Prudence is particularly appropriate, in this context, because default is a rare event. Unlike the asset-pricing setting, where we typically simulate expectations in the central part of the distribution, credit risk operates in the tails. As a consequence, this chapter is dedicated to a closer examination of the intricacies of the Monte-Carlo method. Working from first principles, the importance of convergence analysis and confidence intervals is highlighted. The principal shortcoming of this method, its inherent slowness, is also explained and demonstrated. This naturally leads to a discussion of the set of variance-reduction techniques employed to enhance the speed of these estimators. The chapter concludes with the investigation and implementation of the Glasserman and Li (2005, Management Science, 51(11), 1643–1656.) importance-sampling method to the t-threshold model. This method, which employs the so-called Esscher transform, has close conceptual links to the saddlepoint technique introduced in the previous chapter.

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Vorheriges Kapitel Risk Attribution

Nächstes Kapitel Default Probabilities

Wilde (1997) and Gundlach and Lehrbass (2004) provide more detail on this approach.

The interested reader is referred to Heunis (2011).

One microsecond is one millionth of a second.

In the late 1880, the Comte de Buffon offered an interesting probability problem that led to a surprising geometric result involving π. The renowned mathematician, Pierre-Simon Laplace later remarked that this problem could be used, employing the Monte-Carlo method, to experimentally approximate π. The interested reader is referred to Arnow (1994) and Badger (1994) for more detail.

See Harrison and Kreps (1979) and Harrison and Pliska (1981) for the original references with respect to this fundamental concept. Duffie (1996) also offers an excellent alternative reference.

See Billingsley (1995) and Durrett (1996) for a rigorous and detailed discussion of the results associated with the central-limit theorem.

See Fishman (1995, Chapter 1) and Glasserman (2004a, Chapter 1) for more technical details.

See Box (1981, 1987) for the interesting history of this distribution’s inventor, William Sealy Gosset. We will return to these questions of statistical inference in the following two chapters.

We can easily identify, from the third row and fourth column, the computation from equation 8.30 in Table 8.2.

See Musiela and Rutkowski (1998, Appendix B), Karatzas and Shreve (1991, Section 3.5), or Heunis (2011, Section 5.7) for more background on the Girsanov result.

Importance sampling has interesting links to risk and the notion of entropy. See Reesor and McLeish (2001) for a discussion of these useful concepts.

Important discussion and results regarding the unbiasedness of this estimator along with upper and lower bounds are found in Glasserman and Li (2005).

When X is continuous, the moment-generating function coincides with the Laplace transform of the density function. For a more detailed discussion of moment-generating functions, see Casella and Berger (1990, Chapter 2) or the relevant discussion in Chap. 7.

Recall that the cumulant generating function, or cumulant, is simply the natural logarithm of the moment-generating function.

This is, in fact, the integrand from the inverse Laplace transform used to identify the loss density in Chap. 7.

We employed the modified Powell method from the Python scipy library, which requires only a fraction of a second to converge to a good solution.

See Appendix D for more details on the overall Python library structure and organization.

See Held and Bové (2014, Appendix A) for more information.

See Beck (2014, Chapter 3) for a bit of context on the non-linear least-squares problem.

See Johnson et al. (1994, Chapters 17–18) for more information on the chi-squared and gamma distributions and their properties.

The invVector argument passes the numerically computed inverse cumulative distribution function values for the variance-gamma and generalized-hyperbolic models. This is purely to speed the code by performing this computationally expensive step once rather than many times.

Recall that one microsecond is a millionth of a second.

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Titel: Monte Carlo Methods
verfasst von: David Jamieson Bolder
Verlag: Springer International Publishing
Buch: Credit-Risk Modelling
Print ISBN: 978-3-319-94687-0

Electronic ISBN: 978-3-319-94688-7

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-3-319-94688-7_8